0, 25, 2025, 13225…are squares that remain squares if every digit in the number defining them is augmented by 1.
Let's call them squarish numbers.

a. List two more samples of squarish numbers.
b. Prove that all such numbers are evenly divisible by 25.
c. Why are there neither 3-digit nor 6-digit squarish numbers?
d. Prove that between 10^k and 10^(k+1) there is at most one squarish number.

(In reply to re(2): Part a only (spoiler) by Jer)

Indeed I hadn't noticed that contradiction.  The three numbers do check out, so part d must be wrong.

Posted by Charlie on 2011-03-14 17:38:34