How does this integer encoding work?

Question

In this code golf question, there is a python answer that encodes the lengths of all integers from 1 to 99 in english to a big number:

7886778663788677866389978897746775667552677566755267756675527886778663788677866355644553301220112001

To get the length of n, you just have to calculate 3 + (the_big_number / (10**n)) % 10. How does this work?

Answer 1

(the_big_number / (10^n)) % 10 pulls out the nth least significant digit of the big number, so the lengths are just stored starting with the length of "zero" (1+3=4) at the far right, and following over to the length of "ninety-nine" (7+3=10) at the far left.

The shortest English numbers are three letters ("one", "two", "six", "ten"), so each length is stored with an offset of three. The longest prior to 100 are 9 + 3 = 12 letters (e.g. "seventy-eight"), so each number can be stored as a single digit.

Answer 2

Starting from the right:

• the first digit is how many letters are in "zero" minus 3
• the second digit is how many letters are in "one", minus 3
• the third digit...
• ...the 100^th digit is how many letters are in "ninety nine" minus three.

Note that that the longest number "seventy seven" has only 12 letters, which conveniently fits in a single digit after subtracting 3.