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==Grid 0: Problems 1-99==
==Grid 0: Problems 1-99==


* [[Project Euler/1|Problem 1]]- Multiples of 3 and 5 - printing out all multiples of 3 and 5.
* [[Project Euler/1]]- Multiples of 3 and 5 - printing out all multiples of 3 and 5.
* [[Project Euler/2|Problem 2]] - Even Fibonacci - summing the Fibonacci numbers that are even and less than 4 million
* [[Project Euler/2]] - Even Fibonacci - summing the Fibonacci numbers that are even and less than 4
* [[Project Euler/3|Problem 3]] - Largest Prime Factor - Largest prime factor of a given 12-digit number
* million
* [[Project Euler/4|Problem 4]] - Largest Palindrome Product - Largest palindrome product (extracting substrings and sorting)
* [[Project Euler/3]] - Largest Prime Factor - Largest prime factor of a given 12-digit number
* [[Project Euler/5|Problem 5]] - LCM - Least common multiple of all the integers from 1 to 20
* [[Project Euler/4]] - Largest Palindrome Product - Largest palindrome product (extracting substrings
* [[Project Euler/6|Problem 6]] - SoS - Sum of squares and squares of sums
* and sorting)
* [[Project Euler/7|Problem 7]] - Ten Thousand Primes - Find the 10,001st prime.
* [[Project Euler/5]] - LCM - Least common multiple of all the integers from 1 to 20
* [[Project Euler/8|Problem 8]] - Adjacent Digits - Largest product formed by 13 adjacent digits.
* [[Project Euler/6]] - SoS - Sum of squares and squares of sums
* [[Project Euler/9|Problem 9]] - Pythagorean Triplet Sum - Finding a Pythagorean triplet with a specified sum.
* [[Project Euler/7]] - Ten Thousand Primes - Find the 10,001st prime.
* [[Project Euler/10|Problem 10]] Sum of Primes - Sum of all primes below 2 million.
* [[Project Euler/8]] - Adjacent Digits - Largest product formed by 13 adjacent digits.
* [[Project Euler/9]] - Pythagorean Triplet Sum - Finding a Pythagorean triplet with a specified sum.
* [[Project Euler/10]] Sum of Primes - Sum of all primes below 2 million.


* [[Project Euler/11|Problem 11]] - Greatest Product in Grid - Finding the greatest product of 4 numbers on a grid.
* [[Project Euler/11]] - Greatest Product in Grid - Finding the greatest product of 4 numbers on a grid.
* [[Project Euler/12|Problem 12]] - Highly Factorable Triangular Numbers - Finding highly factorable triangular numbers
* [[Project Euler/12]] - Highly Factorable Triangular Numbers - Finding highly factorable triangular
* [[Project Euler/13|Problem 13]] - Sum of Big Numbers - Work out the first 10 digits of a sum of 100 50-digit numbers
* numbers
* [[Project Euler/14|Problem 14]] - Longest Collatz Sequence - Finding the longest Collatz sequence for starting integers under 1 million
* [[Project Euler/13]] - Sum of Big Numbers - Work out the first 10 digits of a sum of 100 50-digit
* [[Project Euler/15|Problem 15]] - Lattice Paths - Finding the number of variations on a route through a lattice.
* numbers
* [[Project Euler/16|Problem 16]] - Summing the Digits - summing up the digits of a large power of 2, 2**1000
* [[Project Euler/14]] - Longest Collatz Sequence - Finding the longest Collatz sequence for starting
* [[Project Euler/17|Problem 17]] - Number Spelling - spelling out all the numbers from one to a thousand
* integers under 1 million
* [[Project Euler/18|Problem 18]] - Shortest Path through a Triangle - find the path through a triangle of numbers that leads to the smallest sum
* [[Project Euler/15]] - Lattice Paths - Finding the number of variations on a route through a lattice.
* [[Project Euler/19|Problem 19]] - Counting Sundays
* [[Project Euler/16]] - Summing the Digits - summing up the digits of a large power of 2, 2**1000
* [[Project Euler/20|Problem 20]] - Factorial Digit Sum - Sum of the digits in the number 100!
* [[Project Euler/17]] - Number Spelling - spelling out all the numbers from one to a thousand
* [[Project Euler/18]] - Shortest Path through a Triangle - find the path through a triangle of numbers
* that leads to the smallest sum
* [[Project Euler/19]] - Counting Sundays
* [[Project Euler/20]] - Factorial Digit Sum - Sum of the digits in the number 100!


* [[Project Euler/21|Problem 21]] - Amicable Numbers - Sum of all amicable numbers under 10000
* [[Project Euler/21]] - Amicable Numbers - Sum of all amicable numbers under 10000
* [[Project Euler/22|Problem 22]] - Names Scores - Sort 5000+ names alphabetically and compute name scores
* [[Project Euler/22]] - Names Scores - Sort 5000+ names alphabetically and compute name scores
* [[Project Euler/23|Problem 23]] - Non-Abundant Sums - Sum of all positive integers not expressible as the sum of two abundant numbers
* [[Project Euler/23]] - Non-Abundant Sums - Sum of all positive integers not expressible as the sum of
* [[Project Euler/24|Problem 24]] - Lexicographic Permutations - Find the millionth lexicographic permutation of the digits 0-9
* two abundant numbers
* [[Project Euler/25|Problem 25]] - 1000-digit Fibonacci Number - Index of the first term in the Fibonacci sequence to contain 1000 digits
* [[Project Euler/24]] - Lexicographic Permutations - Find the millionth lexicographic permutation of
* [[Project Euler/26|Problem 26]] - Reciprocal Cycles - Find d<1000 for which 1/d contains the longest recurring cycle
* the digits 0-9
* [[Project Euler/25]] - 1000-digit Fibonacci Number - Index of the first term in the Fibonacci sequence
* to contain 1000 digits
* [[Project Euler/26]] - Reciprocal Cycles - Find d<1000 for which 1/d contains the longest recurring
* cycle


* [[Project Euler/27|Problem 27]] - Quadratic Primes - Find the quadratic formula n²+an+b producing the most consecutive primes
* [[Project Euler/27]] - Quadratic Primes - Find the quadratic formula n²+an+b producing the most
* [[Project Euler/28|Problem 28]] - Number Spiral Diagonals
* consecutive primes
* [[Project Euler/29|Problem 29]] - Distinct Terms Generated by Powers
* [[Project Euler/28]] - Number Spiral Diagonals
* [[Project Euler/30|Problem 30]] - Sum of Fifth Power of Digits
* [[Project Euler/29]] - Distinct Terms Generated by Powers
* [[Project Euler/30]] - Sum of Fifth Power of Digits


* [[Project Euler/31|Problem 31]] - Polya - Change for a Dollar
* [[Project Euler/31]] - Polya - Change for a Dollar
* [[Project Euler/32|Problem 32]] - Pandigital Products (A X B = C covering all 9 digits)
* [[Project Euler/32]] - Pandigital Products (A X B = C covering all 9 digits)
* [[Project Euler/33|Problem 33]] - Digit Cancelling Fractions - Find the product of the four non-trivial curious fractions where cancelling a common digit gives the correct simplified value.
* [[Project Euler/33]] - Digit Cancelling Fractions - Find the product of the four non-trivial curious
* [[Project Euler/34|Problem 34]] - Digit Factorials - Find the sum of all numbers equal to the sum of the factorial of their digits.
* fractions where cancelling a common digit gives the correct simplified value.
* [[Project Euler/35|Problem 35]] - Circular Primes - Count how many circular primes are there below one million.
* [[Project Euler/34]] - Digit Factorials - Find the sum of all numbers equal to the sum of the
* [[Project Euler/36|Problem 36]] - Double-base Palindromes - Find the sum of all numbers below one million that are palindromic in both base 10 and base 2.
* factorial of their digits.
* [[Project Euler/37|Problem 37]] - Truncatable Primes - Find the sum of the only eleven primes that are truncatable from left to right and right to left.
* [[Project Euler/35]] - Circular Primes - Count how many circular primes are there below one million.
* [[Project Euler/38|Problem 38]] - Pandigital Multiples - Find the largest 1-to-9 pandigital number that can be formed as the concatenated product of an integer with (1,2,...,n).
* [[Project Euler/36]] - Double-base Palindromes - Find the sum of all numbers below one million that
* [[Project Euler/39|Problem 39]] - Integer Right Triangles - Find the perimeter p ≤ 1000 for which the number of integer-sided right triangles is maximised.
* are palindromic in both base 10 and base 2.
* [[Project Euler/40|Problem 40]] - Champernowne's Constant - Find the product of digits at specific positions in the fractional part of Champernowne's constant.
* [[Project Euler/37]] - Truncatable Primes - Find the sum of the only eleven primes that are
* truncatable from left to right and right to left.
* [[Project Euler/38]] - Pandigital Multiples - Find the largest 1-to-9 pandigital number that can be
* formed as the concatenated product of an integer with (1,2,...,n).
* [[Project Euler/39]] - Integer Right Triangles - Find the perimeter p ≤ 1000 for which the number of
* integer-sided right triangles is maximised.
* [[Project Euler/40]] - Champernowne's Constant - Find the product of digits at specific positions in
* the fractional part of Champernowne's constant.


* [[Project Euler/41|Problem 41]] - Pandigital Prime - Find the largest n-digit pandigital prime that exists.
* [[Project Euler/41]] - Pandigital Prime - Find the largest n-digit pandigital prime that exists.
* [[Project Euler/42|Problem 42]] - Coded Triangle Numbers - Count how many words in a given list are triangle words (where word value equals a triangle number).
* [[Project Euler/42]] - Coded Triangle Numbers - Count how many words in a given list are triangle
* [[Project Euler/43|Problem 43]] - Sub-string Divisibility - Find the sum of all pandigital numbers with an unusual substring divisibility property.
* words (where word value equals a triangle number).
* [[Project Euler/44|Problem 44]] - Pentagon Numbers - Find the pair of pentagonal numbers whose sum and difference are pentagonal, minimising their difference.
* [[Project Euler/43]] - Sub-string Divisibility - Find the sum of all pandigital numbers with an
* [[Project Euler/45|Problem 45]] - Triangular, Pentagonal, and Hexagonal - Find the next triangle number that is also pentagonal and hexagonal after 40755.
* unusual substring divisibility property.
* [[Project Euler/46|Problem 46]] - Goldbach's Other Conjecture - Find the smallest odd composite that cannot be written as the sum of a prime and twice a square.
* [[Project Euler/44]] - Pentagon Numbers - Find the pair of pentagonal numbers whose sum and difference
* [[Project Euler/47|Problem 47]] - Distinct Primes Factors - Find the first four consecutive integers to have four distinct prime factors each.
* are pentagonal, minimising their difference.
* [[Project Euler/48|Problem 48]] - Self Powers - Find the last ten digits of the sum 1^1 + 2^2 + ... + 1000^1000.
* [[Project Euler/45]] - Triangular, Pentagonal, and Hexagonal - Find the next triangle number that is
* [[Project Euler/49|Problem 49]] - Prime Permutations - Find the 12-digit number formed by concatenating three 4-digit primes that are permutations and form an arithmetic sequence.
* also pentagonal and hexagonal after 40755.
* [[Project Euler/50|Problem 50]] - Consecutive Prime Sum - Find the prime below one million that can be written as the sum of the most consecutive primes.
* [[Project Euler/46]] - Goldbach's Other Conjecture - Find the smallest odd composite that cannot be
* written as the sum of a prime and twice a square.
* [[Project Euler/47]] - Distinct Primes Factors - Find the first four consecutive integers to have four
* distinct prime factors each.
* [[Project Euler/48]] - Self Powers - Find the last ten digits of the sum 1^1 + 2^2 + ... + 1000^1000.
* [[Project Euler/49]] - Prime Permutations - Find the 12-digit number formed by concatenating three
* 4-digit primes that are permutations and form an arithmetic sequence.
* [[Project Euler/50]] - Consecutive Prime Sum - Find the prime below one million that can be written as
* the sum of the most consecutive primes.


* [[Project Euler/51|Problem 51]]- Prime Replacement - Finding the number of primes that can be formed by replacing particular digits of a number
* [[Project Euler/51]]- Prime Replacement - Finding the number of primes that can be formed by replacing
* [[Project Euler/52|Problem 52]]- Permuted Multiples - Find a number whose multiples 2x, 3x, 4x, 5x ad 6x are permutations of one another.
* particular digits of a number
* [[Project Euler/53|Problem 53]] - Number of Combinations Over 1M - Find how many different n choose r values are greater than 1 million for n between 1 and 100.
* [[Project Euler/52]]- Permuted Multiples - Find a number whose multiples 2x, 3x, 4x, 5x ad 6x are
* [[Project Euler/54|Problem 54]] - Comparing poker hands to determine a winner
* permutations of one another.
* [[Project Euler/55|Problem 55]] - Lychrel Numbers - Count how many Lychrel numbers (numbers that never form a palindrome through the reverse-and-add process) are there below ten-thousand.
* [[Project Euler/53]] - Number of Combinations Over 1M - Find how many different n choose r values are
* [[Project Euler/56|Problem 56]] - Powerful Digit Sum - For natural numbers of the form a^b where a,b < 100, find the maximum digital sum.
* greater than 1 million for n between 1 and 100.
* [[Project Euler/57|Problem 57]] - Square Root Convergents - In the first one-thousand expansions of the continued fraction for √2, count how many fractions have a numerator with more digits than the denominator.
* [[Project Euler/54]] - Comparing poker hands to determine a winner
* [[Project Euler/58|Problem 58]] - Counting how many composite numbers have exactly 8 factors
* [[Project Euler/55]] - Lychrel Numbers - Count how many Lychrel numbers (numbers that never form a
* [[Project Euler/59|Problem 59]] - Decrypting 3-letter secret key (Vigenere cipher)
* palindrome through the reverse-and-add process) are there below ten-thousand.
* [[Project Euler/60|Problem 60]] - Prime pair sets - finding five primes such that any prime pair can be concatenated to form a new prime
* [[Project Euler/56]] - Powerful Digit Sum - For natural numbers of the form a^b where a,b < 100, find
* the maximum digital sum.
* [[Project Euler/57]] - Square Root Convergents - In the first one-thousand expansions of the continued
* fraction for √2, count how many fractions have a numerator with more digits than the denominator.
* [[Project Euler/58]] - Counting how many composite numbers have exactly 8 factors
* [[Project Euler/59]] - Decrypting 3-letter secret key (Vigenere cipher)
* [[Project Euler/60]] - Prime pair sets - finding five primes such that any prime pair can be
* concatenated to form a new prime


