SOLUTION: You have a 10 cm cube and you cut the cube into 1 cm pieces, what is the largest cube you can build that looks solid from the outside but is hollow on the inside, and how many piec

Question 63537: You have a 10 cm cube and you cut the cube into 1 cm pieces, what is the largest cube you can build that looks solid from the outside but is hollow on the inside, and how many pieces would be left over?

Answer by 303795(602) (Show Source): You can put this solution on YOUR website!
The 10 cm cube contains 10x10x10 = 1000 one cm blocks
The hollow cube of side n will use n*n blocks on the front face and the same number on the backface. (2n^2)
The top of the cube will use n*(n-2). The (n-2) term comes about because the cubes on the front and rear edge were already counted in the front face and rear face. The base of the cube uses the same number as well. (2n(n-2))
The other two sides use (n-2)^2 cubes each (same reason as above for the (n-2) terms. (2(n-2)^2).
The total number of cubes needed to build a hollow cube of side n is therefore
This could be used with trial and error in a spreadsheet or on a calculator to find that a 13 cm cube uses 866 cubes and there must be 134 left over. (A 14 cm cube needs 1016 blocks)
The formula could also be simplified.

Note: The formula finds the area of the faces (6n^2) then subtracts the blocks on each of the edges as they have been counted twice(-12n) then adds on the 8 corner blocks as there were subtracted twice (8).

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