We know 10 x 10 = 100.
Working from there:
11 x 11 = 121
12 x 12 = 144
13x 13 = 169
14 x 14 = 196
15x15 = 225 but that's over 200.
So there are four perfect squares between 100 and 200 (not inclusive of 100)
You can also use a progression.
Say , (note difference ).
How many terms between and ?
Start from the biggest square under , which is . Now many with can you fit in ?
Is there any sum of terms of with and starting from consecutive that fit into ? Sure ! We have , but we cannot fit , so there are only three perfect squares between and .
There ... must be a better way to explain this, though ..
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I know this solution is a bit useless and boring after the previous answers, but eh, this is worth a post