## 2070.43 – Big and Little Cubes

A large cube is made up of $n^3$ little cubes stacked and glued so that the big cube has $n$ little cubes along each edge. If you turn the big cube so as to see the most little cubes, how many little cubes do you see?

## Solution

From the picture, you can see ${n^2}$ cubes on the front face, another $n(n-1)$ on the right-hand face, and $({n-1)^2}$ on top.

Add them up:

$n^2 + n(n-1) + (n-1)^2 = n^2 + n^2 - n + n^2 - 2n + 1 = 3n^2 -3n + 1.$

The answer is $3n^2 -3n + 1.$

Another way to understand this answer is that the $3n^2$ counts the three visible faces of $n^2$ small cubes each, but that this over-counts because each edge of $n$ cubes is counted twice---being in two faces. So we subtract $3n$ to compensate for the double-counting of the edges. Now the single corner cube is miss-counted. It is counted three times in the $3n^2$ then subtracted three times in the $-3n$ and so is not counted at all! We must add it back in, hence the $+1$.