Even Squares; Odd Squares
2006 May 6
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)
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.