Project Euler: Difference between revisions

Revision as of 22:11, 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/27 - Quadratic Primes - Find the quadratic formula n²+an+b producing the most
consecutive primes
Project Euler/28 - Number Spiral Diagonals
Project Euler/29 - Distinct Terms Generated by Powers
Project Euler/30 - Sum of Fifth Power of Digits

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 - Digital Root Sums of Factorisations - sum of digital roots of the individual factors of a number.
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 - Pandigital Concatenating Products - Find the largest 0 to 9 pandigital that can be formed by concatenating products, where the concatenation of the input numbers is also pandigital.
Project Euler/171 - Square Sum of the Digital Squares - Find the sum of all numbers where the sum of the squares of the digits is a perfect square.
Project Euler/172 - Few Repeated Digits - how many 18 digit numbers have no digit occurring more
than 3 times in n?
Project Euler/173 - Hollow Square Laminae - Using up to one million tiles how many different "hollow" square laminae can be formed?
Project Euler/174 - Hollow Square Laminae II - Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements.
Project Euler/175 - Fractions of Powers of Two - Fractions involving the number of different ways a number can be expressed as a sum of powers of 2.
Project Euler/176 - Shared Cathetus Triangles - Right-angled triangles that share a cathetus.
Project Euler/177 - Integer Angled Quadrilaterals.
Project Euler/178 - Step Numbers.
Project Euler/179 - Consecutive Positive Divisors - consecutive integers with the same number of positive divisors.

Project Euler/180 - Rational Zeros of Three Variables - Rational zeros of a function of three variables.
Project Euler/181 - Grouping Two Colours - Investigating in how many ways objects of two different colours can be grouped.
Project Euler/182 - RSA Encryption.
Project Euler/183 - Maximum Product of Parts.
Project Euler/184 - Triangles Containing the Origin.
Project Euler/185 - Number Mind.
Project Euler/186 - Connectedness of a Network.
Project Euler/187 - Semiprimes.
Project Euler/188 - Hyperexponentiation - The hyperexponentiation of a number.
Project Euler/189 - Tri-colouring a Triangular Grid.

Project Euler/190 - Maximising a Weighted Product.
Project Euler/191 - Prize Strings.
Project Euler/192 - Best Approximations.
Project Euler/193 - Squarefree Numbers.
Project Euler/194 - Coloured Configurations.
Project Euler/195 - 60-Degree Triangle Inscribed Circles - Inscribed circles of triangles with one angle of 60 degrees.
Project Euler/196 - Prime Triplets.
Project Euler/197 - Recursively Defined Sequence - Investigating the behaviour of a recursively defined sequence.
Project Euler/198 - Ambiguous Numbers.
Project Euler/199 - Iterative Circle Packing.

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

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

@@ Line 299: / Line 299: @@
 * [[Project Euler/158]] - Strings of various lengths, with exactly one character lexicographically out
 * of sorts
-* [[Project Euler/159]]
+* [[Project Euler/159]] - Digital Root Sums of Factorisations - sum of digital roots of the individual factors of a number.
 * [[Project Euler/160]] - Factorial Trailing Digits - find the last five non-zero digits of
 * 1,000,000,000,000!
@@ Line 323: / Line 323: @@
 * [[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/173]] - Hollow Square Laminae - Using up to one million tiles how many different "hollow" square laminae can be formed?
-* [[Project Euler/174]]
+* [[Project Euler/174]] - Hollow Square Laminae II - Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements.
-* [[Project Euler/175]]
+* [[Project Euler/175]] - Fractions of Powers of Two - Fractions involving the number of different ways a number can be expressed as a sum of powers of 2.
-* [[Project Euler/176]]
+* [[Project Euler/176]] - Shared Cathetus Triangles - Right-angled triangles that share a cathetus.
-* [[Project Euler/177]]
+* [[Project Euler/177]] - Integer Angled Quadrilaterals.
-* [[Project Euler/178]]
+* [[Project Euler/178]] - Step Numbers.
-* [[Project Euler/179]]
+* [[Project Euler/179]] - Consecutive Positive Divisors - consecutive integers with the same number of positive divisors.
-* [[Project Euler/180]]
+* [[Project Euler/180]] - Rational Zeros of Three Variables - Rational zeros of a function of three variables.
-* [[Project Euler/181]]
+* [[Project Euler/181]] - Grouping Two Colours - Investigating in how many ways objects of two different colours can be grouped.
-* [[Project Euler/182]]
+* [[Project Euler/182]] - RSA Encryption.
-* [[Project Euler/183]]
+* [[Project Euler/183]] - Maximum Product of Parts.
-* [[Project Euler/184]]
+* [[Project Euler/184]] - Triangles Containing the Origin.
-* [[Project Euler/185]]
+* [[Project Euler/185]] - Number Mind.
-* [[Project Euler/186]]
+* [[Project Euler/186]] - Connectedness of a Network.
-* [[Project Euler/187]]
+* [[Project Euler/187]] - Semiprimes.
-* [[Project Euler/188]]
+* [[Project Euler/188]] - Hyperexponentiation - The hyperexponentiation of a number.
-* [[Project Euler/189]]
+* [[Project Euler/189]] - Tri-colouring a Triangular Grid.
-* [[Project Euler/190]]
+* [[Project Euler/190]] - Maximising a Weighted Product.
-* [[Project Euler/191]]
+* [[Project Euler/191]] - Prize Strings.
-* [[Project Euler/192]]
+* [[Project Euler/192]] - Best Approximations.
-* [[Project Euler/193]]
+* [[Project Euler/193]] - Squarefree Numbers.
-* [[Project Euler/194]]
+* [[Project Euler/194]] - Coloured Configurations.
-* [[Project Euler/195]]
+* [[Project Euler/195]] - 60-Degree Triangle Inscribed Circles - Inscribed circles of triangles with one angle of 60 degrees.
-* [[Project Euler/196]]
+* [[Project Euler/196]] - Prime Triplets.
-* [[Project Euler/197]]
+* [[Project Euler/197]] - Recursively Defined Sequence - Investigating the behaviour of a recursively defined sequence.
-* [[Project Euler/198]]
+* [[Project Euler/198]] - Ambiguous Numbers.
-* [[Project Euler/199]]
+* [[Project Euler/199]] - Iterative Circle Packing.
 ==Grid 2: Problems 200-299==

Project Euler: Difference between revisions

From charlesreid1

Revision as of 22:11, 24 June 2026

Contents

Grid 0: Problems 1-99

Grid 1: Problems 100-199

Grid 2: Problems 200-299

Grid 3: Problems 300-399

Grid 4: Problems 400-499

Grid 5: Problems 500-599

Grid 6: Problems 600-699

Grid 7: Problems 700-799

Grid 8: Problems 800-899

Grid 9: Problems 900-999