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Simply calculate the cubes of a few small numbers, and check which of them are in the desired range.
1, 8, 27, 64, 125
13 cubes 4,9,16,25,36,46,64,81,100,121,144,169,196
There are infinitely many cubes between any two numbers - no matter how close together they are. However, there may be a more useful answer in terms of "perfect" cubes: 43 = 64 < 100 < 53 = 125 and 83 = 512 < 600 < 93 = 729 So there are 4 perfect cubes in the range - those of 5 6, 7 and 8.
If you have one cube in the first solid than six cubes in the second solid and than fifteen cubes in the third solid, than you would get 215 cubes in the eighth solid.