Is there a largest perfect square with all even digits?  What is it?
Is there a largest perfect square with all odd digits?  What is it?

(2401 falls into neither category, since the "1" is odd and the "2" "4" and "0" are even)

1. May 6, 2006 pm31 3:44 pm 3:44 pm

For even digits, no. Consider (2*10^n)^2, for n = 0, 1, 2, … .

For odd digits, yes. The units digit of the square of an even number is always even. The tens digit of the square of an odd number greater than 3 is always even. (Consider modulo 20.) So 9 is the largest such perfect square.

Nice puzzle! Where did you find it? (Or is it original?)

2. May 6, 2006 pm31 4:03 pm 4:03 pm

Nick,

I believe that I took this one 5 or 10 years ago from Compuserve’s “The Science Math Forum,” which I no longer visit. There was a man named Bertie Taylor (from England maybe) who used to post amazingly engaging puzzles, including this one. He disappeared, and I can no longer find him.

It’s a great question, btw. I recently coauthored a print piece on logic puzzles that includes:

Graham, Knuth and Patashnik (1989) in their book Concrete Mathematics – A Foundation for COmputer Science wrote: “Mathematicians have unfortunately developed a tradition of borrowing exercises without any acknowledgement; we believe that the opposite tradition, practiced for example by books and magazines about chess (where names, dates, and locations of original chess problems are routinely specified) is far superior.” We agree and would like to give credit to the creators of the problems in this article but the sources are unknown to us. If any reader can provide us with information about the source of these problems we would be pleased to receive it.

I should practice what I preach, and at least offer the place where I found my puzzles. Thank you for asking.