What is the maximum volume of $N$-D slice of an $M$-D hypercube? Consider a unit hypercube of M dimensions. We wish to make a cut of dimension N through it. What is the largest N-D size (length, area, volume, ...) we can achieve, $S(M, N)$, and what cut gives it?
Examples:

*

*$S(M=2, N=1) = \sqrt2$

*

*Linear cut through a square

*Largest slice is the diagonal

*Length $\sqrt{1^2 + 1^2}$



*$S(M=3, N=1) = \sqrt3$

*

*Linear cut through a cube

*Largest slice is the space diagonal

*Length $\sqrt{1^2 + 1^2 + 1^2}$



*$S(M=3, N=2) = \sqrt 2$

*

*Planar cut through a cube

*Largest slice is the plane through the diagonal


*Area $\sqrt 2 \cdot 1$
Bits of this function are easy to generalize:
$$
S(M, N) = \left\{
\begin{array}{rl}
   \sqrt M & : N = 1\\
   ? & : 1 < N < M\\
   1 & : N = M\\
   0 & : N > M\\
 \end{array}
\right.
$$
Can we generalize for $N = 2, 3, ...$?

Just to put into words, without proof, the idea that has been floating around here.
Let $M=aN+b$.
Split the $M$ dimensions into $b$ groups of $a+1$ and $N-b$ groups of $a$.
Take the diagonal in each group, of length $\sqrt a$ or $\sqrt{a+1}$.
The product of all these diagonals is
$$\sqrt{a^{N-b}(a+1)^b}$$
