Project Euler/40/Solution

Main article: Project Euler/40

Solution

Here is a calculation of the product $d_{1}\times d_{10}\times d_{100}\times d_{1000}\times d_{10000}\times d_{100000}\times d_{1000000}$, where $d_{n}$ is the $n$-th digit of the Champernowne constant $C_{10}=0.123456789101112...$.

We use the established method involving cumulative digit counts ($C_{k}$) for blocks of $k$-digit numbers.

• $C_{1}=9$
• $C_{2}=189$
• $C_{3}=2889$
• $C_{4}=38889$
• $C_{5}=488889$
• $C_{6}=5888889$

Finding the Digits

• Finding $d_{1}$

The first digit ($n=1$) is the first digit of '1'.

$d_{1}=1$

• Finding $d_{10}$

$n=10$. Falls into the 2-digit block ($k=2$), since $C_{1}<10\leq C_{2}$.

Position within block: $m=n-C_{1}=10-9=1$.

Integer index ($0$-based): $num\_index=\lfloor (m-1)/k\rfloor =\lfloor (1-1)/2\rfloor =0$.

Integer $N=10^{k-1}+num\_index=10^{2-1}+0=10$.

Digit position within $N$ ($1$-based): $digit\_pos=(m-1){\pmod {k}}+1=(1-1){\pmod {2}}+1=1$. The 1st digit of 10.

$d_{10}=1$

• Finding $d_{100}$

$n=100$. Falls into the 2-digit block ($k=2$), since Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C_{1}<100\leq C_{2}} .

Position within block: $m=100-C_{1}=100-9=91$.

Integer index: $num\_index=\lfloor (91-1)/2\rfloor =\lfloor 90/2\rfloor =45$.

Integer $N=10^{2-1}+45=10+45=55$.

Digit position: $digit\_pos=(91-1){\pmod {2}}+1=90{\pmod {2}}+1=1$. The 1st digit of 55.

$d_{100}=5$

• Finding $d_{1000}$

$n=1000$. Falls into the 3-digit block ($k=3$), since $C_{2}<1000\leq C_{3}$.

Position within block: $m=1000-C_{2}=1000-189=811$.

Integer index: $num\_index=\lfloor (811-1)/3\rfloor =\lfloor 810/3\rfloor =270$.

Integer $N=10^{3-1}+270=100+270=370$.

Digit position: $digit\_pos=(811-1){\pmod {3}}+1=810{\pmod {3}}+1=1$. The 1st digit of 370.

$d_{1000}=3$

• Finding $d_{10000}$

$n=10000$. Falls into the 4-digit block ($k=4$), since $C_{3}<10000\leq C_{4}$.

Position within block: $m=10000-C_{3}=10000-2889=7111$.

Integer index: $num\_index=\lfloor (7111-1)/4\rfloor =\lfloor 7110/4\rfloor =1777$.

Integer Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N=10^{4-1}+1777=1000+1777=2777} .

Digit position: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle digit\_pos=(7111-1){\pmod {4}}+1=7110{\pmod {4}}+1=2+1=3} . The 3rd digit of 2777.

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{10000}=7}

• Finding Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{100000}}

$n=100000$. Falls into the 5-digit block (Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k=5} ), since Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C_{4}<100000\leq C_{5}} .

Position within block: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m=100000-C_{4}=100000-38889=61111} .

Integer index: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle num\_index=\lfloor (61111-1)/5\rfloor =\lfloor 61110/5\rfloor =12222} .

Integer Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle N=10^{5-1}+12222=10000+12222=22222} .

Digit position: Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle digit\_pos=(61111-1){\pmod {5}}+1=61110{\pmod {5}}+1=0+1=1} . The 1st digit of 22222.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_{100000} = 2}

• Finding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_{1000000}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=1000000} . Falls into the 6-digit block (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=6} ), since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_5 < 1000000 \le C_6} .

Position within block: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m = 1000000 - C_5 = 1000000 - 488889 = 511111} .

Integer index: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle num\_index = \lfloor (511111 - 1) / 6 \rfloor = \lfloor 511110 / 6 \rfloor = 85185} .

Integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N = 10^{6-1} + 85185 = 100000 + 85185 = 185185} .

Digit position: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle digit\_pos = (511111 - 1) \pmod 6 + 1 = 511110 \pmod 6 + 1 = 0 + 1 = 1} . The 1st digit of 185185.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_{1000000} = 1}

Calculating the Product

We need to compute the product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P = d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P = 1 \times 1 \times 5 \times 3 \times 7 \times 2 \times 1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P = 105 \times 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P = 210}

The product of the specified digits is 210.