* [[Project Euler/61|Problem 61]] - Six cyclic 4-digit numbers, each of which are polygonal numbers (triangle, square, pentagonal, hexagonal, heptagonal, octagonal)
* [[Project Euler/61]] - Six cyclic 4-digit numbers, each of which are polygonal numbers (triangle,
* [[Project Euler/62|Problem 62]] - Cyclic permutations of cubes - find cubes that permute to other cubes.
* square, pentagonal, hexagonal, heptagonal, octagonal)
* [[Project Euler/63|Problem 63]] - Powerful digit counts - finding n-digit numbers that are n-th powers
* [[Project Euler/62]] - Cyclic permutations of cubes - find cubes that permute to other cubes.
* [[Project Euler/64|Problem 64]] - Continued Fractions - Odd period square roots - finding the continued fraction representation of an odd number, and determining if it has an odd period. First 1,000 numbers, so these sequences get LONG.
* [[Project Euler/63]] - Powerful digit counts - finding n-digit numbers that are n-th powers
* [[Project Euler/65|Problem 65]] - Convergents of e - computing the 100th convergent (rational representation of continued fraction) for e and the square root of 2.
* [[Project Euler/64]] - Continued Fractions - Odd period square roots - finding the continued fraction
* [[Project Euler/66|Problem 66]] - Diophantine equation - a nice problem involving quadratic Diphantine equations called Pell equations. These equations can be solved using the technique of continued fraction representations. It is much easier to solve this problem, then 64 and 65, rather than the other way around.
* representation of an odd number, and determining if it has an odd period. First 1,000 numbers, so these sequences
* [[Project Euler/67|Problem 67]] - Maximum path sum - a retake on [[Project Euler/18]] with a larger triangle for which a brute force solution technique is impossible.
* get LONG.
* [[Project Euler/68|Problem 68]] - Magic 5-gon Ring - Using numbers 1 to 10, find the maximum 16-digit string for a "magic" 5-gon ring.
* [[Project Euler/65]] - Convergents of e - computing the 100th convergent (rational representation of
* [[Project Euler/69|Problem 69]] - Totient Maximum - Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
* continued fraction) for e and the square root of 2.
* [[Project Euler/70|Problem 70]] - Totient Permutation - Find n < 10^7 for which φ(n) is a permutation of n and the ratio n/φ(n) is minimized.
* [[Project Euler/66]] - Diophantine equation - a nice problem involving quadratic Diphantine equations
* called Pell equations. These equations can be solved using the technique of continued fraction representations.
* It is much easier to solve this problem, then 64 and 65, rather than the other way around.
* [[Project Euler/67]] - Maximum path sum - a retake on [[Project Euler/18]] with a larger triangle for
* which a brute force solution technique is impossible.
* [[Project Euler/68]] - Magic 5-gon Ring - Using numbers 1 to 10, find the maximum 16-digit string for
* a "magic" 5-gon ring.
* [[Project Euler/69]] - Totient Maximum - Find the value of n ≤ 1,000,000 for which n/φ(n) is a
* maximum.
* [[Project Euler/70]] - Totient Permutation - Find n < 10^7 for which φ(n) is a permutation of n and
* the ratio n/φ(n) is minimized.


* [[Project Euler/71|Problem 71]] - Ordered Fractions - Find the numerator of the fraction immediately to the left of 3/7 for denominators ≤ 1,000,000.
* [[Project Euler/71]] - Ordered Fractions - Find the numerator of the fraction immediately to the left
* [[Project Euler/72|Problem 72]] - Counting Fractions - Count the number of reduced proper fractions with denominator ≤ 1,000,000.
* of 3/7 for denominators ≤ 1,000,000.
* [[Project Euler/73|Problem 73]] - Counting Fractions in a Range - Count reduced proper fractions between 1/3 and 1/2 with denominator ≤ 12,000.
* [[Project Euler/72]] - Counting Fractions - Count the number of reduced proper fractions with
* [[Project Euler/74|Problem 74]] - Digit Factorial Chains - Find the sum of all numbers that produce a chain of exactly 60 non-repeating terms of digit factorial sums.
* denominator ≤ 1,000,000.
* [[Project Euler/75|Problem 75]] - Singular Integer Right Triangles - Find the number of perimeters ≤ 1,500,000 for which exactly one integer-sided right triangle exists.
* [[Project Euler/73]] - Counting Fractions in a Range - Count reduced proper fractions between 1/3 and
* [[Project Euler/76|Problem 76]] - Counting Summations - How many ways can 100 be written as a sum of at least two positive integers?
* 1/2 with denominator ≤ 12,000.
* [[Project Euler/77|Problem 77]] - Prime Summations - Find the first value that can be written as the sum of primes in over 5,000 different ways.
* [[Project Euler/74]] - Digit Factorial Chains - Find the sum of all numbers that produce a chain of
* [[Project Euler/78|Problem 78]] - Coin Partitions - Find the least value of n for which the partition function p(n) is divisible by 1,000,000.
* exactly 60 non-repeating terms of digit factorial sums.
* [[Project Euler/79|Problem 79]] - Passcode Derivation - Derive the shortest possible secret passcode from a list of successful keylog entries.
* [[Project Euler/75]] - Singular Integer Right Triangles - Find the number of perimeters ≤ 1,500,000
* [[Project Euler/80|Problem 80]] - Square Root Digital Expansion - Sum of the first 100 decimal digits for all irrational square roots up to 100.
* for which exactly one integer-sided right triangle exists.
* [[Project Euler/76]] - Counting Summations - How many ways can 100 be written as a sum of at least two
* positive integers?
* [[Project Euler/77]] - Prime Summations - Find the first value that can be written as the sum of
* primes in over 5,000 different ways.
* [[Project Euler/78]] - Coin Partitions - Find the least value of n for which the partition function
* p(n) is divisible by 1,000,000.
* [[Project Euler/79]] - Passcode Derivation - Derive the shortest possible secret passcode from a list
* of successful keylog entries.
* [[Project Euler/80]] - Square Root Digital Expansion - Sum of the first 100 decimal digits for all
* irrational square roots up to 100.


* [[Project Euler/81|Problem 81]] - Path Sum: Two Ways - Find the minimal path sum from top left to bottom right in an 80×80 matrix, moving only right and down.
* [[Project Euler/81]] - Path Sum: Two Ways - Find the minimal path sum from top left to bottom right in
* [[Project Euler/82|Problem 82]] - Path Sum: Three Ways - Find the minimal path sum from any cell in the left column to any cell in the right column, moving right, up, or down.
* an 80×80 matrix, moving only right and down.
* [[Project Euler/83|Problem 83]] - Path Sum: Four Ways - Find the minimal path sum from top left to bottom right moving up, down, left, or right.
* [[Project Euler/82]] - Path Sum: Three Ways - Find the minimal path sum from any cell in the left
* [[Project Euler/84|Problem 84]] - Monopoly Odds - Find the three most popular squares in Monopoly when using two 4-sided dice.
* column to any cell in the right column, moving right, up, or down.
* [[Project Euler/85|Problem 85]] - Counting Rectangles - Find the rectangular grid area whose number of contained rectangles is closest to 2 million.
* [[Project Euler/83]] - Path Sum: Four Ways - Find the minimal path sum from top left to bottom right
* [[Project Euler/86|Problem 86]] - Cuboid Route - Find the least M such that the number of distinct cuboids with an integer shortest route exceeds 1 million.
* moving up, down, left, or right.
* [[Project Euler/87|Problem 87]] - Prime Power Triples - Count numbers below 50 million expressible as the sum of a prime square, prime cube, and prime fourth power.
* [[Project Euler/84]] - Monopoly Odds - Find the three most popular squares in Monopoly when using two
* [[Project Euler/88|Problem 88]] - Product-Sum Numbers - Find the sum of all minimal product-sum numbers for 2 ≤ k ≤ 12,000.
* 4-sided dice.
* [[Project Euler/89|Problem 89]] - Roman Numerals - Find the number of characters saved by writing each Roman numeral in its minimal form.
* [[Project Euler/85]] - Counting Rectangles - Find the rectangular grid area whose number of contained
* [[Project Euler/90|Problem 90]] - Cube Digit Pairs - Count distinct arrangements of digits on two cubes that can display all square numbers from 01 to 99.
* rectangles is closest to 2 million.
* [[Project Euler/86]] - Cuboid Route - Find the least M such that the number of distinct cuboids with
* an integer shortest route exceeds 1 million.
* [[Project Euler/87]] - Prime Power Triples - Count numbers below 50 million expressible as the sum of
* a prime square, prime cube, and prime fourth power.
* [[Project Euler/88]] - Product-Sum Numbers - Find the sum of all minimal product-sum numbers for 2 ≤ k
* ≤ 12,000.
* [[Project Euler/89]] - Roman Numerals - Find the number of characters saved by writing each Roman
* numeral in its minimal form.
* [[Project Euler/90]] - Cube Digit Pairs - Count distinct arrangements of digits on two cubes that can
* display all square numbers from 01 to 99.


* [[Project Euler/91|Problem 91]] - Right Triangles with Integer Coordinates - Count right triangles with vertices on integer grid points in a 50×50 grid.
* [[Project Euler/91]] - Right Triangles with Integer Coordinates - Count right triangles with vertices
* [[Project Euler/92|Problem 92]] - Square Digit Chains - Count numbers below 10 million whose square digit chain arrives at 89.
* on integer grid points in a 50×50 grid.
* [[Project Euler/93|Problem 93]] - Arithmetic Expressions - Find the longest set of consecutive integers obtainable using four distinct digits and arithmetic operators.
* [[Project Euler/92]] - Square Digit Chains - Count numbers below 10 million whose square digit chain
* [[Project Euler/94|Problem 94]] - Almost Equilateral Triangles - Sum of perimeters of almost equilateral integer triangles with integral area and perimeter ≤ 1 billion.
* arrives at 89.
* [[Project Euler/95|Problem 95]] - Amicable Chains - Find the smallest member of the longest amicable chain with no element exceeding 1 million.
* [[Project Euler/93]] - Arithmetic Expressions - Find the longest set of consecutive integers
* [[Project Euler/96|Problem 96]] - Su Doku - Solve 50 Sudoku puzzles and sum the 3-digit numbers found in the top-left corner of each solution.
* obtainable using four distinct digits and arithmetic operators.
* [[Project Euler/97|Problem 97]] - Large Non-Mersenne Prime - Find the last ten digits of the non-Mersenne prime 28433×2^7830457+1.
* [[Project Euler/94]] - Almost Equilateral Triangles - Sum of perimeters of almost equilateral integer
* [[Project Euler/98|Problem 98]] - Anagramic Squares - Find the largest square number formed by anagramic pairs of dictionary words.
* triangles with integral area and perimeter ≤ 1 billion.
* [[Project Euler/99|Problem 99]] - Largest Exponential - Determine which line number gives the numerically largest value from a list of base/exponent pairs.
* [[Project Euler/95]] - Amicable Chains - Find the smallest member of the longest amicable chain with
* no element exceeding 1 million.
* [[Project Euler/96]] - Su Doku - Solve 50 Sudoku puzzles and sum the 3-digit numbers found in the
* top-left corner of each solution.
* [[Project Euler/97]] - Large Non-Mersenne Prime - Find the last ten digits of the non-Mersenne prime
* 28433×2^7830457+1.
* [[Project Euler/98]] - Anagramic Squares - Find the largest square number formed by anagramic pairs of
* dictionary words.
* [[Project Euler/99]] - Largest Exponential - Determine which line number gives the numerically largest
* value from a list of base/exponent pairs.


==Grid 1: Problems 100-199==
==Grid 1: Problems 100-199==


* [[Project Euler/100|Problem 100]] - Combinations of Red and Blue Discs - find arrangements of blue and red discs that lead to a probability of exactly 50% that a blue disc is removed, two times in a row.
* [[Project Euler/100]] - Combinations of Red and Blue Discs - find arrangements of blue and red discs
* [[Project Euler/101|Problem 101]] - Bad Optimal Polynomials - Lagrangian polynomial interpolation for a sequence of numbers, interpolation of an optimal N-1 polynomial given N points of data.
* that lead to a probability of exactly 50% that a blue disc is removed, two times in a row.
* [[Project Euler/102|Problem 102]] - Triangles Containing Origin - given 3 endpoints, determine if a triangle contains the origin.
* [[Project Euler/101]] - Bad Optimal Polynomials - Lagrangian polynomial interpolation for a sequence
* [[Project Euler/103|Problem 103]] - Special Subset Sums: Optimum - finding the optimum special sum set with n=7.
* of numbers, interpolation of an optimal N-1 polynomial given N points of data.
* [[Project Euler/104|Problem 104]] - Pandigital Fibonacci Ends - finding Fibonacci numbers with pandigital beginnings and endings.
* [[Project Euler/102]] - Triangles Containing Origin - given 3 endpoints, determine if a triangle
* [[Project Euler/105|Problem 105]] - Special Subset Sums: Testing - testing sets for the special sum property.
* contains the origin.
* [[Project Euler/106|Problem 106]] - Special Subset Sums: Meta-testing - counting subset pairs that need to be tested.
* [[Project Euler/103]] - Special Subset Sums: Optimum - finding the optimum special sum set with n=7.
* [[Project Euler/107|Problem 107]] - Minimal Network - finding the minimal network connecting all vertices (minimum spanning tree).
* [[Project Euler/104]] - Pandigital Fibonacci Ends - finding Fibonacci numbers with pandigital
* [[Project Euler/108|Problem 108]] - Diophantine Reciprocals I - solving 1/x + 1/y = 1/n for distinct solutions.
* beginnings and endings.
* [[Project Euler/109|Problem 109]] - Darts - counting the number of distinct ways to check out in darts with a score less than 100.
* [[Project Euler/105]] - Special Subset Sums: Testing - testing sets for the special sum property.
* [[Project Euler/106]] - Special Subset Sums: Meta-testing - counting subset pairs that need to be
* tested.
* [[Project Euler/107]] - Minimal Network - finding the minimal network connecting all vertices
* (minimum spanning tree).
* [[Project Euler/108]] - Diophantine Reciprocals I - solving 1/x + 1/y = 1/n for distinct solutions.
* [[Project Euler/109]] - Darts - counting the number of distinct ways to check out in darts with a
* score less than 100.


