X+Y=100. X is a square number. Y is a prime number. Work out the values of X and Y

You can brute force all possibilities easily, for example using Excel.

The square numbers below 100 are 1

, 4

, 9

, 16

, 25

, 36

, 49

, 64,

81.

Their counterparts to add up to 100 are: 99

, 96

, 91

, 84

, 75

, 64

, 51

, 36, 19.

Of these, only 19 is a prime.

So the (only) solution is: X=81 (9*9) and Y=19.