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Try this... really interesting

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A cube is cut into 1000 identical small cubes each of 1unit side length. Then the problem here is, to find the total number of cubes that can be formed using these 1000 cubes.

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With 1 unit sized cube each, 1000 cubes can be formed, orWith 8 unit sized cubes each, 125 cubes can be formed, orWith 27 unit sized cubes each, 37 cubes can be formed, orWith 64 unit sized cubes each, 15 cubes can be formed, orWith 125 unit sized cubes each, 8 cubes can be formed, orWith 216 unit sized cubes each, 4 cubes can be formed, orWith 343 unit sized cubes each, 2 cubes can be formed, orWith 512 unit sized cubes each, 1 cube can be formed, orWith 729 unit sized cubes each, 1 cube can be formed, orWith 1000 unit sized cubes each, 1 cube can be formed.So, on the whole 1000+125+37+15+8+4+2+1+1+1 = 1194 different types of cubes can be formed.If the unit sized cubes position is not changed after cutting the big cube, then With 1 unit sized cube each, 1000 cubes can be formed, orWith 8 unit sized cubes each, 125 cubes can be formed, orWith 27 unit sized cubes each, 27 cubes can be formed, orWith 64 unit sized cubes each, 8 cubes can be formed, orWith 125 unit sized cubes each, 8 cubes can be formed, orWith 216 unit sized cubes each, 1 cube can be formed, orWith 343 unit sized cubes each, 1 cube can be formed, orWith 512 unit sized cubes each, 1 cube can be formed, orWith 729 unit sized cubes each, 1 cube can be formed, orWith 1000 unit sized cubes each, 1 cube can be formed.So, on the whole 1000+125+27+8+8+1+1+1+1+1 = 1173 different types of cubes can be formed.Here, I used only combination method, but if permutation method is used the result would be something else & is very complicated I think.If my answer is wrong then please let me know, my e-mail ID is vuppala.praveen@gmail.com

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Thanks for the explanation. It is really helpful.

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