* [[Project Euler/110|Problem 110]] - Diophantine Reciprocals II - finding the smallest n with over 4 million solutions to 1/x + 1/y = 1/n.
* [[Project Euler/110]] - Diophantine Reciprocals II - finding the smallest n with over 4 million
* [[Project Euler/111|Problem 111]] - Primes with Runs - finding primes with maximum runs of repeated digits.
* solutions to 1/x + 1/y = 1/n.
* [[Project Euler/112|Problem 112]] - Bouncy Numbers - counting numbers whose digits are neither increasing nor decreasing.
* [[Project Euler/111]] - Primes with Runs - finding primes with maximum runs of repeated digits.
* [[Project Euler/113|Problem 113]] - Non-bouncy Numbers - counting numbers below a googol that are not bouncy.
* [[Project Euler/112]] - Bouncy Numbers - counting numbers whose digits are neither increasing nor
* [[Project Euler/114|Problem 114]] - Counting Block Combinations I - counting ways to fill a row with red and grey blocks.
* decreasing.
* [[Project Euler/115|Problem 115]] - Counting Block Combinations II - finding the minimum row length for over 1 million fill combinations.
* [[Project Euler/113]] - Non-bouncy Numbers - counting numbers below a googol that are not bouncy.
* [[Project Euler/116|Problem 116]] - Red, Green or Blue Tiles - counting ways to replace tiles with colored blocks.
* [[Project Euler/114]] - Counting Block Combinations I - counting ways to fill a row with red and grey
* [[Project Euler/117|Problem 117]] - Red, Green, and Blue Tiles - counting ways to place colored tiles of various lengths.
* blocks.
* [[Project Euler/118|Problem 118]] - Pandigital Prime Sets - partitioning the digits 1-9 into sets of prime numbers.
* [[Project Euler/115]] - Counting Block Combinations II - finding the minimum row length for over 1
* [[Project Euler/119|Problem 119]] - Digit Power Sum - finding numbers equal to the sum of their digits raised to some power.
* million fill combinations.
* [[Project Euler/116]] - Red, Green or Blue Tiles - counting ways to replace tiles with colored
* blocks.
* [[Project Euler/117]] - Red, Green, and Blue Tiles - counting ways to place colored tiles of various
* lengths.
* [[Project Euler/118]] - Pandigital Prime Sets - partitioning the digits 1-9 into sets of prime
* numbers.
* [[Project Euler/119]] - Digit Power Sum - finding numbers equal to the sum of their digits raised to
* some power.


* [[Project Euler/120|Problem 120]] - Square Remainders - sum of maximum remainders when (a−1)^n + (a+1)^n is divided by a^2.
* [[Project Euler/120]] - Square Remainders - sum of maximum remainders when (a−1)^n + (a+1)^n is
* [[Project Euler/121|Problem 121]] - Disc Game Prize Fund - finding max prize fund for a disc game with changing probabilities.
* divided by a^2.
* [[Project Euler/122|Problem 122]] - Efficient Exponentiation - computing n^15 using minimal multiplications (addition chains).
* [[Project Euler/121]] - Disc Game Prize Fund - finding max prize fund for a disc game with changing
* [[Project Euler/123|Problem 123]] - Prime Square Remainders - finding the prime where the maximum remainder exceeds 10^10.
* probabilities.
* [[Project Euler/124|Problem 124]] - Ordered Radicals - finding the k-th element when numbers are sorted by their radical (product of prime factors).
* [[Project Euler/122]] - Efficient Exponentiation - computing n^15 using minimal multiplications
* [[Project Euler/125|Problem 125]] - Palindromic Sums - sums of consecutive squares that are palindromic numbers.
* (addition chains).
* [[Project Euler/126|Problem 126]] - Cuboid Layers - counting the number of cubes needed to cover visible faces of cuboids in successive layers.
* [[Project Euler/123]] - Prime Square Remainders - finding the prime where the maximum remainder
* [[Project Euler/127|Problem 127]] - abc-hits - counting triples where rad(abc) < c and a and b are coprime.
* exceeds 10^10.
* [[Project Euler/128|Problem 128]] - Hexagonal Tile Differences - finding tiles in a hexagonal spiral where all neighbors have prime differences.
* [[Project Euler/124]] - Ordered Radicals - finding the k-th element when numbers are sorted by their
* [[Project Euler/129|Problem 129]] - Repunit Divisibility - finding the least n such that a repunit R(n) is divisible by a given number.
* radical (product of prime factors).
* [[Project Euler/130|Problem 130]] - Composites with Prime Repunit Property - composite numbers where n divides the repunit R(n−1).
* [[Project Euler/125]] - Palindromic Sums - sums of consecutive squares that are palindromic numbers.
* [[Project Euler/126]] - Cuboid Layers - counting the number of cubes needed to cover visible faces of
* cuboids in successive layers.
* [[Project Euler/127]] - abc-hits - counting triples where rad(abc) < c and a and b are coprime.
* [[Project Euler/128]] - Hexagonal Tile Differences - finding tiles in a hexagonal spiral where all
* neighbors have prime differences.
* [[Project Euler/129]] - Repunit Divisibility - finding the least n such that a repunit R(n) is
* divisible by a given number.
* [[Project Euler/130]] - Composites with Prime Repunit Property - composite numbers where n divides
* the repunit R(n−1).


* [[Project Euler/131|Problem 131]] - Prime Cube Partnership - primes p for which n^3 + n^2·p is a perfect cube.
* [[Project Euler/131]] - Prime Cube Partnership - primes p for which n^3 + n^2·p is a perfect cube.
* [[Project Euler/132|Problem 132]] - Large Repunit Factors - sum of the first forty prime factors of R(10^9).
* [[Project Euler/132]] - Large Repunit Factors - sum of the first forty prime factors of R(10^9).
* [[Project Euler/133|Problem 133]] - Repunit Nonfactors - primes that will never divide any repunit R(10^n).
* [[Project Euler/133]] - Repunit Nonfactors - primes that will never divide any repunit R(10^n).
* [[Project Euler/134|Problem 134]] - Prime Pair Connection - connecting consecutive primes p1, p2 to form a number divisible by p2.
* [[Project Euler/134]] - Prime Pair Connection - connecting consecutive primes p1, p2 to form a number
* [[Project Euler/135|Problem 135]] - Same Differences - solving x^2 − y^2 − z^2 = n where x, y, z form an arithmetic progression.
* divisible by p2.
* [[Project Euler/136|Problem 136]] - Singleton Difference - finding n with exactly one solution to x^2 − y^2 − z^2 = n.
* [[Project Euler/135]] - Same Differences - solving x^2 − y^2 − z^2 = n where x, y, z form an
* [[Project Euler/137|Problem 137]] - Fibonacci Golden Nuggets - Fibonacci numbers appearing as solutions to a Pell-type Diophantine equation.
* arithmetic progression.
* [[Project Euler/138|Problem 138]] - Special Isosceles Triangles - isosceles triangles with integer height and half-base differing by 1.
* [[Project Euler/136]] - Singleton Difference - finding n with exactly one solution to x^2 − y^2 − z^2
* [[Project Euler/139|Problem 139]] - Pythagorean Tiles - Pythagorean triangles that allow tiling of a square of side equal to the hypotenuse.
* = n.
* [[Project Euler/140|Problem 140]] - Modified Fibonacci Golden Nuggets - golden nuggets from a modified Fibonacci sequence.
* [[Project Euler/137]] - Fibonacci Golden Nuggets - Fibonacci numbers appearing as solutions to a
* Pell-type Diophantine equation.
* [[Project Euler/138]] - Special Isosceles Triangles - isosceles triangles with integer height and
* half-base differing by 1.
* [[Project Euler/139]] - Pythagorean Tiles - Pythagorean triangles that allow tiling of a square of
* side equal to the hypotenuse.
* [[Project Euler/140]] - Modified Fibonacci Golden Nuggets - golden nuggets from a modified Fibonacci
* sequence.


* [[Project Euler/141|Problem 141]] - Square Progressive Numbers - perfect squares that are also progressive (geometric progression of digits).
* [[Project Euler/141]] - Square Progressive Numbers - perfect squares that are also progressive
* [[Project Euler/142|Problem 142]] - Perfect Square Collection - finding x+y+z where x>y>z>0, all pairwise sums/differences are squares.
* (geometric progression of digits).
* [[Project Euler/143|Problem 143]] - Torricelli Triangles - triangles whose Torricelli point has integer distances to the vertices.
* [[Project Euler/142]] - Perfect Square Collection - finding x+y+z where x>y>z>0, all pairwise
* [[Project Euler/144|Problem 144]] - Laser Beam Reflections - reflecting a laser beam inside an elliptical mirror until it exits.
* sums/differences are squares.
* [[Project Euler/145|Problem 145]] - Reversible Numbers - counting numbers n below 1 billion where n + reverse(n) has all odd digits.
* [[Project Euler/143]] - Torricelli Triangles - triangles whose Torricelli point has integer distances
* [[Project Euler/146|Problem 146]] - Investigating a Prime Pattern - finding n where n^2+1, n^2+3, n^2+7, n^2+9, n^2+13, n^2+27 are consecutive primes.
* to the vertices.
* [[Project Euler/147|Problem 147]] - Rectangles in Cross-hatched Grids - counting all rectangles in a cross-hatched rectangular grid.
* [[Project Euler/144]] - Laser Beam Reflections - reflecting a laser beam inside an elliptical mirror
* [[Project Euler/148|Problem 148]] - Exploring Pascal's Triangle - counting entries in the first billion rows of Pascal's triangle not divisible by 7.
* until it exits.
* [[Project Euler/149|Problem 149]] - Maximum-sum Subsequence - finding the maximum sum of adjacent subsequences in a generated 2000×2000 array.
* [[Project Euler/145]] - Reversible Numbers - counting numbers n below 1 billion where n + reverse(n)
* [[Project Euler/150|Problem 150]] - Sub-triangle Sums - finding the minimum-sum sub-triangle in a triangular array of 1000 rows.
* has all odd digits.
* [[Project Euler/146]] - Investigating a Prime Pattern - finding n where n^2+1, n^2+3, n^2+7, n^2+9,
* n^2+13, n^2+27 are consecutive primes.
* [[Project Euler/147]] - Rectangles in Cross-hatched Grids - counting all rectangles in a
* cross-hatched rectangular grid.
* [[Project Euler/148]] - Exploring Pascal's Triangle - counting entries in the first billion rows of
* Pascal's triangle not divisible by 7.
* [[Project Euler/149]] - Maximum-sum Subsequence - finding the maximum sum of adjacent subsequences in
* a generated 2000×2000 array.
* [[Project Euler/150]] - Sub-triangle Sums - finding the minimum-sum sub-triangle in a triangular
* array of 1000 rows.


* [[Project Euler/151|Problem 151]] - Paper Sheets of Standard Sizes - expected number of times (excluding first and last batch) that the supervisor finds a single sheet of paper in the envelope, when randomly drawing and cutting A1→A5 paper sheets across 16 weekly batches.
* [[Project Euler/151]] - Paper Sheets of Standard Sizes - expected number of times (excluding first
* [[Project Euler/152|Problem 152]] - Sums of Square Reciprocals - count the number of ways to write 1/2 as a sum of reciprocals of squares using distinct integers between 2 and 80 inclusive.
* and last batch) that the supervisor finds a single sheet of paper in the envelope, when randomly drawing and
* [[Project Euler/153|Problem 153]] - Investigating Gaussian Integers - sum of all Gaussian integer divisors (with positive real part) for all rational integers n up to 10^8.
* cutting A1→A5 paper sheets across 16 weekly batches.
* [[Project Euler/154|Problem 154]] - Exploring Pascal's Pyramid - count how many coefficients in the trinomial expansion (x + y + z)^200000 are multiples of 10^12.
* [[Project Euler/152]] - Sums of Square Reciprocals - count the number of ways to write 1/2 as a sum
* [[Project Euler/155|Problem 155]] - Counting Capacitor Circuits - number of distinct total capacitance values D(n) obtainable using up to n=18 equal-valued capacitors in series and parallel combinations.
* of reciprocals of squares using distinct integers between 2 and 80 inclusive.
* [[Project Euler/156|Problem 156]] - Counting Digits - sum over digits d=1..9 of the sum of all solutions n where the total count of digit d written from 0 to n equals n (i.e., f(n,d)=n).
* [[Project Euler/153]] - Investigating Gaussian Integers - sum of all Gaussian integer divisors (with
* [[Project Euler/157|Problem 157]] - Base-10 Diophantine Reciprocal - count the number of positive integer solutions to 1/a + 1/b = p/10^n with a ≤ b, for 1 ≤ n ≤ 9.
* positive real part) for all rational integers n up to 10^8.
* [[Project Euler/158|Problem 158]] - Strings of various lengths, with exactly one character lexicographically out of sorts
* [[Project Euler/154]] - Exploring Pascal's Pyramid - count how many coefficients in the trinomial
* [[Project Euler/159|Problem 159]]
* expansion (x + y + z)^200000 are multiples of 10^12.
* [[Project Euler/160|Problem 160]] - Factorial Trailing Digits - find the last five non-zero digits of 1,000,000,000,000!
* [[Project Euler/155]] - Counting Capacitor Circuits - number of distinct total capacitance values
* [[Project Euler/161|Problem 161]] - Triominoes - count the number of ways a 9×12 grid can be tiled with triominoes.
* D(n) obtainable using up to n=18 equal-valued capacitors in series and parallel combinations.
* [[Project Euler/162|Problem 162]] - Hexadecimal Numbers - count hex numbers with ≤16 digits containing 0, 1, and A at least once.
* [[Project Euler/156]] - Counting Digits - sum over digits d=1..9 of the sum of all solutions n where
* [[Project Euler/163|Problem 163]] - Cross-hatched Triangles - count triangles in a size 36 cross-hatched equilateral triangle.
* the total count of digit d written from 0 to n equals n (i.e., f(n,d)=n).
* [[Project Euler/164|Problem 164]] - Three Consecutive Digital Sum Limit - count 20-digit numbers where no three consecutive digits sum to more than 9.
* [[Project Euler/157]] - Base-10 Diophantine Reciprocal - count the number of positive integer
* [[Project Euler/165|Problem 165]] - Intersections - count distinct true intersection points among 5000 line segments.
* solutions to 1/a + 1/b = p/10^n with a ≤ b, for 1 ≤ n ≤ 9.
* [[Project Euler/166|Problem 166]] - Criss Cross - count 4×4 digit grids where each row, column, and both diagonals share the same sum.
* [[Project Euler/158]] - Strings of various lengths, with exactly one character lexicographically out
* [[Project Euler/167|Problem 167]] - Investigating Ulam Sequences - sum of U(2,2n+1)_k for n=2..10, where k=10^11.
* of sorts
* [[Project Euler/168|Problem 168]] - Number Rotations - find the last 5 digits of the sum of all n (10<n<10^100) that divide their own right-rotation.
* [[Project Euler/159]]
* [[Project Euler/169|Problem 169]] - Sums of Powers of Two - count ways to express 10^25 as a sum of powers of 2 using each power at most twice.
* [[Project Euler/160]] - Factorial Trailing Digits - find the last five non-zero digits of
* [[Project Euler/170|Problem 170]]
* 1,000,000,000,000!
* [[Project Euler/171|Problem 171]]
* [[Project Euler/161]] - Triominoes - count the number of ways a 9×12 grid can be tiled with
* [[Project Euler/172|Problem 172]] - Few Repeated Digits - how many 18 digit numbers have no digit occurring more than 3 times in n?
* triominoes.
* [[Project Euler/173|Problem 173]]
* [[Project Euler/162]] - Hexadecimal Numbers - count hex numbers with ≤16 digits containing 0, 1, and
* [[Project Euler/174|Problem 174]]
* A at least once.
* [[Project Euler/175|Problem 175]]
* [[Project Euler/163]] - Cross-hatched Triangles - count triangles in a size 36 cross-hatched
* [[Project Euler/176|Problem 176]]
* equilateral triangle.
* [[Project Euler/177|Problem 177]]
* [[Project Euler/164]] - Three Consecutive Digital Sum Limit - count 20-digit numbers where no three
* [[Project Euler/178|Problem 178]]
* consecutive digits sum to more than 9.
* [[Project Euler/179|Problem 179]]
* [[Project Euler/165]] - Intersections - count distinct true intersection points among 5000 line
* segments.
* [[Project Euler/166]] - Criss Cross - count 4×4 digit grids where each row, column, and both
* diagonals share the same sum.
* [[Project Euler/167]] - Investigating Ulam Sequences - sum of U(2,2n+1)_k for n=2..10, where k=10^11.
* [[Project Euler/168]] - Number Rotations - find the last 5 digits of the sum of all n (10<n<10^100)
* that divide their own right-rotation.
* [[Project Euler/169]] - Sums of Powers of Two - count ways to express 10^25 as a sum of powers of 2
* using each power at most twice.
* [[Project Euler/170]]
* [[Project Euler/171]]
* [[Project Euler/172]] - Few Repeated Digits - how many 18 digit numbers have no digit occurring more
* than 3 times in n?
* [[Project Euler/173]]
* [[Project Euler/174]]
* [[Project Euler/175]]
* [[Project Euler/176]]
* [[Project Euler/177]]
* [[Project Euler/178]]
* [[Project Euler/179]]


