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Reversed Ages

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.

Table of reversed ages 1 - 98
Table

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Copyright (C) Anthony P. Stone 2009. This material may be freely used, provided the author is acknowledged.

Last updated: 18 March 2009