Reversible Digits Problem for Mother and Child

My daughter turns 36 on November 23rd and her mother, who is born on the same day, turns 63. I was struck by the reversible digits and immediately wondered whether this had happened before. A little investigation revealed that it had: at 25 and 52, at 14 and 41 and (if allowing leading zeroes) at 03 and 30. In fact it happens every 11 years. I wondered whether this was true for every mother and child. It's not: such reversibility of digits can only happen if the mother's age when she gives birth is a multiple of 9 e.g. 9, 18, 27, 36 etc. Here is my proof of this assertion. 9 is a little young of course but it has happened I think.

Let \(n\) be the age of the mother when the child is born.

Suppose mother and child turn \(x \, y\) and \(y \,x\) years old in a certain year.

Now the mother's age of \(x \, y\) has numerical value of \(10x+y\).

Similarly the child's age of \(y \, x\) has a numerical value of \(10y+x\).

We know that \( (10x+y)-(10y+x) =n\).

This means that \(9(x-y)=n\) where \(x>y\).

\(x\) and \(y\) have integer values from 0 to 9 and thus \(n\) must be a multiple of 9.

Suppose \(n=9\), this means \(x-y=1\) and there are various solutions.

Let's consider \(x=1\) and \(y=0\), giving the mother's age as 10 and the child's as 01.

Next consider \(x=2\) and \(y=1\), giving the mother's age as 21 and the child's as 12.

Next consider \(x=3\) and \(y=2\), giving the mother's age as 32 and the child's as 23.

We see can see clearly when the ages of mother and child share the same digits and why these match ups occur every 11 years.

ADDENDUM:  June 19th 2022

Here is some SageMath code that will spit out the above data (change the offset from 27 to some other number and experiment):