* [[Project Euler/190|Problem 180]]
* [[Project Euler/180]]
* [[Project Euler/191|Problem 181]]
* [[Project Euler/181]]
* [[Project Euler/192|Problem 182]]
* [[Project Euler/182]]
* [[Project Euler/193|Problem 183]]
* [[Project Euler/183]]
* [[Project Euler/194|Problem 184]]
* [[Project Euler/184]]
* [[Project Euler/195|Problem 185]]
* [[Project Euler/185]]
* [[Project Euler/196|Problem 186]]
* [[Project Euler/186]]
* [[Project Euler/197|Problem 187]]
* [[Project Euler/187]]
* [[Project Euler/198|Problem 188]]
* [[Project Euler/188]]
* [[Project Euler/199|Problem 189]]
* [[Project Euler/189]]


* [[Project Euler/190|Problem 190]]
* [[Project Euler/190]]
* [[Project Euler/191|Problem 191]]
* [[Project Euler/191]]
* [[Project Euler/192|Problem 192]]
* [[Project Euler/192]]
* [[Project Euler/193|Problem 193]]
* [[Project Euler/193]]
* [[Project Euler/194|Problem 194]]
* [[Project Euler/194]]
* [[Project Euler/195|Problem 195]]
* [[Project Euler/195]]
* [[Project Euler/196|Problem 196]]
* [[Project Euler/196]]
* [[Project Euler/197|Problem 197]]
* [[Project Euler/197]]
* [[Project Euler/198|Problem 198]]
* [[Project Euler/198]]
* [[Project Euler/199|Problem 199]]
* [[Project Euler/199]]


==Grid 2: Problems 200-299==
==Grid 2: Problems 200-299==


* [[Project Euler/254|Problem 254]] - Maximum Source of Sums of Digits of Sums of Digits of Sums of Factorial Digit Sums
* [[Project Euler/254]] - Maximum Source of Sums of Digits of Sums of Digits of Sums of Factorial Digit
* Sums


* [[Project Euler/255|Problem 255]] - Rounded Square Roots - computing rounded-square-roots using an iterative integer method (Heron's method adapted to integer arithmetic).
* [[Project Euler/255]] - Rounded Square Roots - computing rounded-square-roots using an iterative
* [[Project Euler/256|Problem 256]] - Tatami-Free Rooms - counting even-sized rectangular rooms that cannot be covered by 1×2 tatami mats without a forbidden cross pattern.
* integer method (Heron's method adapted to integer arithmetic).
* [[Project Euler/257|Problem 257]] - Angular Bisectors - integer-sided triangles whose angular bisector segments are also integers.
* [[Project Euler/256]] - Tatami-Free Rooms - counting even-sized rectangular rooms that cannot be
* [[Project Euler/258|Problem 258]] - A Lagged Fibonacci Sequence - finding values from a lagged Fibonacci generator defined by a cubic formula.
* covered by 1×2 tatami mats without a forbidden cross pattern.
* [[Project Euler/259|Problem 259]] - Reachable Numbers - numbers expressible as arithmetic expressions using digits 1 through 9 in order, each exactly once.
* [[Project Euler/257]] - Angular Bisectors - integer-sided triangles whose angular bisector segments
* [[Project Euler/260|Problem 260]] - Stone Game - a three-pile Nim-like game; counting winning configurations for the first player.
* are also integers.
* [[Project Euler/261|Problem 261]] - Pivotal Square Sums - finding square-pivot integers where a sum of consecutive squares equals a perfect square.
* [[Project Euler/258]] - A Lagged Fibonacci Sequence - finding values from a lagged Fibonacci
* [[Project Euler/262|Problem 262]] - Mountain Range - finding the shortest continuous path between two points across a mountainous terrain.
* generator defined by a cubic formula.
* [[Project Euler/263|Problem 263]] - An Engineers' Dream Come True - finding numbers with special properties relating consecutive primes and practical numbers.
* [[Project Euler/259]] - Reachable Numbers - numbers expressible as arithmetic expressions using
* [[Project Euler/264|Problem 264]] - Triangle Centres - integer-coordinate triangles whose centroid and orthocenter are also on integer coordinates.
* digits 1 through 9 in order, each exactly once.
* [[Project Euler/265|Problem 265]] - Binary Circles - placing 2^N binary digits in a circle such that all N-digit clockwise subsequences are distinct.
* [[Project Euler/260]] - Stone Game - a three-pile Nim-like game; counting winning configurations for
* [[Project Euler/266|Problem 266]] - Pseudo Square Root - finding the product of pseudo square roots (largest divisor ≤ √n) of primes below 190.
* the first player.
* [[Project Euler/267|Problem 267]] - Billionaire - maximizing the chance of reaching £1 billion through optimal betting on 1000 fair coin tosses.
* [[Project Euler/261]] - Pivotal Square Sums - finding square-pivot integers where a sum of
* [[Project Euler/268|Problem 268]] - Counting Numbers with at Least Four Distinct Prime Factors Less than 100.
* consecutive squares equals a perfect square.
* [[Project Euler/269|Problem 269]] - Polynomials with at Least One Integer Root - polynomials whose coefficients are the digits of n in base 10.
* [[Project Euler/262]] - Mountain Range - finding the shortest continuous path between two points
* [[Project Euler/270|Problem 270]] - Cutting Squares - counting ways to cut an N×N square into pieces with integer side lengths.
* across a mountainous terrain.
* [[Project Euler/271|Problem 271]] - Modular Cubes, Part 1 - summing x (1 < x < n) for which x³ ≡ 1 (mod n), for a specific n.
* [[Project Euler/263]] - An Engineers' Dream Come True - finding numbers with special properties
* [[Project Euler/272|Problem 272]] - Modular Cubes, Part 2 - extending the modular cubes problem to a larger modulus.
* relating consecutive primes and practical numbers.
* [[Project Euler/273|Problem 273]] - Sum of Squares - summing values of a in a² + b² = N for squarefree N with all prime factors of the form 4k+1.
* [[Project Euler/264]] - Triangle Centres - integer-coordinate triangles whose centroid and
* [[Project Euler/274|Problem 274]] - Divisibility Multipliers - finding positive multipliers m < p that preserve divisibility by p for primes p coprime to 10.
* orthocenter are also on integer coordinates.
* [[Project Euler/275|Problem 275]] - Balanced Sculptures - counting polyomino-based sculptures of order n whose combined centre of mass has x-coordinate zero.
* [[Project Euler/265]] - Binary Circles - placing 2^N binary digits in a circle such that all N-digit
* [[Project Euler/276|Problem 276]] - Primitive Triangles - integer-sided triangles with integer area where the greatest common divisor of sides is 1.
* clockwise subsequences are distinct.
* [[Project Euler/277|Problem 277]] - A Modified Collatz Sequence - a Collatz-like sequence with three possible steps: divide by 3, or apply floor-based rules.
* [[Project Euler/266]] - Pseudo Square Root - finding the product of pseudo square roots (largest
* [[Project Euler/278|Problem 278]] - Linear Combinations of Semiprimes - counting numbers expressible as linear combinations of pairs of semiprimes.
* divisor ≤ √n) of primes below 190.
* [[Project Euler/279|Problem 279]] - Triangles with Integral Sides and an Integral Angle - triangles where at least one angle measured in degrees is an integer.
* [[Project Euler/267]] - Billionaire - maximizing the chance of reaching £1 billion through optimal
* [[Project Euler/280|Problem 280]] - Ant and Seeds - an ant walking on a 5×5 grid carrying seeds; finding the expected number of steps to complete the task.
* betting on 1000 fair coin tosses.
* [[Project Euler/281|Problem 281]] - Pizza Toppings - counting distinct ways to place m toppings on m·n pizza slices, considering rotational symmetry.
* [[Project Euler/268]] - Counting Numbers with at Least Four Distinct Prime Factors Less than 100.
* [[Project Euler/282|Problem 282]] - The Ackermann Function - computing sums of values of the Ackermann function modulo large numbers.
* [[Project Euler/269]] - Polynomials with at Least One Integer Root - polynomials whose coefficients
* [[Project Euler/283|Problem 283]] - Integer-sided Triangles for Which the Area/Perimeter Ratio is Integral.
* are the digits of n in base 10.
* [[Project Euler/284|Problem 284]] - Steady Squares - numbers in base 14 whose square ends with the number itself.
* [[Project Euler/270]] - Cutting Squares - counting ways to cut an N×N square into pieces with integer
* [[Project Euler/285|Problem 285]] - Pythagorean Odds - expected value in a game involving random points and the probability of a Pythagorean distance.
* side lengths.
* [[Project Euler/286|Problem 286]] - Scoring Probabilities - basketball shooting probability as a function of distance; finding the constant q that yields a 50% scoring chance.
* [[Project Euler/271]] - Modular Cubes, Part 1 - summing x (1 < x < n) for which x³ ≡ 1 (mod n), for a
* [[Project Euler/287|Problem 287]] - Quadtree Encoding - bit-length of the quadtree encoding of a 2^N × 2^N disk-shaped black-and-white image.
* specific n.
* [[Project Euler/288|Problem 288]] - An Enormous Factorial - computing N(p,q) modulo large powers of p for a specifically defined sequence.
* [[Project Euler/272]] - Modular Cubes, Part 2 - extending the modular cubes problem to a larger
* [[Project Euler/289|Problem 289]] - Eulerian Cycles - counting non-crossing Eulerian cycles on a grid formed by arranging circles.
* modulus.
* [[Project Euler/290|Problem 290]] - Digital Signature - sum of digits of all numbers expressible as a particular form up to 10^18.
* [[Project Euler/273]] - Sum of Squares - summing values of a in a² + b² = N for squarefree N with all
* [[Project Euler/291|Problem 291]] - Panaitopol Primes - primes expressible as (x⁴ − y⁴) / (x³ + y³) for positive integers x and y.
* prime factors of the form 4k+1.
* [[Project Euler/292|Problem 292]] - Pythagorean Polygons - convex polygons with integer perimeter formed from at least three edge-disjoint right triangles.
* [[Project Euler/274]] - Divisibility Multipliers - finding positive multipliers m < p that preserve
* [[Project Euler/293|Problem 293]] - Pseudo-Fortunate Numbers - for admissible numbers N, the smallest integer m > 1 such that N + m is prime.
* divisibility by p for primes p coprime to 10.
* [[Project Euler/294|Problem 294]] - Sum of Digits — Experience #23 - summing digits of multiples of 23.
* [[Project Euler/275]] - Balanced Sculptures - counting polyomino-based sculptures of order n whose
* [[Project Euler/295|Problem 295]] - Lenticular Holes - convex areas enclosed by two circles whose centers and intersection points are on lattice points.
* combined centre of mass has x-coordinate zero.
* [[Project Euler/296|Problem 296]] - Angular Bisector and Tangent - integer-sided triangles where the angular bisector is tangent to an inscribed circle.
* [[Project Euler/276]] - Primitive Triangles - integer-sided triangles with integer area where the
* [[Project Euler/297|Problem 297]] - Zeckendorf Representation - sum of the number of terms in Zeckendorf representations of all numbers below 10^17.
* greatest common divisor of sides is 1.
* [[Project Euler/298|Problem 298]] - Selective Amnesia - a memory game with random numbers; expected absolute difference in scores after 50 turns.
* [[Project Euler/277]] - A Modified Collatz Sequence - a Collatz-like sequence with three possible
* [[Project Euler/299|Problem 299]] - Three Similar Triangles - integer-sided triangles containing three similar right triangles.
* steps: divide by 3, or apply floor-based rules.
* [[Project Euler/278]] - Linear Combinations of Semiprimes - counting numbers expressible as linear
* combinations of pairs of semiprimes.
* [[Project Euler/279]] - Triangles with Integral Sides and an Integral Angle - triangles where at
* least one angle measured in degrees is an integer.
* [[Project Euler/280]] - Ant and Seeds - an ant walking on a 5×5 grid carrying seeds; finding the
* expected number of steps to complete the task.
* [[Project Euler/281]] - Pizza Toppings - counting distinct ways to place m toppings on m·n pizza
* slices, considering rotational symmetry.
* [[Project Euler/282]] - The Ackermann Function - computing sums of values of the Ackermann function
* modulo large numbers.
* [[Project Euler/283]] - Integer-sided Triangles for Which the Area/Perimeter Ratio is Integral.
* [[Project Euler/284]] - Steady Squares - numbers in base 14 whose square ends with the number itself.
* [[Project Euler/285]] - Pythagorean Odds - expected value in a game involving random points and the
* probability of a Pythagorean distance.
* [[Project Euler/286]] - Scoring Probabilities - basketball shooting probability as a function of
* distance; finding the constant q that yields a 50% scoring chance.
* [[Project Euler/287]] - Quadtree Encoding - bit-length of the quadtree encoding of a 2^N × 2^N
* disk-shaped black-and-white image.
* [[Project Euler/288]] - An Enormous Factorial - computing N(p,q) modulo large powers of p for a
* specifically defined sequence.
* [[Project Euler/289]] - Eulerian Cycles - counting non-crossing Eulerian cycles on a grid formed by
* arranging circles.
* [[Project Euler/290]] - Digital Signature - sum of digits of all numbers expressible as a particular
* form up to 10^18.
* [[Project Euler/291]] - Panaitopol Primes - primes expressible as (x⁴ − y⁴) / (x³ + y³) for positive
* integers x and y.
* [[Project Euler/292]] - Pythagorean Polygons - convex polygons with integer perimeter formed from at
* least three edge-disjoint right triangles.
* [[Project Euler/293]] - Pseudo-Fortunate Numbers - for admissible numbers N, the smallest integer m >
* 1 such that N + m is prime.
* [[Project Euler/294]] - Sum of Digits — Experience #23 - summing digits of multiples of 23.
* [[Project Euler/295]] - Lenticular Holes - convex areas enclosed by two circles whose centers and
* intersection points are on lattice points.
* [[Project Euler/296]] - Angular Bisector and Tangent - integer-sided triangles where the angular
* bisector is tangent to an inscribed circle.
* [[Project Euler/297]] - Zeckendorf Representation - sum of the number of terms in Zeckendorf
* representations of all numbers below 10^17.
* [[Project Euler/298]] - Selective Amnesia - a memory game with random numbers; expected absolute
* difference in scores after 50 turns.
* [[Project Euler/299]] - Three Similar Triangles - integer-sided triangles containing three similar
* right triangles.


