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| Main Author: | |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2110.05416 |
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Table of Contents:
- For any odd integer $n\geq3$ a board (of size $n$) is a square array of $n\times n$ positions with a simple rule of how to move between positions. The goal of the game we introduce is to find a path from the upper left corner of a board to the center of the square. If there exists such a path we say that the board is solvable, and we say that the length of this board is the length of a shortest such path. There are $8^{n^2}$ different boards. We discuss various properties of these boards and present some questions and conjectures. In particular, we show that for $n\gg1$ roughly $\frac{1}{3}$ of the boards are solvable, and that the expected length of a random solvable board tends to $\frac{209}{96}$, i.e., very big solvable boards tend to have extremely short solutions.