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1, 3, 5, 7, 9 are the possible digits for the tens digit. Notice they are all odd.
My reasoning:
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When you multiply any 2 numbers together, the last digit of the result will be the last digit of the product of the original last digits (ex. 12*7 = 84: 7*2 = 14 : 4 is last digit)
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The last TWO digits of the result will be the same as the last 2 digits of the product of just the last two digits. (ex, 1422*234 = 332,748: 22*48 = 748 : 48 are last two digits )
If is a perfect square ends in 6, the last digit squared must end in 6 as well.
The only single digit numbers squared that end in 6 are: 4*4=16 and 6*6=36
They are the only 2 so the number that is being squared must end in 4 or 6
Let the second last digit be X. X can be any digit.
So the last 2 digits of our number that is being squared are either X6 or X4
That is, the last 2 digits of the number are either (10X+6) or (10X+4),
this will result in X always being an odd number.
