Why does twelve plus one equal one?

Answer 1

Normally it does not. It only does if you are working with congruence numbers, modulo 12. That is a rather technical way of saying you are using "clock" arithmetic.

There are other such examples:

modulo 7 for days of the week

modulo 2 for ON/OFF

are another two that most people are familiar with, even if they don't know that they are using modulo arithmetic!