==Grid 3: Problems 300-399==
==Grid 3: Problems 300-399==


* [[Project Euler/301|Problem 301]] - Nim - Counting losing positions in three-heap normal-play Nim for n ≤ 2^30.
* [[Project Euler/301]] - Nim - Counting losing positions in three-heap normal-play Nim for n ≤ 2^30.
* [[Project Euler/302|Problem 302]] - Strong Achilles Numbers - Count how many strong Achilles numbers are below 10^18.
* [[Project Euler/302]] - Strong Achilles Numbers - Count how many strong Achilles numbers are below
* [[Project Euler/303|Problem 303]] - Multiples with Small Digits - Sum of least positive multiples using only digits ≤ 2.
* 10^18.
* [[Project Euler/304|Problem 304]] - Primonacci - Sum of Fibonacci numbers at prime indices starting after 10^14.
* [[Project Euler/303]] - Multiples with Small Digits - Sum of least positive multiples using only
* [[Project Euler/305|Problem 305]] - Reflexive Position - Starting positions of the n-th occurrence of n in the concatenated infinite integer string.
* digits ≤ 2.
* [[Project Euler/306|Problem 306]] - Paper-strip Game - Combinatorial game: pick two contiguous white squares and paint them black.
* [[Project Euler/304]] - Primonacci - Sum of Fibonacci numbers at prime indices starting after 10^14.
* [[Project Euler/307|Problem 307]] - Chip Defects - Probability of at least one chip having 3+ defects when distributing defects randomly.
* [[Project Euler/305]] - Reflexive Position - Starting positions of the n-th occurrence of n in the
* [[Project Euler/308|Problem 308]] - An Amazing Prime-generating Automaton - Find the 10^15th prime generated by Conway's Fractran program.
* concatenated infinite integer string.
* [[Project Euler/309|Problem 309]] - Integer Ladders - Count integer solutions to the classic crossing ladders problem.
* [[Project Euler/306]] - Paper-strip Game - Combinatorial game: pick two contiguous white squares and
* [[Project Euler/310|Problem 310]] - Nim Square - Nim variant where players may only remove a square number of stones.
* paint them black.
* [[Project Euler/311|Problem 311]] - Biclinic Integral Quadrilaterals - Count biclinic integral quadrilaterals with bounded sum of squared sides.
* [[Project Euler/307]] - Chip Defects - Probability of at least one chip having 3+ defects when
* [[Project Euler/312|Problem 312]] - Cyclic Paths on Sierpiński Graphs - Counting Hamiltonian cycles on Sierpiński graphs.
* distributing defects randomly.
* [[Project Euler/313|Problem 313]] - Sliding Game - Minimum moves to slide a counter across an m×n grid.
* [[Project Euler/308]] - An Amazing Prime-generating Automaton - Find the 10^15th prime generated by
* [[Project Euler/314|Problem 314]] - The Mouse on the Moon - Maximizing enclosed-area/wall-length ratio on a grid of posts.
* Conway's Fractran program.
* [[Project Euler/315|Problem 315]] - Digital Root Clocks - 7-segment display power consumption for digital root clocks.
* [[Project Euler/309]] - Integer Ladders - Count integer solutions to the classic crossing ladders
* [[Project Euler/316|Problem 316]] - Numbers in Decimal Expansions - Expected position of a number in a random infinite decimal sequence.
* problem.
* [[Project Euler/317|Problem 317]] - Firecracker - Volume of the region through which firecracker fragments travel.
* [[Project Euler/310]] - Nim Square - Nim variant where players may only remove a square number of
* [[Project Euler/318|Problem 318]] - 2011 Nines - Count consecutive nines in fractional parts of powers of sqrt(p)+sqrt(q).
* stones.
* [[Project Euler/319|Problem 319]] - Bounded Sequences - Count sequences of length n satisfying x_i^j < (x_j+1)^i.
* [[Project Euler/311]] - Biclinic Integral Quadrilaterals - Count biclinic integral quadrilaterals
* [[Project Euler/320|Problem 320]] - Factorials Divisible by a Huge Integer - Smallest n such that n! is divisible by (i!)^1234567890.
* with bounded sum of squared sides.
* [[Project Euler/321|Problem 321]] - Swapping Counters - Minimum moves to swap n red and n blue counters on a row.
* [[Project Euler/312]] - Cyclic Paths on Sierpiński Graphs - Counting Hamiltonian cycles on Sierpiński
* [[Project Euler/322|Problem 322]] - Binomial Coefficients Divisible by 10 - Count binomial coefficients divisible by 10 in a given range.
* graphs.
* [[Project Euler/323|Problem 323]] - Bitwise-OR Operations on Random Integers - Expected number of random 32-bit integers to fill all bits via bitwise-OR.
* [[Project Euler/313]] - Sliding Game - Minimum moves to slide a counter across an m×n grid.
* [[Project Euler/324|Problem 324]] - Building a Tower - Number of ways to fill a 3×3×n tower with 2×1×1 blocks.
* [[Project Euler/314]] - The Mouse on the Moon - Maximizing enclosed-area/wall-length ratio on a grid
* [[Project Euler/325|Problem 325]] - Stone Game II - Two-pile game where removal must be a multiple of the smaller pile.
* of posts.
* [[Project Euler/326|Problem 326]] - Modulo Summations - Sequence defined by recursive modular sums, count zero-sum subarrays.
* [[Project Euler/315]] - Digital Root Clocks - 7-segment display power consumption for digital root
* [[Project Euler/327|Problem 327]] - Rooms of Doom - Minimum security cards to traverse rooms with limited carrying capacity.
* clocks.
* [[Project Euler/328|Problem 328]] - Lowest-cost Search - Optimal strategy to find a hidden number where each guess costs the value guessed.
* [[Project Euler/316]] - Numbers in Decimal Expansions - Expected position of a number in a random
* [[Project Euler/329|Problem 329]] - Prime Frog - Probability of a frog's croaking sequence when jumping on prime and non-prime squares.
* infinite decimal sequence.
* [[Project Euler/330|Problem 330]] - Euler's Number - Infinite sequence defined via Euler's number e, find A(10^9)+B(10^9).
* [[Project Euler/317]] - Firecracker - Volume of the region through which firecracker fragments
* travel.
* [[Project Euler/318]] - 2011 Nines - Count consecutive nines in fractional parts of powers of
* sqrt(p)+sqrt(q).
* [[Project Euler/319]] - Bounded Sequences - Count sequences of length n satisfying x_i^j < (x_j+1)^i.
* [[Project Euler/320]] - Factorials Divisible by a Huge Integer - Smallest n such that n! is divisible
* by (i!)^1234567890.
* [[Project Euler/321]] - Swapping Counters - Minimum moves to swap n red and n blue counters on a row.
* [[Project Euler/322]] - Binomial Coefficients Divisible by 10 - Count binomial coefficients divisible
* by 10 in a given range.
* [[Project Euler/323]] - Bitwise-OR Operations on Random Integers - Expected number of random 32-bit
* integers to fill all bits via bitwise-OR.
* [[Project Euler/324]] - Building a Tower - Number of ways to fill a 3×3×n tower with 2×1×1 blocks.
* [[Project Euler/325]] - Stone Game II - Two-pile game where removal must be a multiple of the smaller
* pile.
* [[Project Euler/326]] - Modulo Summations - Sequence defined by recursive modular sums, count
* zero-sum subarrays.
* [[Project Euler/327]] - Rooms of Doom - Minimum security cards to traverse rooms with limited
* carrying capacity.
* [[Project Euler/328]] - Lowest-cost Search - Optimal strategy to find a hidden number where each
* guess costs the value guessed.
* [[Project Euler/329]] - Prime Frog - Probability of a frog's croaking sequence when jumping on prime
* and non-prime squares.
* [[Project Euler/330]] - Euler's Number - Infinite sequence defined via Euler's number e, find
* A(10^9)+B(10^9).


* [[Project Euler/331|Problem 331]] - Cross Flips - Minimal turns to flip all disks to white on an N×N board with cross-flipping moves.
* [[Project Euler/331]] - Cross Flips - Minimal turns to flip all disks to white on an N×N board with
* [[Project Euler/332|Problem 332]] - Spherical Triangles - Area of the smallest spherical triangle with integer-coordinate vertices.
* cross-flipping moves.
* [[Project Euler/333|Problem 333]] - Special Partitions - Count partitions of integers into terms of the form 2^i × 3^j.
* [[Project Euler/332]] - Spherical Triangles - Area of the smallest spherical triangle with
* [[Project Euler/334|Problem 334]] - Spilling the Beans - Game where removing two beans from a bowl puts one bean in each adjacent bowl.
* integer-coordinate vertices.
* [[Project Euler/333]] - Special Partitions - Count partitions of integers into terms of the form 2^i
* × 3^j.
* [[Project Euler/334]] - Spilling the Beans - Game where removing two beans from a bowl puts one bean
* in each adjacent bowl.


* [[Project Euler/335]]
* [[Project Euler/335]]
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==Grid 4: Problems 400-499==
==Grid 4: Problems 400-499==


* [[Project Euler/400|Problem 400]] - Fibonacci Tree Game - A take-away game on a Fibonacci tree; find the number of winning moves for the first player on T(10000).
* [[Project Euler/400]] - Fibonacci Tree Game - A take-away game on a Fibonacci tree; find the number
* [[Project Euler/401|Problem 401]] - Sum of Squares of Divisors - Find the sum of σ₂(i) for i=1 to n, where σ₂ is the sum of squares of divisors.
* of winning moves for the first player on T(10000).
* [[Project Euler/402|Problem 402]] - Integer-valued Polynomials - Sum of M(a,b,c) over all a,b,c ≤ N, where M is the maximum m such that n⁴+an³+bn²+cn is always a multiple of m.
* [[Project Euler/401]] - Sum of Squares of Divisors - Find the sum of σ₂(i) for i=1 to n, where σ₂ is
* [[Project Euler/403|Problem 403]] - Lattice Points Enclosed by Parabola and Line - Count lattice points in the region bounded by y = x²/k and y = ax + b.
* the sum of squares of divisors.
* [[Project Euler/404|Problem 404]] - Crisscross Ellipses - Count lattice points inside the intersection of two ellipses x²+4y²=4a² and its rotated copy.
* [[Project Euler/402]] - Integer-valued Polynomials - Sum of M(a,b,c) over all a,b,c ≤ N, where M is
* [[Project Euler/405|Problem 405]] - A Rectangular Tiling - Count the number of ways to tile a 2×n rectangle with 1×1 and 1×2 tiles.
* the maximum m such that n⁴+an³+bn²+cn is always a multiple of m.
* [[Project Euler/406|Problem 406]] - Guessing Game - Find the minimal total cost for a guessing game with three possible answers per question.
* [[Project Euler/403]] - Lattice Points Enclosed by Parabola and Line - Count lattice points in the
* [[Project Euler/407|Problem 407]] - Idempotents - Sum of the largest a ≤ n such that a² ≡ a (mod n) for all n up to 10⁷.
* region bounded by y = x²/k and y = ax + b.
* [[Project Euler/408|Problem 408]] - Admissible Paths Through a Grid - Count admissible north/east paths avoiding points where x, y, and x+y are all perfect squares.
* [[Project Euler/404]] - Crisscross Ellipses - Count lattice points inside the intersection of two
* [[Project Euler/409|Problem 409]] - Nim Extreme - Count winning nim positions with n non-empty piles of distinct sizes less than 2ⁿ.
* ellipses x²+4y²=4a² and its rotated copy.
* [[Project Euler/410|Problem 410]] - Circle and Tangent Line - Find the sum of all radii r for which a circle and a tangent line satisfy certain integer conditions.
* [[Project Euler/405]] - A Rectangular Tiling - Count the number of ways to tile a 2×n rectangle with
* [[Project Euler/411|Problem 411]] - Uphill Paths - Find the maximum number of stations on an uphill path where stations are defined by powers of 2 modulo n.
* 1×1 and 1×2 tiles.
* [[Project Euler/412|Problem 412]] - Gnomon Numbering - Count valid numberings of an m×m grid with an n×n corner removed, where each cell is smaller than those below and left.
* [[Project Euler/406]] - Guessing Game - Find the minimal total cost for a guessing game with three
* [[Project Euler/413|Problem 413]] - One-child Numbers - Count d-digit numbers where exactly one substring is divisible by d.
* possible answers per question.
* [[Project Euler/414|Problem 414]] - Kaprekar Constant - Sum of constants reached by the Kaprekar routine across different bases and digit lengths.
* [[Project Euler/407]] - Idempotents - Sum of the largest a ≤ n such that a² ≡ a (mod n) for all n up
* [[Project Euler/415|Problem 415]] - Titanic Sets - Count titanic sets of lattice points, where some line passes through exactly two points of the set.
* to 10⁷.
* [[Project Euler/416|Problem 416]] - A Frog's Trip - Count the number of ways a frog can travel from the leftmost to the rightmost square and back, jumping 1-3 squares.
* [[Project Euler/408]] - Admissible Paths Through a Grid - Count admissible north/east paths avoiding
* [[Project Euler/417|Problem 417]] - Reciprocal Cycles II - Sum of the lengths of reciprocal cycles for unit fractions 1/d with denominators d up to 10⁸.
* points where x, y, and x+y are all perfect squares.
* [[Project Euler/418|Problem 418]] - Factorisation Triples - Count integer triples (a,b,c) with a·b·c=n and a≤b≤c for n up to a large value.
* [[Project Euler/409]] - Nim Extreme - Count winning nim positions with n non-empty piles of distinct
* [[Project Euler/419|Problem 419]] - Look and Say Sequence - Count the occurrences of digits 1, 2, and 3 in the 10¹²th term of the look-and-say sequence.
* sizes less than 2ⁿ.
* [[Project Euler/420|Problem 420]] - 2×2 Positive Integer Matrix - Count 2×2 positive integer matrices with trace < N that can be expressed as a square in two different ways.
* [[Project Euler/410]] - Circle and Tangent Line - Find the sum of all radii r for which a circle and
* a tangent line satisfy certain integer conditions.
* [[Project Euler/411]] - Uphill Paths - Find the maximum number of stations on an uphill path where
* stations are defined by powers of 2 modulo n.
* [[Project Euler/412]] - Gnomon Numbering - Count valid numberings of an m×m grid with an n×n corner
* removed, where each cell is smaller than those below and left.
* [[Project Euler/413]] - One-child Numbers - Count d-digit numbers where exactly one substring is
* divisible by d.
* [[Project Euler/414]] - Kaprekar Constant - Sum of constants reached by the Kaprekar routine across
* different bases and digit lengths.
* [[Project Euler/415]] - Titanic Sets - Count titanic sets of lattice points, where some line passes
* through exactly two points of the set.
* [[Project Euler/416]] - A Frog's Trip - Count the number of ways a frog can travel from the leftmost
* to the rightmost square and back, jumping 1-3 squares.
* [[Project Euler/417]] - Reciprocal Cycles II - Sum of the lengths of reciprocal cycles for unit
* fractions 1/d with denominators d up to 10⁸.
* [[Project Euler/418]] - Factorisation Triples - Count integer triples (a,b,c) with a·b·c=n and a≤b≤c
* for n up to a large value.
* [[Project Euler/419]] - Look and Say Sequence - Count the occurrences of digits 1, 2, and 3 in the
* 10¹²th term of the look-and-say sequence.
* [[Project Euler/420]] - 2×2 Positive Integer Matrix - Count 2×2 positive integer matrices with trace
* < N that can be expressed as a square in two different ways.


