Problem 40 - Finding the nth digit of the fractional part of the irrational number.

링크

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

Solved by pencil

1~9	9 of 1-digits numbers, total 9 digits	0
10 ~ 99	90 of 2-digits numbers, total 180 digits	9
100 ~ 999	900 of 3-digits numbers, total 2700 digits	189
1000 ~ 9999	9000 of 4-digits numbers, total 36000 digits	2889
10000 ~ 99999	90000 of 5-digits numbers, total 450000 digits	38889
100000 ~ 999999	900000 of 6-digits numbers, total 5400000 digits	488889

d1 = 1

d10 = 1

d100 = 5

100 - 9 = 91th digit in 2-digits number block -> zero based index is 90

90 / 2 = 45th number in 2-digits number block = 55

90 % 2 = 0th digit in 55 = 5

d1000 = 3

1000 - 189 = 811th digit in 3-digits number block -> zero based index is 810

810 / 3 = 270th number in 3-digits number block = 370

810 % 3 = 0th digit in 370 = 3

and so on...