How many square numbers have no even digits?
The answer is: two (1 and 9).
Bolaji Atanda won the prize. He experimented with a calculator, and concluded:
a. there are no other square numbers with no even digits; and, even more:
b. in every square number one of the last two digits, the tens digit or the units digit, is even.
You can see by looking that this is true for square numbers up to 100.
Every square number bigger than 100 is (10m+n)2 for some m and some n<10
(10m+n)2 = 100m2 + 20mn + n2
so the last-but-one digit of the square number is even (because the last digit of 2mn is even) unless the tens digit of n2 is odd.
But 5, 7, and 9 are the only values of n which can give us an odd last digit of the square number.
52=25, 72=49, 92=81. In each case the tens digit is even.
Therefore either the last digit, or the last-but-one digit, of every square number is even. ▇