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Reversal of Age Digits Every Eleven Years

Date: 11/06/2007 at 14:32:16
From: Cheryl
Subject: Reversal of age numbers every 11 years

Is there an explanation or name for something I accidentally came 
upon some years ago?  Every 11 years, my age is the exact reverse of 
my mother's age, and vice-versa.  For example, in 1963 I was 13 and
she was 31.  In 1974, I was 24 and she was 42.  In 1985, I was 35 and
she was 53, etc.

I'm curious as to how to explain such an occurrence.  The only work
I've done is to chart out this occurrence, starting from the time I was 1.

Date: 11/06/2007 at 17:44:23
From: Doctor Rick
Subject: Re: Reversal of age numbers every 11 years

Hi, Cheryl.

I know the pattern, but I have never thought of it in terms of ages. 
Here is the more general pattern of which this is part:

  If you take any two-digit number and switch the digits, the
  difference between the two numbers is a multiple of 9.  In
  particular, the difference is 9 times the difference between
  the two digits.

For example, 42 - 24 = 18, which is 9 times 2, and 4 - 2 = 2.  For 
another example, 52 - 25 = 27, which is 9 times 3, and 5 - 2 = 3.

The reason the same thing happens every 11 years is that when you add 
11 to a two-digit number, you add 1 to each digit (unless a digit is 
9--when that happens, the pattern ends).  If you add the same amount 
to two numbers, the difference between them stays the same.  Thus the 
difference between the digits of 24 is 2; when we add 11 to get 35, 
the difference between the digits is 2 again.

And whenever the difference between the digits is 2, the difference 
between your age and the number with digits reversed is 18.  Therefore 
the number with digits reversed is your mother's age, since she is 
always 18 years older than you.

But why is the difference between one number and the number with 
digits reversed equal to 9 times the difference in the digits?  I'll 
explain briefly.  Let's say I start with the number 35.  If I move the 
tens digit to the ones place, I change 30 to 3, thus reducing the 
number by 30 - 3 = 27.  At the same time I move the ones digit to the 
tens place, increasing the number by 50 - 5 = 45.  Both of these are 
multiples of 9; their difference is 45 - 27 = 18.  That's where the 
18 comes from.  I could go farther in my explanation if you're 
comfortable with some algebra.

My own father was 26 1/2 years older than I.  Thus, for half of each 
year my age was 27 more than his.  On my 14th birthday, in 1966, Dad 
was 41.  On my 25th birthday, Dad was 52.  On my 47th birthday, Dad 
was 74.  Do you see how this is an example of the same phenomenon you 
noticed?  It happens because the difference of our ages is also a 
multiple of 9, though it's 27 rather than 18.  Yet I never noticed 
this phenomenon until now!  Thanks for pointing it out.

- Doctor Rick, The Math Forum 
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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