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?
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 (a1)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 nearcube with sides (a1), a, (a+1): Take off that side with one too few rows, it is a 1xax(a1). The remaining block is axax(a1). These pieces can be put together to make the (a1)xax(a+1).

Posted by Jer
on 20160516 09:19:28 