2070.43 – Big and Little Cubes

A large cube is made up of n3n^3 little cubes stacked and glued so that the big cube has nn 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 n2{n^2} cubes on the front face, another n(n−1)n(n-1) on the right-hand face, and (n−1)2({n-1)^2} on top.

Add them up:

n2+n(n−1)+(n−1)2=n2+n2−n+n2−2n+1=3n2−3n+1\begin{align*}n^2 + n(n-1) + (n-1)^2 &= n^2 + n^2 - n + n^2 - 2n + 1 \\&= 3n^2 -3n + 1\end{align*}

The answer is 3n2−3n+13n^2 -3n + 1

Another way to understand this answer is that the 3n23n^2 counts the three visible faces of n2n^2 small cubes each, but that this over-counts because each edge of nn cubes is counted twice – being in two faces. So we subtract 3n3n to compensate for the double-counting of the edges. Now the single corner cube is miss-counted. It is counted three times in the 3n23n^2 then subtracted three times in the −3n-3n and so is not counted at all! We must add it back in, hence the +1+1.