==Grid 5: Problems 500-599==
==Grid 5: Problems 500-599==


* [[Project Euler/500|Problem 500]] - Smallest Number with 2n Factors - Finding the smallest number with 2^n divisors
* [[Project Euler/500]] - Smallest Number with 2n Factors - Finding the smallest number with 2^n
* [[Project Euler/501|Problem 501]] - Eight Divisors - Finding numbers with exactly 8 divisors, less than 1 trillion
* divisors
* [[Project Euler/502|Problem 502]] - Castles - finding the maximum number of castles that can be formed on extremely large grids
* [[Project Euler/501]] - Eight Divisors - Finding numbers with exactly 8 divisors, less than 1
* trillion
* [[Project Euler/502]] - Castles - finding the maximum number of castles that can be formed on
* extremely large grids


==Grid 6: Problems 600-699==
==Grid 6: Problems 600-699==
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==Grid 9: Problems 900-999==
==Grid 9: Problems 900-999==


* [[Project Euler/900|Problem 900]] - DistribuNim II
* [[Project Euler/900]] - DistribuNim II
* [[Project Euler/901|Problem 901]] - Well Drilling
* [[Project Euler/901]] - Well Drilling
* [[Project Euler/902|Problem 902]] - Permutation Powers
* [[Project Euler/902]] - Permutation Powers
* [[Project Euler/903|Problem 903]] - Total Permutation Powers
* [[Project Euler/903]] - Total Permutation Powers
* [[Project Euler/904|Problem 904]] - Pythagorean Angle
* [[Project Euler/904]] - Pythagorean Angle
* [[Project Euler/905|Problem 905]] - Now I Know
* [[Project Euler/905]] - Now I Know
* [[Project Euler/906|Problem 906]] - A Collective Decision
* [[Project Euler/906]] - A Collective Decision
* [[Project Euler/907|Problem 907]] - Stacking Cups
* [[Project Euler/907]] - Stacking Cups
* [[Project Euler/908|Problem 908]] - Clock Sequence II
* [[Project Euler/908]] - Clock Sequence II
* [[Project Euler/909|Problem 909]] - L-expressions I
* [[Project Euler/909]] - L-expressions I
* [[Project Euler/910|Problem 910]] - L-expressions II
* [[Project Euler/910]] - L-expressions II
* [[Project Euler/911|Problem 911]] - Khinchin Exceptions
* [[Project Euler/911]] - Khinchin Exceptions
* [[Project Euler/912|Problem 912]] - Where are the Odds?
* [[Project Euler/912]] - Where are the Odds?
* [[Project Euler/913|Problem 913]] - Row-major vs Column-major
* [[Project Euler/913]] - Row-major vs Column-major
* [[Project Euler/914|Problem 914]] - Triangles inside Circles
* [[Project Euler/914]] - Triangles inside Circles
* [[Project Euler/915|Problem 915]] - Giant GCDs
* [[Project Euler/915]] - Giant GCDs
* [[Project Euler/916|Problem 916]] - Restricted Permutations
* [[Project Euler/916]] - Restricted Permutations
* [[Project Euler/917|Problem 917]] - Minimal Path Using Additive Cost
* [[Project Euler/917]] - Minimal Path Using Additive Cost
* [[Project Euler/918|Problem 918]] - Recursive Sequence Summation
* [[Project Euler/918]] - Recursive Sequence Summation
* [[Project Euler/919|Problem 919]] - Fortunate Triangles
* [[Project Euler/919]] - Fortunate Triangles
* [[Project Euler/920|Problem 920]] - Tau Numbers
* [[Project Euler/920]] - Tau Numbers
* [[Project Euler/932|Problem 932]]
* [[Project Euler/932]]


{{ProjectEulerFlag}}
{{ProjectEulerFlag}}

Revision as of 22:03, 24 June 2026

Grid 0: Problems 1-99

  • Project Euler/1- Multiples of 3 and 5 - printing out all multiples of 3 and 5.
  • Project Euler/2 - Even Fibonacci - summing the Fibonacci numbers that are even and less than 4
  • million
  • Project Euler/3 - Largest Prime Factor - Largest prime factor of a given 12-digit number
  • Project Euler/4 - Largest Palindrome Product - Largest palindrome product (extracting substrings
  • and sorting)
  • Project Euler/5 - LCM - Least common multiple of all the integers from 1 to 20
  • Project Euler/6 - SoS - Sum of squares and squares of sums
  • Project Euler/7 - Ten Thousand Primes - Find the 10,001st prime.
  • Project Euler/8 - Adjacent Digits - Largest product formed by 13 adjacent digits.
  • Project Euler/9 - Pythagorean Triplet Sum - Finding a Pythagorean triplet with a specified sum.
  • Project Euler/10 Sum of Primes - Sum of all primes below 2 million.
  • Project Euler/11 - Greatest Product in Grid - Finding the greatest product of 4 numbers on a grid.
  • Project Euler/12 - Highly Factorable Triangular Numbers - Finding highly factorable triangular
  • numbers
  • Project Euler/13 - Sum of Big Numbers - Work out the first 10 digits of a sum of 100 50-digit
  • numbers
  • Project Euler/14 - Longest Collatz Sequence - Finding the longest Collatz sequence for starting
  • integers under 1 million
  • Project Euler/15 - Lattice Paths - Finding the number of variations on a route through a lattice.
  • Project Euler/16 - Summing the Digits - summing up the digits of a large power of 2, 2**1000
  • Project Euler/17 - Number Spelling - spelling out all the numbers from one to a thousand
  • Project Euler/18 - Shortest Path through a Triangle - find the path through a triangle of numbers
  • that leads to the smallest sum
  • Project Euler/19 - Counting Sundays
  • Project Euler/20 - Factorial Digit Sum - Sum of the digits in the number 100!
  • Project Euler/21 - Amicable Numbers - Sum of all amicable numbers under 10000
  • Project Euler/22 - Names Scores - Sort 5000+ names alphabetically and compute name scores
  • Project Euler/23 - Non-Abundant Sums - Sum of all positive integers not expressible as the sum of
  • two abundant numbers
  • Project Euler/24 - Lexicographic Permutations - Find the millionth lexicographic permutation of
  • the digits 0-9
  • Project Euler/25 - 1000-digit Fibonacci Number - Index of the first term in the Fibonacci sequence
  • to contain 1000 digits
  • Project Euler/26 - Reciprocal Cycles - Find d<1000 for which 1/d contains the longest recurring
  • cycle
  • Project Euler/31 - Polya - Change for a Dollar
  • Project Euler/32 - Pandigital Products (A X B = C covering all 9 digits)
  • Project Euler/33 - Digit Cancelling Fractions - Find the product of the four non-trivial curious
  • fractions where cancelling a common digit gives the correct simplified value.
  • Project Euler/34 - Digit Factorials - Find the sum of all numbers equal to the sum of the
  • factorial of their digits.
  • Project Euler/35 - Circular Primes - Count how many circular primes are there below one million.
  • Project Euler/36 - Double-base Palindromes - Find the sum of all numbers below one million that
  • are palindromic in both base 10 and base 2.
  • Project Euler/37 - Truncatable Primes - Find the sum of the only eleven primes that are
  • truncatable from left to right and right to left.
  • Project Euler/38 - Pandigital Multiples - Find the largest 1-to-9 pandigital number that can be
  • formed as the concatenated product of an integer with (1,2,...,n).
  • Project Euler/39 - Integer Right Triangles - Find the perimeter p ≤ 1000 for which the number of
  • integer-sided right triangles is maximised.
  • Project Euler/40 - Champernowne's Constant - Find the product of digits at specific positions in
  • the fractional part of Champernowne's constant.
  • Project Euler/41 - Pandigital Prime - Find the largest n-digit pandigital prime that exists.
  • Project Euler/42 - Coded Triangle Numbers - Count how many words in a given list are triangle
  • words (where word value equals a triangle number).
  • Project Euler/43 - Sub-string Divisibility - Find the sum of all pandigital numbers with an
  • unusual substring divisibility property.
  • Project Euler/44 - Pentagon Numbers - Find the pair of pentagonal numbers whose sum and difference
  • are pentagonal, minimising their difference.
  • Project Euler/45 - Triangular, Pentagonal, and Hexagonal - Find the next triangle number that is
  • also pentagonal and hexagonal after 40755.
  • Project Euler/46 - Goldbach's Other Conjecture - Find the smallest odd composite that cannot be
  • written as the sum of a prime and twice a square.
  • Project Euler/47 - Distinct Primes Factors - Find the first four consecutive integers to have four
  • distinct prime factors each.
  • Project Euler/48 - Self Powers - Find the last ten digits of the sum 1^1 + 2^2 + ... + 1000^1000.
  • Project Euler/49 - Prime Permutations - Find the 12-digit number formed by concatenating three
  • 4-digit primes that are permutations and form an arithmetic sequence.
  • Project Euler/50 - Consecutive Prime Sum - Find the prime below one million that can be written as
  • the sum of the most consecutive primes.
  • Project Euler/51- Prime Replacement - Finding the number of primes that can be formed by replacing
  • particular digits of a number
  • Project Euler/52- Permuted Multiples - Find a number whose multiples 2x, 3x, 4x, 5x ad 6x are
  • permutations of one another.
  • Project Euler/53 - Number of Combinations Over 1M - Find how many different n choose r values are
  • greater than 1 million for n between 1 and 100.
  • Project Euler/54 - Comparing poker hands to determine a winner
  • Project Euler/55 - Lychrel Numbers - Count how many Lychrel numbers (numbers that never form a
  • palindrome through the reverse-and-add process) are there below ten-thousand.
  • Project Euler/56 - Powerful Digit Sum - For natural numbers of the form a^b where a,b < 100, find
  • the maximum digital sum.
  • Project Euler/57 - Square Root Convergents - In the first one-thousand expansions of the continued
  • fraction for √2, count how many fractions have a numerator with more digits than the denominator.
  • Project Euler/58 - Counting how many composite numbers have exactly 8 factors
  • Project Euler/59 - Decrypting 3-letter secret key (Vigenere cipher)
  • Project Euler/60 - Prime pair sets - finding five primes such that any prime pair can be
  • concatenated to form a new prime
  • Project Euler/61 - Six cyclic 4-digit numbers, each of which are polygonal numbers (triangle,
  • square, pentagonal, hexagonal, heptagonal, octagonal)
  • Project Euler/62 - Cyclic permutations of cubes - find cubes that permute to other cubes.
  • Project Euler/63 - Powerful digit counts - finding n-digit numbers that are n-th powers
  • Project Euler/64 - Continued Fractions - Odd period square roots - finding the continued fraction
  • representation of an odd number, and determining if it has an odd period. First 1,000 numbers, so these sequences
  • get LONG.
  • Project Euler/65 - Convergents of e - computing the 100th convergent (rational representation of
  • continued fraction) for e and the square root of 2.
  • Project Euler/66 - Diophantine equation - a nice problem involving quadratic Diphantine equations
  • called Pell equations. These equations can be solved using the technique of continued fraction representations.
  • It is much easier to solve this problem, then 64 and 65, rather than the other way around.
  • Project Euler/67 - Maximum path sum - a retake on Project Euler/18 with a larger triangle for
  • which a brute force solution technique is impossible.
  • Project Euler/68 - Magic 5-gon Ring - Using numbers 1 to 10, find the maximum 16-digit string for
  • a "magic" 5-gon ring.
  • Project Euler/69 - Totient Maximum - Find the value of n ≤ 1,000,000 for which n/φ(n) is a
  • maximum.
  • Project Euler/70 - Totient Permutation - Find n < 10^7 for which φ(n) is a permutation of n and
  • the ratio n/φ(n) is minimized.
  • Project Euler/71 - Ordered Fractions - Find the numerator of the fraction immediately to the left
  • of 3/7 for denominators ≤ 1,000,000.
  • Project Euler/72 - Counting Fractions - Count the number of reduced proper fractions with
  • denominator ≤ 1,000,000.
  • Project Euler/73 - Counting Fractions in a Range - Count reduced proper fractions between 1/3 and
  • 1/2 with denominator ≤ 12,000.
  • Project Euler/74 - Digit Factorial Chains - Find the sum of all numbers that produce a chain of
  • exactly 60 non-repeating terms of digit factorial sums.
  • Project Euler/75 - Singular Integer Right Triangles - Find the number of perimeters ≤ 1,500,000
  • for which exactly one integer-sided right triangle exists.
  • Project Euler/76 - Counting Summations - How many ways can 100 be written as a sum of at least two
  • positive integers?
  • Project Euler/77 - Prime Summations - Find the first value that can be written as the sum of
  • primes in over 5,000 different ways.
  • Project Euler/78 - Coin Partitions - Find the least value of n for which the partition function
  • p(n) is divisible by 1,000,000.
  • Project Euler/79 - Passcode Derivation - Derive the shortest possible secret passcode from a list
  • of successful keylog entries.
  • Project Euler/80 - Square Root Digital Expansion - Sum of the first 100 decimal digits for all
  • irrational square roots up to 100.
  • Project Euler/81 - Path Sum: Two Ways - Find the minimal path sum from top left to bottom right in
  • an 80×80 matrix, moving only right and down.
  • Project Euler/82 - Path Sum: Three Ways - Find the minimal path sum from any cell in the left
  • column to any cell in the right column, moving right, up, or down.
  • Project Euler/83 - Path Sum: Four Ways - Find the minimal path sum from top left to bottom right
  • moving up, down, left, or right.
  • Project Euler/84 - Monopoly Odds - Find the three most popular squares in Monopoly when using two
  • 4-sided dice.
  • Project Euler/85 - Counting Rectangles - Find the rectangular grid area whose number of contained
  • rectangles is closest to 2 million.
  • Project Euler/86 - Cuboid Route - Find the least M such that the number of distinct cuboids with
  • an integer shortest route exceeds 1 million.
  • Project Euler/87 - Prime Power Triples - Count numbers below 50 million expressible as the sum of
  • a prime square, prime cube, and prime fourth power.
  • Project Euler/88 - Product-Sum Numbers - Find the sum of all minimal product-sum numbers for 2 ≤ k
  • ≤ 12,000.
  • Project Euler/89 - Roman Numerals - Find the number of characters saved by writing each Roman
  • numeral in its minimal form.
  • Project Euler/90 - Cube Digit Pairs - Count distinct arrangements of digits on two cubes that can
  • display all square numbers from 01 to 99.
  • Project Euler/91 - Right Triangles with Integer Coordinates - Count right triangles with vertices
  • on integer grid points in a 50×50 grid.
  • Project Euler/92 - Square Digit Chains - Count numbers below 10 million whose square digit chain
  • arrives at 89.
  • Project Euler/93 - Arithmetic Expressions - Find the longest set of consecutive integers
  • obtainable using four distinct digits and arithmetic operators.
  • Project Euler/94 - Almost Equilateral Triangles - Sum of perimeters of almost equilateral integer
  • triangles with integral area and perimeter ≤ 1 billion.
  • Project Euler/95 - Amicable Chains - Find the smallest member of the longest amicable chain with
  • no element exceeding 1 million.
  • Project Euler/96 - Su Doku - Solve 50 Sudoku puzzles and sum the 3-digit numbers found in the
  • top-left corner of each solution.
  • Project Euler/97 - Large Non-Mersenne Prime - Find the last ten digits of the non-Mersenne prime
  • 28433×2^7830457+1.
  • Project Euler/98 - Anagramic Squares - Find the largest square number formed by anagramic pairs of
  • dictionary words.
  • Project Euler/99 - Largest Exponential - Determine which line number gives the numerically largest
  • value from a list of base/exponent pairs.

