Project Euler Problem 40: Champernowne’s Constant

The source code for this problem can be found here.

Problem Statement

An irrational decimal fraction is created by concatenating the positive integers:

\[0.12345678910\color{red}{1}112131415161718192021 \dots\]

It can be seen that the \(12^{th}\) digit of the fractional part is \(1\).

If \(d_n\) represents the \(n^{th}\) digit of the fractional part, find the value of the following expression.

\[d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}\]

Solution Discussion

Explicitly building this decimal representation is not clever but will work, a \(1000000\) digit number is not large in the scheme of things.

Use Python to build the fractional part as a string of decimal digits and then simply index into the relevant points to extract each \(d_i\) needed. Finally, multiply these values to compute the answer.

Solution Implementation

from functools import reduce
from operator import mul

def solve():
    """ Compute the answer to Project Euler's problem #40 """

    upper_limit = 200000  # large enough s.t. the total string length exceeds the highest index

    fractional_string = "".join(["{}".format(i + 1) for i in range(upper_limit)])  # build the string representation

    # Extract the relevant d_i values
    indices = [1, 10, 100, 1000, 10000, 100000, 1000000]
    digits = [fractional_string[i - 1] for i in indices]  # Python using 0-based indices

    # Multiply each d_i together to get the answer
    answer = reduce(mul, map(int, digits))

    return answer

expected_answer = 210

Compute the answer to Project Euler’s problem #40