Problem Description

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

0.123456789101112131415161718192021…

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.

d₁ × d₁₀ × d₁₀₀ × d₁₀₀₀ × d₁₀₀₀₀ × d₁₀₀₀₀₀ × d_1000000

Analysis

This is known as Champernowne’s constant (OEIS A033307).

We concatenate the counting numbers to a string from 1 until the length reaches at least 1 million digits, then multiply the digit at the appropriate positions together. We know from the problem description that the first two terms d₁ & d₁₀ will evaluate to 1 and, therefore, serve no consequence on the product.

Solution

Runs < 1 second in Perl.

my $s = '.';
my $n = 1;
 
$s.=$n++ while(length($s) <= 1_000_000);
 
print "Answer to PE40 = ",
substr($s,1e2,1) * substr($s,1e3,1) * substr($s,1e4,1) * substr($s,1e5,1) * substr($s,1e6,1);

Comments

• We initialize $s with a character to start the string index at 1 instead of 0.
• Mathematica: Times @@ Flatten[IntegerDigits@Range@2*^5][[10^Range@6]]