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A cube with missing row (Posted on 2016-05-16) Difficulty: 1 of 5
I have n cubical building blocks. I try to arrange them into the largest possible solid cube, but I find that there are not enough blocks: One side of the large cube has exactly one row too few.

What can be said about n?

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts What are you asking for? (Possible spoiler) Comment 1 of 1
Some possible answers:
Since it would take a^3 blocks to make a cube axaxa but there is a row missing n = a^3 - a. 

If could be said that n is of the form a^3 - a.

This expression factors to (a-1)a(a+1) so it could be said that n is the product of three consecutive integers.  It can also be said that n is a multiple of 6.

Although you could not make a cube, it is easy to rearrange the blocks to make the near-cube with sides (a-1), a, (a+1):  Take off that side with one too few rows, it is a 1xax(a-1).  The remaining block is axax(a-1).  These pieces can be put together to make the (a-1)xax(a+1).


  Posted by Jer on 2016-05-16 09:19:28
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