Reversible Ages

Copyright 1997 by Paul Niquette.  All rights reserved.

If the mother's age in the puzzle is written as AB in decimal, she is 10 A + B years old, and her daughter is 10 B + A years old.

The difference...

    (10 A + B) - (10 B + A)
...is the age of the mother when her daughter was born. Let's call that x, and since...
    x = 9 (A - B)
...we know that x is divisible by 9.

It requires little sophistication indeed to rule out...

  • A - B = 0, unless mother and daughter are twins;
  • A - B = 1, unless the mother gave birth at age 9;
  • A - B > 6, limited by 'the biological clock.'
Thus, the mother is older than her daughter by...
    x = 18 or 27 or 36 or 45 or 54 years.
For any given value of x, the reversibility phenomenon will be repeated only on birthdays in which both A and B have increased by the same amount. The smallest such amount is 1, so every 11 years, mother and daughter will have reversible ages.

As noted elsewhere, the youngest two-digit age, is 10. The youngest reversible two-digit age is 12, but that makes x = 9, quite young for motherhood. Accordingly, the youngest age for the daughter that will meet the conditions of the puzzle is 13.
 

13 years old

...or 31 years old, if you assumed 
the note was written by a son.

Reversiblity in ages will be observed when...
 

Daughter:
24
35
46
57
68
79
Mother:
42
53
64
75
86
97

...exactly six more times, as called for in the puzzle.

It is easy to see that taking any higher value of x will permit fewer recurrences. The sophisticated solver will be able to show that there are only 26 pairs of ages for mothers and daughters that are reversible.


Bonus Puzzle Solution

Here is the idle question asked in the Bonus Puzzle...

Is there a temperature in Fahrenheit degrees that can be converted to Celsius degrees -- by simply reversing its digits?
The formula for converting Fahrenheit to Celsius is C = (5/9*(F -32).  If we let C = 10 x + y and F = 10 y + x then solve for x = (41 y - 160) / 85.  Of course, x and y are each required to be decimal ciphers: 1, 2, 3, ... 9. 

By trial substitutions we find two solutions...
x = 1 ; y = 6 and x = 2 ; y = 8
Thus...
  • 16 degrees Celsius = 61 degrees Fahrenheit.  Slightly warm -- but quite comfortable.
  • 28 degrees Celsius = 82 degrees Fahrenheit.  Quite warm -- but something cool to know.
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