Sum of squares puzzle
An observation I learned from Steven Spallone, and presumably due to him.
The sum of squares
$$\sum_{k=0}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$
can be represented by a puzzle:
you can make $6$ pyramids that fit together to form
a $n \times (n+1) \times (2n+1)$ box.
(This is not a given: the formula could hold, but it might not
be realizable in this way.)
To be precise, the ``pyramids'' are composed of $n$ layers,
each a $k \times k$ square ($1 \times 1, 2 \times 2, \dots, n \times n$),
with the vertex over one of the corners of the bottom.
Steven calls this a ``staggered step pyramid''; I call it a ``step yangma''.
(picture!)
\section{Assembling the pyramids}
It's easiest to assemble in $2$ pieces, each of $3$ pyramids.
Each side will be an $n \times n \times n+1$ block, and in the middle there will be a staircase sticking out, which'll fit with the other side (there's an embedded $\sum k = n(n+1)/2$).
To assemble each side, it helps to understand why $\int x^2 = \frac{1}{3} x^3$:
you can cut the cube into $3$ pyramids (as described elsewhere),
so when piecing the puzzle together, arrange as if there were no steps (put $3$ vertices together) and then slide a block or two over so that the gaps are eliminated.
Also, the $2$ sides have opposite chirality, else the staircases don't fit together.
\section{Higher dimensional generalizations}
This also works for triangular numbers in $2$ dimensions, as noted above
(and trivially for sum of $0$th powers in $1$ dimension).
I don't think it generalizes to higher dimensions.
Notably, the fifth power sum ($\sum k^5$) doesn't even factor into integral linear factors!
\section{Construction}
It's an easy (but somewhat time-consuming) project to make such a puzzle.
$n=3$ is sufficient, $n=4$ is convincing, and $n=5$ is plenty.
Ingredients:
\begin{itemize}
\item wooden cubes\\
From a craft store, say. $2$~cm (3/4~in) is ok for demonstration, but smaller is easier to handle. Compute how many you need! (and get a few extra, in case of defects).
\item wood glue\\
(such as Elmer's)
\item wood primer\\
(else the paint doesn't work well)
\item paint\\
Spray-paint works fine; $6$ colors: I suggest black/white, red/green, blue/yellow for contrast.
\end{itemize}
Steps:
\begin{itemize}
\item assemble pyramids.\\
Probably best fit if do each pyramid all at once, but easier to assemble $1$ dimension at a time: rods, then layers, then whole pyramids.
\item prime. dry.
\item paint. dry.
\end{itemize}
Note that the pyramids can be tipped over and rest on the edges of the layers, which makes painting easier. (Touch up tips for extra class.)