Problem of the Week 1170

Happy Birthdays

Some people are happy if their n'th birthday occurs in a year for which the last two digits are the reverse of the digits of n. For example, someone born in 1945 would be happy on their 38th birthday, which occurs in 1983.

Devise a formula that tells, for any year n (n ≥ 0), how often a person who is born in year n will experience a happy birthday. Assume the person lives to have a 99th birthday, but not a 100th; and also count the true date of birth as the 00th birthday. That is, birthdays run from 00 to 99 (always two digits). Ignore leap days.

In particular, in what year should one be born to maximize happy birthdays?

Source: Frances Meier and Joan Hutchinson.

© Copyright 2013 Stan Wagon. Reproduced with permission.

[View the solution]