## Is there a method to cut a hypercube into disjoint cubes

Since Borsuk conjecture hold for centrally symmetric convex sets in $$\mathbb{R}^n$$ so we can cut a hypercube into at least $$n+1$$ disjoint parts.

Is there a method how can one do that?

## Products and sum of cubes in Fibonacci

Consider the familiar sequence of Fibonacci numbers: $$F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$$.

Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a different approach. Hence,

QUESTION. Is there a combinatorial or more conceptual reason for this “pretty” identity? $$F_nF_{n-1}F_{n-2}=\frac{F_n^3-F_{n-1}^3-F_{n-2}^3}3.$$

Caveat. I’m open to as many alternative replies, of course.

Remark. The motivation comes as follows. Define $$F_n!=F_1\cdots F_n$$ and $$F_0!=1$$. Further, $$\binom{n}k_F:=\frac{F_n!}{F_k!\cdot F_{n-k}!}$$. Then, I was studying these coefficients and was lead to $$\binom{n}3_F=\frac{F_n^3-F_{n-1}^3-F_{n-2}^3}{3!}.$$

## Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)

Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?

## How to find the vole of a cube remaining after drilling a cylinder with a diameter larger than the cube’s side length?

So, for example, say there’s a cube with side length L, and you drill a cylinder with a diameter larger than L through the cube, what volume remains?