What Are The Most Common Digits?

Riddler Express

From Oscar Lanzi comes a matter of tricky triangular numbers:

Suppose you are adding up whole numbers in order. You start with 1, then add 2 to get 3. Then you add 3 to get 6. Then you add 4 to get 10, and so on.

If you keep going — adding larger and larger numbers in this fashion, and looking at the resulting sums — you might notice a few patterns among the last two digits. One such pattern is that no sum ever ends with the digits “23.” Meanwhile, some pairs of final digits are more common than others.

Which pairs of final digits are the most common? (Here, “23” is a distinct final pair of digits from “32.”)

Solution

The pattern of the last two digits of triangular numbers repeats at most every 100 numbers.

I looked at the last two digits of the 1^st - 100^th triangular numbers and achieved the following:

Answer

The numbers $\equiv 3 \mod 25$ appear five times each.

Rohan Lewis

2023.03.06

Code can be found here.