Grid 1: Problems 100-199

  • Project Euler/100 - Combinations of Red and Blue Discs - find arrangements of blue and red discs
  • that lead to a probability of exactly 50% that a blue disc is removed, two times in a row.
  • Project Euler/101 - Bad Optimal Polynomials - Lagrangian polynomial interpolation for a sequence
  • of numbers, interpolation of an optimal N-1 polynomial given N points of data.
  • Project Euler/102 - Triangles Containing Origin - given 3 endpoints, determine if a triangle
  • contains the origin.
  • Project Euler/103 - Special Subset Sums: Optimum - finding the optimum special sum set with n=7.
  • Project Euler/104 - Pandigital Fibonacci Ends - finding Fibonacci numbers with pandigital
  • beginnings and endings.
  • Project Euler/105 - Special Subset Sums: Testing - testing sets for the special sum property.
  • Project Euler/106 - Special Subset Sums: Meta-testing - counting subset pairs that need to be
  • tested.
  • Project Euler/107 - Minimal Network - finding the minimal network connecting all vertices
  • (minimum spanning tree).
  • Project Euler/108 - Diophantine Reciprocals I - solving 1/x + 1/y = 1/n for distinct solutions.
  • Project Euler/109 - Darts - counting the number of distinct ways to check out in darts with a
  • score less than 100.
  • Project Euler/110 - Diophantine Reciprocals II - finding the smallest n with over 4 million
  • solutions to 1/x + 1/y = 1/n.
  • Project Euler/111 - Primes with Runs - finding primes with maximum runs of repeated digits.
  • Project Euler/112 - Bouncy Numbers - counting numbers whose digits are neither increasing nor
  • decreasing.
  • Project Euler/113 - Non-bouncy Numbers - counting numbers below a googol that are not bouncy.
  • Project Euler/114 - Counting Block Combinations I - counting ways to fill a row with red and grey
  • blocks.
  • Project Euler/115 - Counting Block Combinations II - finding the minimum row length for over 1
  • million fill combinations.
  • Project Euler/116 - Red, Green or Blue Tiles - counting ways to replace tiles with colored
  • blocks.
  • Project Euler/117 - Red, Green, and Blue Tiles - counting ways to place colored tiles of various
  • lengths.
  • Project Euler/118 - Pandigital Prime Sets - partitioning the digits 1-9 into sets of prime
  • numbers.
  • Project Euler/119 - Digit Power Sum - finding numbers equal to the sum of their digits raised to
  • some power.
  • Project Euler/120 - Square Remainders - sum of maximum remainders when (a−1)^n + (a+1)^n is
  • divided by a^2.
  • Project Euler/121 - Disc Game Prize Fund - finding max prize fund for a disc game with changing
  • probabilities.
  • Project Euler/122 - Efficient Exponentiation - computing n^15 using minimal multiplications
  • (addition chains).
  • Project Euler/123 - Prime Square Remainders - finding the prime where the maximum remainder
  • exceeds 10^10.
  • Project Euler/124 - Ordered Radicals - finding the k-th element when numbers are sorted by their
  • radical (product of prime factors).
  • Project Euler/125 - Palindromic Sums - sums of consecutive squares that are palindromic numbers.
  • Project Euler/126 - Cuboid Layers - counting the number of cubes needed to cover visible faces of
  • cuboids in successive layers.
  • Project Euler/127 - abc-hits - counting triples where rad(abc) < c and a and b are coprime.
  • Project Euler/128 - Hexagonal Tile Differences - finding tiles in a hexagonal spiral where all
  • neighbors have prime differences.
  • Project Euler/129 - Repunit Divisibility - finding the least n such that a repunit R(n) is
  • divisible by a given number.
  • Project Euler/130 - Composites with Prime Repunit Property - composite numbers where n divides
  • the repunit R(n−1).
  • Project Euler/131 - Prime Cube Partnership - primes p for which n^3 + n^2·p is a perfect cube.
  • Project Euler/132 - Large Repunit Factors - sum of the first forty prime factors of R(10^9).
  • Project Euler/133 - Repunit Nonfactors - primes that will never divide any repunit R(10^n).
  • Project Euler/134 - Prime Pair Connection - connecting consecutive primes p1, p2 to form a number
  • divisible by p2.
  • Project Euler/135 - Same Differences - solving x^2 − y^2 − z^2 = n where x, y, z form an
  • arithmetic progression.
  • Project Euler/136 - Singleton Difference - finding n with exactly one solution to x^2 − y^2 − z^2
  • = n.
  • Project Euler/137 - Fibonacci Golden Nuggets - Fibonacci numbers appearing as solutions to a
  • Pell-type Diophantine equation.
  • Project Euler/138 - Special Isosceles Triangles - isosceles triangles with integer height and
  • half-base differing by 1.
  • Project Euler/139 - Pythagorean Tiles - Pythagorean triangles that allow tiling of a square of
  • side equal to the hypotenuse.
  • Project Euler/140 - Modified Fibonacci Golden Nuggets - golden nuggets from a modified Fibonacci
  • sequence.
  • Project Euler/141 - Square Progressive Numbers - perfect squares that are also progressive
  • (geometric progression of digits).
  • Project Euler/142 - Perfect Square Collection - finding x+y+z where x>y>z>0, all pairwise
  • sums/differences are squares.
  • Project Euler/143 - Torricelli Triangles - triangles whose Torricelli point has integer distances
  • to the vertices.
  • Project Euler/144 - Laser Beam Reflections - reflecting a laser beam inside an elliptical mirror
  • until it exits.
  • Project Euler/145 - Reversible Numbers - counting numbers n below 1 billion where n + reverse(n)
  • has all odd digits.
  • Project Euler/146 - Investigating a Prime Pattern - finding n where n^2+1, n^2+3, n^2+7, n^2+9,
  • n^2+13, n^2+27 are consecutive primes.
  • Project Euler/147 - Rectangles in Cross-hatched Grids - counting all rectangles in a
  • cross-hatched rectangular grid.
  • Project Euler/148 - Exploring Pascal's Triangle - counting entries in the first billion rows of
  • Pascal's triangle not divisible by 7.
  • Project Euler/149 - Maximum-sum Subsequence - finding the maximum sum of adjacent subsequences in
  • a generated 2000×2000 array.
  • Project Euler/150 - Sub-triangle Sums - finding the minimum-sum sub-triangle in a triangular
  • array of 1000 rows.
  • Project Euler/151 - Paper Sheets of Standard Sizes - expected number of times (excluding first
  • and last batch) that the supervisor finds a single sheet of paper in the envelope, when randomly drawing and
  • cutting A1→A5 paper sheets across 16 weekly batches.
  • Project Euler/152 - Sums of Square Reciprocals - count the number of ways to write 1/2 as a sum
  • of reciprocals of squares using distinct integers between 2 and 80 inclusive.
  • Project Euler/153 - Investigating Gaussian Integers - sum of all Gaussian integer divisors (with
  • positive real part) for all rational integers n up to 10^8.
  • Project Euler/154 - Exploring Pascal's Pyramid - count how many coefficients in the trinomial
  • expansion (x + y + z)^200000 are multiples of 10^12.
  • Project Euler/155 - Counting Capacitor Circuits - number of distinct total capacitance values
  • D(n) obtainable using up to n=18 equal-valued capacitors in series and parallel combinations.
  • Project Euler/156 - Counting Digits - sum over digits d=1..9 of the sum of all solutions n where
  • the total count of digit d written from 0 to n equals n (i.e., f(n,d)=n).
  • Project Euler/157 - Base-10 Diophantine Reciprocal - count the number of positive integer
  • solutions to 1/a + 1/b = p/10^n with a ≤ b, for 1 ≤ n ≤ 9.
  • Project Euler/158 - Strings of various lengths, with exactly one character lexicographically out
  • of sorts
  • Project Euler/159
  • Project Euler/160 - Factorial Trailing Digits - find the last five non-zero digits of
  • 1,000,000,000,000!
  • Project Euler/161 - Triominoes - count the number of ways a 9×12 grid can be tiled with
  • triominoes.
  • Project Euler/162 - Hexadecimal Numbers - count hex numbers with ≤16 digits containing 0, 1, and
  • A at least once.
  • Project Euler/163 - Cross-hatched Triangles - count triangles in a size 36 cross-hatched
  • equilateral triangle.
  • Project Euler/164 - Three Consecutive Digital Sum Limit - count 20-digit numbers where no three
  • consecutive digits sum to more than 9.
  • Project Euler/165 - Intersections - count distinct true intersection points among 5000 line
  • segments.
  • Project Euler/166 - Criss Cross - count 4×4 digit grids where each row, column, and both
  • diagonals share the same sum.
  • Project Euler/167 - Investigating Ulam Sequences - sum of U(2,2n+1)_k for n=2..10, where k=10^11.
  • Project Euler/168 - Number Rotations - find the last 5 digits of the sum of all n (10<n<10^100)
  • that divide their own right-rotation.
  • Project Euler/169 - Sums of Powers of Two - count ways to express 10^25 as a sum of powers of 2
  • using each power at most twice.
  • Project Euler/170
  • Project Euler/171
  • Project Euler/172 - Few Repeated Digits - how many 18 digit numbers have no digit occurring more
  • than 3 times in n?
  • Project Euler/173
  • Project Euler/174
  • Project Euler/175
  • Project Euler/176
  • Project Euler/177
  • Project Euler/178
  • Project Euler/179

