My mother was 27 when I was born. 27, as we know, is a multiple of 9, and the thing about multiples of 9 is that for any number of two different digits, call it AB, the difference between AB and BA will always be a multiple of 9.
So, when I turned (zero) 3, she was thirty; then, eleven years later, when I turned 14, she was 41; and when – on ad infinitum (until it kind of breaks down when I turn seventy and she is 107, though the lower two digit pattern seems to continue).
My question is, why is it 11 years? I know that the reciprocal of nine is point-one-one-bar, but that seems reaching. Is there a mathematical way to describe this, or is it just what it is?
One way to visualise this is to look at what the reversed digits of each number is doing for every case between 01 and 99 - here’s a quick and dirty graph: https://1drv.ms/i/s!Ar4eOUAx-yGwh_UrVYm5ZOiWdrGEqg
As you can see, the condition where the digits are the same (22, 33, 44 etc) occurs with an interval of 11 - because, whereas the original numbers (blue dots) are just rising on a line, the reverse-digit numbers (orange dots) are completing a series of increments of 10, then resetting back to 1 more than where they started.
Any line you plot with the gradient of the blue dots, if it intersects any orange dots at all, will only do so every 11th increment.
… so if we also plot your age plus 27 as a series of green dots, it intersects the pattern of orange dots every 11 years. If your mother was 28 or 26 years older than you, there would be no collisions at all. https://1drv.ms/i/s!Ar4eOUAx-yGwh_U6t-M41ON-2b6djA
The reason 9 and 11 are special is that we are using a base ten number system.
If one would use a base twelve system, the same things would happen with eleven and thirteen.
If one would use a base eight system, the same things would happen with seven and nine.
and so on…