Math puzzle: A sequence of odd events

From left: Snezhana Kudryavtseva/iStock/Getty Images Plus; Cavan Images/Cavan/Getty Images Plus; Tatiana Sviridova/Moment/Getty Images Plus

In Spring 2025, we — Lilly Fang, Lester Mackey and our two children — noticed that something odd was happening to our family. First, Noah turned the odd age of 9 in May. Then Isaiah turned odd — or, more specifically, 7 — in August. Next, Lilly turned odd, 41, in November and finally, Lester turned odd, also 41, in February. This led us to wonder…

  1. Does every family experience an “odd cycle,” in which the age of each member flips from even to odd in turn?
  2. With a list of birthdays and birth years for all members of a given family, what is a simple way to determine whether (a) the family will experience an odd cycle and (b) in what order the members will flip from an even age to an odd one?
  3. Why was Noah the first to turn an odd age and not Lilly (who was born first) or Lester (who has the earliest birthday each year)?
  4. Is there a simple way to check 2(a) and 2(b) without calculating anyone’s age?

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