If two people have ages **87** and **78**, these are *reversed ages*. Reversed ages start at 1 year old (or 01) with **01, 10**, and go on until **98, 89**; after which they do not occur. (If both ages are 77, say, they are of course not reversed, but equal.) A given pair of reversed ages lasts only between two particular birthdays. The basic properties of reversed ages are as follows.

** When reversed ages occur for two people, their ages differ by a multiple of 9**

*Proof.* Let the reversed ages be a = 10u + v, b = 10v + u, where u, v may be 0, 1, ..., 9. Then a - b = 9(u -v).

*Example.* For **82, 28**, the difference is 54 = 9 × 6. (If a < b, the multiplying factor in a - b will be negative.)

**If reversed ages occur, this repeats every 11 years**

*Proof.* With the same notation, a + 11 = 10(u+1) + (v+1), b + 11 = 10(v+1) + (u+1), and a, b are reversed.

*Example.* If the ages are **23, 32**, then 11 years later they are **34, 43**.

** If reversed ages occur for two people, this lasts until one has their next birthday.**

*Example.*For reversed ages **87, 78**, if the younger person has a birthday first, the ages become 87, 79. If the older person's birthday is first, the ages become 88, 78.

**If, at some time, two people have ages differing by a multiple of 9, they have reversed ages at some stage.**

*Proof.* Consider a person 1 year old, and other people 9, 18, ..., 81 years older. The pairs of ages are:

1, 10; already reversed (as **01, 10**)

1, 19; after 1 year this becomes 2, 20 (or **02, 20**)

1, 28; after 2 years **3, 30**. In general,

1, 9n + 1; after n -1 years **n, 10n**. This can go on until n = 9:

1, 82; after 8 years **9, 90**.

After 11, 22, ... years these ages are:

from 1, 10: **12, 21; 23, 32**, ... , and 88 years later **89, 98**

from 2, 20: **13, 31; 24, 42**, ...

etc. Finally,

from 8, 80; **19, 91**. This is the only occasion in this case.

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Last updated: 18 March 2009