This is a really lovely question.

The even/odd discussion is a good one.

But let's also just think about what this question is asking. We have 3 consecutive integers: z, y, x.

We square each. Subtract. And multiply the difference. Also the answers don't get too big, so let's write out the perfect squares up to 100, and in between each, the difference:

1

___3

4

___5

9

___7

16

___9

25

___11

36

___13

49

___15

64

___17

81

___19

100

Interesting. The difference between consecutive perfect squares increases by consecutive odd numbers (so this is also a good place to realize the answer cannot be even).

But you could also multiply

1

___3

4___________15

___5

9___________35

___7

16__________63

___9

25__________99

___11

36__________143

___13

49 _________...hey wait these are al 1 less than an even perfect square: 195

___15

64

___17

81__________ 323

___19

100

These are each 1 less than an even perfect square because multiply consecutive odd integers gets us in an: (E + 1)(E - 1) = (E² - 1)