In this post, we were introduced to the game of Numerical Boggle on a $6 \times 6$ board, the rules of which are as follows
- Each cell must contain a single digit from $0$ to $9$.
- Starting in one cell you collect digits as you move to neighbouring cells (in all 8 directions). As the digits are collected they are concatenated left to right, to form a single number. Note that the starting digit is collected too and you can revisit cells.
The task then was to construct such a grid (of size $6 \times 6$) such that the smallest positive number that cannot be constructed is as large as possible.
Obviously, this game and subsequent optimisation can be generalised to square grids of any size, $n \times n$.
Moreover, we need not restrict ourselves to base $10$. Given any positive integer $b$, we can decree that each cell must contain a single digit from $0$ to $b-1$ and pose the optimisation with respect to this new base (with the exception of unary which only uses $1$).
Motivated by this generalisation, we can look at the problem in smaller bases.
In particular, if we look at the case $n=2$ and $b=2$, our optimisation task may result in something like the following
It turns out that, for this grid (or indeed any $2 \times 2$ grid with two $0$s and two $1$s) it is possible to construct every binary number according to the rules of Numerical Boggle (try it yourself). We will say that such a grid has infinite extent in base $b$.
Furthermore, we will say that a base $b$ admits a grid of infinite extent is there is some square grid of finite size ($n \times n$) which has infinite extent in base $b$. This leads us to our puzzle.
What is the largest positive integer base $b$ which admits a finite square grid of infinite extent or does such a $b$ exist? Please provide proof of your answer.