Some integers are one more than the sum of the squares of their digits in base 10.

Prove that the set of such integers is finite and list all of them.

Well, of course it is finite.

It must be positive, because it is the sum of squares.

And it must be less than 4 digits, because 4*9^2 + 1 is only 3 digits.

Further improvements to the upper and lower bounds are easy, but these observations are sufficient to prove that the set is finite.