Tallest Jenga

This is a Garden Post. It may be unfinished or changing over time.

02 August 2025

I wrote this post entirely in HTML/MathML for HTML Day 2025.

While playing Jenga, I wondered, as I'm sure many have - what's the tallest tower that could be created in a game of Jenga?

Let's do some maths to work it out!

Initial Thoughts

Lets declare $L (k)$ as the number of blocks in the layer $k$ .

$N$ shall be the number of layers in our tower.

In the initial game state, $N = 18$ , and $\sum_{k = 1}^{N} L (k) = 54$ (i.e. there are 54 blocks overall).

The smallest number of blocks a layer can have is 1 (of course). But for the top two layers, this is different: it is generally accepted that a player cannot take a block from the top two layers (as the top layer is the one you're building, and the layer below that is structurally necessary to build on top of). So:

\begin{matrix} \forall k : k < N - 1 & L (k) & = & 1 \\ L (N - 1) & = & 3 \\ L (N) & = & 3 \end{matrix}

Therefore, I think that the tallest tower would be a stack of 1s, with 3 on the top two layers.

\begin{matrix} \sum_{k = 1}^{N} L (k) & = & 54 \\ \sum_{k = 1}^{N - 2} L (k) + L (N - 1) + L (N) & = & 54 \\ \sum_{k = 1}^{N - 2} (1) + 3 + 3 & = & 54 \\ (N - 2) + 3 + 3 & = & 54 \\ (N - 2) + 3 + 3 & = & 54 \\ N - 2 & = & 48 \\ N & = & 50 \end{matrix}

So.. 50 layers. That sounds about right? But also this isn't really a proof that it is:

Possible within the rules
The Tallest legal outcome (maybe there's something taller which would be possible with 1 on the top layer, for example)

Maybe I'll look into Lean or something to try and prove it.