Grid 2: Problems 200-299

  • Project Euler/254 - Maximum Source of Sums of Digits of Sums of Digits of Sums of Factorial Digit
  • Sums
  • Project Euler/255 - Rounded Square Roots - computing rounded-square-roots using an iterative
  • integer method (Heron's method adapted to integer arithmetic).
  • Project Euler/256 - Tatami-Free Rooms - counting even-sized rectangular rooms that cannot be
  • covered by 1×2 tatami mats without a forbidden cross pattern.
  • Project Euler/257 - Angular Bisectors - integer-sided triangles whose angular bisector segments
  • are also integers.
  • Project Euler/258 - A Lagged Fibonacci Sequence - finding values from a lagged Fibonacci
  • generator defined by a cubic formula.
  • Project Euler/259 - Reachable Numbers - numbers expressible as arithmetic expressions using
  • digits 1 through 9 in order, each exactly once.
  • Project Euler/260 - Stone Game - a three-pile Nim-like game; counting winning configurations for
  • the first player.
  • Project Euler/261 - Pivotal Square Sums - finding square-pivot integers where a sum of
  • consecutive squares equals a perfect square.
  • Project Euler/262 - Mountain Range - finding the shortest continuous path between two points
  • across a mountainous terrain.
  • Project Euler/263 - An Engineers' Dream Come True - finding numbers with special properties
  • relating consecutive primes and practical numbers.
  • Project Euler/264 - Triangle Centres - integer-coordinate triangles whose centroid and
  • orthocenter are also on integer coordinates.
  • Project Euler/265 - Binary Circles - placing 2^N binary digits in a circle such that all N-digit
  • clockwise subsequences are distinct.
  • Project Euler/266 - Pseudo Square Root - finding the product of pseudo square roots (largest
  • divisor ≤ √n) of primes below 190.
  • Project Euler/267 - Billionaire - maximizing the chance of reaching £1 billion through optimal
  • betting on 1000 fair coin tosses.
  • Project Euler/268 - Counting Numbers with at Least Four Distinct Prime Factors Less than 100.
  • Project Euler/269 - Polynomials with at Least One Integer Root - polynomials whose coefficients
  • are the digits of n in base 10.
  • Project Euler/270 - Cutting Squares - counting ways to cut an N×N square into pieces with integer
  • side lengths.
  • Project Euler/271 - Modular Cubes, Part 1 - summing x (1 < x < n) for which x³ ≡ 1 (mod n), for a
  • specific n.
  • Project Euler/272 - Modular Cubes, Part 2 - extending the modular cubes problem to a larger
  • modulus.
  • Project Euler/273 - Sum of Squares - summing values of a in a² + b² = N for squarefree N with all
  • prime factors of the form 4k+1.
  • Project Euler/274 - Divisibility Multipliers - finding positive multipliers m < p that preserve
  • divisibility by p for primes p coprime to 10.
  • Project Euler/275 - Balanced Sculptures - counting polyomino-based sculptures of order n whose
  • combined centre of mass has x-coordinate zero.
  • Project Euler/276 - Primitive Triangles - integer-sided triangles with integer area where the
  • greatest common divisor of sides is 1.
  • Project Euler/277 - A Modified Collatz Sequence - a Collatz-like sequence with three possible
  • steps: divide by 3, or apply floor-based rules.
  • Project Euler/278 - Linear Combinations of Semiprimes - counting numbers expressible as linear
  • combinations of pairs of semiprimes.
  • Project Euler/279 - Triangles with Integral Sides and an Integral Angle - triangles where at
  • least one angle measured in degrees is an integer.
  • Project Euler/280 - Ant and Seeds - an ant walking on a 5×5 grid carrying seeds; finding the
  • expected number of steps to complete the task.
  • Project Euler/281 - Pizza Toppings - counting distinct ways to place m toppings on m·n pizza
  • slices, considering rotational symmetry.
  • Project Euler/282 - The Ackermann Function - computing sums of values of the Ackermann function
  • modulo large numbers.
  • Project Euler/283 - Integer-sided Triangles for Which the Area/Perimeter Ratio is Integral.
  • Project Euler/284 - Steady Squares - numbers in base 14 whose square ends with the number itself.
  • Project Euler/285 - Pythagorean Odds - expected value in a game involving random points and the
  • probability of a Pythagorean distance.
  • Project Euler/286 - Scoring Probabilities - basketball shooting probability as a function of
  • distance; finding the constant q that yields a 50% scoring chance.
  • Project Euler/287 - Quadtree Encoding - bit-length of the quadtree encoding of a 2^N × 2^N
  • disk-shaped black-and-white image.
  • Project Euler/288 - An Enormous Factorial - computing N(p,q) modulo large powers of p for a
  • specifically defined sequence.
  • Project Euler/289 - Eulerian Cycles - counting non-crossing Eulerian cycles on a grid formed by
  • arranging circles.
  • Project Euler/290 - Digital Signature - sum of digits of all numbers expressible as a particular
  • form up to 10^18.
  • Project Euler/291 - Panaitopol Primes - primes expressible as (x⁴ − y⁴) / (x³ + y³) for positive
  • integers x and y.
  • Project Euler/292 - Pythagorean Polygons - convex polygons with integer perimeter formed from at
  • least three edge-disjoint right triangles.
  • Project Euler/293 - Pseudo-Fortunate Numbers - for admissible numbers N, the smallest integer m >
  • 1 such that N + m is prime.
  • Project Euler/294 - Sum of Digits — Experience #23 - summing digits of multiples of 23.
  • Project Euler/295 - Lenticular Holes - convex areas enclosed by two circles whose centers and
  • intersection points are on lattice points.
  • Project Euler/296 - Angular Bisector and Tangent - integer-sided triangles where the angular
  • bisector is tangent to an inscribed circle.
  • Project Euler/297 - Zeckendorf Representation - sum of the number of terms in Zeckendorf
  • representations of all numbers below 10^17.
  • Project Euler/298 - Selective Amnesia - a memory game with random numbers; expected absolute
  • difference in scores after 50 turns.
  • Project Euler/299 - Three Similar Triangles - integer-sided triangles containing three similar
  • right triangles.

Grid 3: Problems 300-399

  • Project Euler/301 - Nim - Counting losing positions in three-heap normal-play Nim for n ≤ 2^30.
  • Project Euler/302 - Strong Achilles Numbers - Count how many strong Achilles numbers are below
  • 10^18.
  • Project Euler/303 - Multiples with Small Digits - Sum of least positive multiples using only
  • digits ≤ 2.
  • Project Euler/304 - Primonacci - Sum of Fibonacci numbers at prime indices starting after 10^14.
  • Project Euler/305 - Reflexive Position - Starting positions of the n-th occurrence of n in the
  • concatenated infinite integer string.
  • Project Euler/306 - Paper-strip Game - Combinatorial game: pick two contiguous white squares and
  • paint them black.
  • Project Euler/307 - Chip Defects - Probability of at least one chip having 3+ defects when
  • distributing defects randomly.
  • Project Euler/308 - An Amazing Prime-generating Automaton - Find the 10^15th prime generated by
  • Conway's Fractran program.
  • Project Euler/309 - Integer Ladders - Count integer solutions to the classic crossing ladders
  • problem.
  • Project Euler/310 - Nim Square - Nim variant where players may only remove a square number of
  • stones.
  • Project Euler/311 - Biclinic Integral Quadrilaterals - Count biclinic integral quadrilaterals
  • with bounded sum of squared sides.
  • Project Euler/312 - Cyclic Paths on Sierpiński Graphs - Counting Hamiltonian cycles on Sierpiński
  • graphs.
  • Project Euler/313 - Sliding Game - Minimum moves to slide a counter across an m×n grid.
  • Project Euler/314 - The Mouse on the Moon - Maximizing enclosed-area/wall-length ratio on a grid
  • of posts.
  • Project Euler/315 - Digital Root Clocks - 7-segment display power consumption for digital root
  • clocks.
  • Project Euler/316 - Numbers in Decimal Expansions - Expected position of a number in a random
  • infinite decimal sequence.
  • Project Euler/317 - Firecracker - Volume of the region through which firecracker fragments
  • travel.
  • Project Euler/318 - 2011 Nines - Count consecutive nines in fractional parts of powers of
  • sqrt(p)+sqrt(q).
  • Project Euler/319 - Bounded Sequences - Count sequences of length n satisfying x_i^j < (x_j+1)^i.
  • Project Euler/320 - Factorials Divisible by a Huge Integer - Smallest n such that n! is divisible
  • by (i!)^1234567890.
  • Project Euler/321 - Swapping Counters - Minimum moves to swap n red and n blue counters on a row.
  • Project Euler/322 - Binomial Coefficients Divisible by 10 - Count binomial coefficients divisible
  • by 10 in a given range.
  • Project Euler/323 - Bitwise-OR Operations on Random Integers - Expected number of random 32-bit
  • integers to fill all bits via bitwise-OR.
  • Project Euler/324 - Building a Tower - Number of ways to fill a 3×3×n tower with 2×1×1 blocks.
  • Project Euler/325 - Stone Game II - Two-pile game where removal must be a multiple of the smaller
  • pile.
  • Project Euler/326 - Modulo Summations - Sequence defined by recursive modular sums, count
  • zero-sum subarrays.
  • Project Euler/327 - Rooms of Doom - Minimum security cards to traverse rooms with limited
  • carrying capacity.
  • Project Euler/328 - Lowest-cost Search - Optimal strategy to find a hidden number where each
  • guess costs the value guessed.
  • Project Euler/329 - Prime Frog - Probability of a frog's croaking sequence when jumping on prime
  • and non-prime squares.
  • Project Euler/330 - Euler's Number - Infinite sequence defined via Euler's number e, find
  • A(10^9)+B(10^9).
  • Project Euler/331 - Cross Flips - Minimal turns to flip all disks to white on an N×N board with
  • cross-flipping moves.
  • Project Euler/332 - Spherical Triangles - Area of the smallest spherical triangle with
  • integer-coordinate vertices.
  • Project Euler/333 - Special Partitions - Count partitions of integers into terms of the form 2^i
  • × 3^j.
  • Project Euler/334 - Spilling the Beans - Game where removing two beans from a bowl puts one bean
  • in each adjacent bowl.

Grid 4: Problems 400-499

  • Project Euler/400 - Fibonacci Tree Game - A take-away game on a Fibonacci tree; find the number
  • of winning moves for the first player on T(10000).
  • Project Euler/401 - Sum of Squares of Divisors - Find the sum of σ₂(i) for i=1 to n, where σ₂ is
  • the sum of squares of divisors.
  • Project Euler/402 - Integer-valued Polynomials - Sum of M(a,b,c) over all a,b,c ≤ N, where M is
  • the maximum m such that n⁴+an³+bn²+cn is always a multiple of m.
  • Project Euler/403 - Lattice Points Enclosed by Parabola and Line - Count lattice points in the
  • region bounded by y = x²/k and y = ax + b.
  • Project Euler/404 - Crisscross Ellipses - Count lattice points inside the intersection of two
  • ellipses x²+4y²=4a² and its rotated copy.
  • Project Euler/405 - A Rectangular Tiling - Count the number of ways to tile a 2×n rectangle with
  • 1×1 and 1×2 tiles.
  • Project Euler/406 - Guessing Game - Find the minimal total cost for a guessing game with three
  • possible answers per question.
  • Project Euler/407 - Idempotents - Sum of the largest a ≤ n such that a² ≡ a (mod n) for all n up
  • to 10⁷.
  • Project Euler/408 - Admissible Paths Through a Grid - Count admissible north/east paths avoiding
  • points where x, y, and x+y are all perfect squares.
  • Project Euler/409 - Nim Extreme - Count winning nim positions with n non-empty piles of distinct
  • sizes less than 2ⁿ.
  • Project Euler/410 - Circle and Tangent Line - Find the sum of all radii r for which a circle and
  • a tangent line satisfy certain integer conditions.
  • Project Euler/411 - Uphill Paths - Find the maximum number of stations on an uphill path where
  • stations are defined by powers of 2 modulo n.
  • Project Euler/412 - Gnomon Numbering - Count valid numberings of an m×m grid with an n×n corner
  • removed, where each cell is smaller than those below and left.
  • Project Euler/413 - One-child Numbers - Count d-digit numbers where exactly one substring is
  • divisible by d.
  • Project Euler/414 - Kaprekar Constant - Sum of constants reached by the Kaprekar routine across
  • different bases and digit lengths.
  • Project Euler/415 - Titanic Sets - Count titanic sets of lattice points, where some line passes
  • through exactly two points of the set.
  • Project Euler/416 - A Frog's Trip - Count the number of ways a frog can travel from the leftmost
  • to the rightmost square and back, jumping 1-3 squares.
  • Project Euler/417 - Reciprocal Cycles II - Sum of the lengths of reciprocal cycles for unit
  • fractions 1/d with denominators d up to 10⁸.
  • Project Euler/418 - Factorisation Triples - Count integer triples (a,b,c) with a·b·c=n and a≤b≤c
  • for n up to a large value.
  • Project Euler/419 - Look and Say Sequence - Count the occurrences of digits 1, 2, and 3 in the
  • 10¹²th term of the look-and-say sequence.
  • Project Euler/420 - 2×2 Positive Integer Matrix - Count 2×2 positive integer matrices with trace
  • < N that can be expressed as a square in two different ways.

Grid 5: Problems 500-599

  • Project Euler/500 - Smallest Number with 2n Factors - Finding the smallest number with 2^n
  • divisors
  • Project Euler/501 - Eight Divisors - Finding numbers with exactly 8 divisors, less than 1
  • trillion
  • Project Euler/502 - Castles - finding the maximum number of castles that can be formed on
  • extremely large grids

Grid 6: Problems 600-699

Grid 7: Problems 700-799

Grid 8: Problems 800-899

Grid 9: Problems 900-999



Euler.jpg
Project Euler
project euler notes

Project Euler

Grid 0: Problems 1-99

Problem 1  · Problem 2  · Problem 3  · Problem 4  · Problem 5  · Problem 6  · Problem 7  · Problem 8  · Problem 9  · Problem 10

Problem 11  · Problem 12  · Problem 13  · Problem 14  · Problem 15  · Problem 16  · Problem 17  · Problem 18  · Problem 19  · Problem 20

Problem 27  · Problem 28  · Problem 29  · Problem 30

Problem 31  · Problem 32  · Problem 33  · Problem 40

Problem 41  · Problem 42  · Problem 43  · Problem 44  · Problem 45  · Problem 46  · Problem 47  · Problem 48  · Problem 49  · Problem 50

Problem 51  · Problem 52  · Problem 53  · Problem 54  · Problem 55  · Problem 56  · Problem 57  · Problem 58  · Problem 59  · Problem 60

Problem 61  · Problem 62  · Problem 63  · Problem 64  · Problem 65  · Problem 66  · Problem 67  · Problem 68  · Problem 69  · Problem 70

Grid 1: Problems 100-199

...

Problem 110  · Problem 111  · Problem 112  · Problem 113  · Problem 114  · Problem 115  · Problem 116  · Problem 117  · Problem 118  · Problem 119

Problem 150  · Problem 151  · Problem 152  · Problem 153  · Problem 154  · Problem 155  · Problem 156  · Problem 157  · Problem 158  · Problem 159

...

Problem 170  · Problem 171  · Problem 172  · Problem 173  · Problem 174  · Problem 175  · Problem 176  · Problem 177  · Problem 178  · Problem 179

...


Problem 190  · Problem 191  · Problem 192  · Problem 193  · Problem 194  · Problem 195  · Problem 196  · Problem 197  · Problem 198  · Problem 199


Grid 2: Problems 200-299

Problem 227  · Problem 254


Grid 3: Problems 300-399


Problem 301  · Problem 310

Problem 312  · Problem 319

Problem 323  · Problem 328

Problem 334  · Problem 337

Problem 345  · Problem 346

Problem 355  · Problem 356

Problem 364  · Problem 367

Problem 373  · Problem 378

Problem 382  · Problem 389

Problem 391

Grid 4: Problems 400-499

Problem 400


Grid 5: Problems 500-599

Problem 500  · Problem 501  · Problem 502


Grid 9: Problems 900-999

Problem 932


* = in progress

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