Reverse Twins

Harold Cooper

2nd October 2025

Fun Fact

$$37|73$$

If two people’s ages differ by a multiple of 9, then every 11 years their ages will be mirrored.

For example, my brother turned 37 while my mom was 73. My mom, being a lover of numbers, immediately pointed out this fun symmetry.

And we realized this “reverse twin” phenomenon has happened every 11 years, when their ages were:

$$\begin{gathered} 04|40 \\ 15|51 \\ 26|62 \\ 37|73 \end{gathered}$$

But my age and my mom’s age always differ by either 35 years (between our birthdays) or 34 years, neither of which is a multiple of 9, so we will never be reverse twins.

$$😭|😭$$

Proof

$$XY|YX$$

The base-10 number “XY” represents the value $10X+Y$ and its reverse “YX” represents $10Y+X$.

So the difference between “XY” and “YX” is:

$$(10X+Y)-(10Y+X)=9X-9Y=9(X-Y)$$

That is, the difference between reversed decimal numbers is always a multiple of 9.

Meanwhile, changing a number by 11 changes its tens digit and ones digit by the same amount, so the symmetry remains.

$$11|11$$

Good News

It turns out my dad and I are reverse twins! Our alignments so far:

$$\begin{gathered} 03|30 \\ 14|41 \\ 25|52 \\ 36|63 \end{gathered}$$

We just hadn’t notice because he’s not as into numbers.

$$👯|👯$$