Why It Works

The split-and-add trick looks like magic, but the reason is plain: 11 = 10 + 1. Multiplying by 11 is the same as multiplying by 10 (shift the digits left) and then adding the original number back.

Write a two-digit number as 10a + b. Times 11 it becomes 110a + 11b, which lines up as 100a + 10(a+b) + b — the first digit, the digit sum in the middle, and the last digit. That is exactly what "split and insert the sum" does, and the carry handles any middle sum above 9.