# Even Squares; Odd Squares

May 6, 2006 pm31 1:20 pm

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)

3 Comments
leave one →

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?)

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:

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

Is there a largest perfect square with alternating odd and even digits?

For example, 9916 squared is 98,327,056. Are there infinitely many more?