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May 06, 2005

How many grandsons

One of the things I miss in the Commentary columns at James Randi's site is the puzzles he used to publish every week. Regrettably, he stopped doing that a few years back.

Since I have a collection of puzzles, I'm thinking of posting one once a week or so - I know I'm not the only person who likes them. Here's a simple one to start with. The answer's in the "Continue reading" section.

Mr. Gubbins, whose age is somewhere between 50 and 70, is fond of telling his friends, "Each of my sons has as many sons as brothers, and the combined number of my sons and grandsons is the same as my number of years."

How old was Gubbins, and how many grandsons did he have?

Gubbins was 64; he had 56 grandsons.

Let N equal the number of sons Gubbins had. Each son had N-1 brothers and, therefore, N-1 sons. Thus Gubbins had N*(N-1) - or (N**2 - N) - grandsons. His age equals the number of sons, N, plus the number of grandsons, (N**2 - N).

But N + N**2 - N is equal to N**2, and so Gubbins' age is a perfect square lying between 50 and 70. Since 64 is the only perfect square between 50 and 70, N**2 = 64 and N = 8. So Gubbins' age is 64 and he had N**2 - N, or 64 - 8 = 56, grandsons.

From 101 Mathematical Puzzles (1977 edition) by Don Reinfeld and David Rice.

Posted by joke du jour at May 6, 2005 09:00 PM

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