### John Valentine

Shared publicly -**How do we know what math research is worthwhile?**[tessellations]

Every so often, I take a peek at what's happening in mathematics or physics. I'm probably not alone when I think, "someone wrote a paper on

*that*?". After a while, usually while cooking food for the family, I think a bit, around what I imagine to be the abstract and practical problems, and it's pleasantly interesting, but I wouldn't want to formally cover every nuance of the problem. A little while later, I might remember that I did something similar that was part of the wider category of associated problems, and wonder whether I could or should have explored it in more detail.

One of my feeds regularly reports on

**shape-packing**: how to pack polygons into planes or surfaces, how to fit regular solids into each other, the polynomial solution that maximally crosses its own path, and so on.

This leads me on to an example:

**Minesweeper**. It's not the P-NP problem that people think of; rather I was briefly interested in the way that differently-shaped minesweeper cells affected game balance. The case of a square is reasonably trivial: each square (not on the edge of the board) has 8 neighbours, and when you count the distribution of the squares that have 0, 1, 2, 3, … neighbouring mines, you can predict approximately how playable the game will be.

However, I didn't just write Minesweeper. I introduced variations: holes (inactive parts of the grid), linked cells (which tied two non-adjacent cells together for counting neighbouring mines), and shaped cells (which grouped contiguous cells into bigger areas).

Shaped cells proved most interesting, because using areas composed of squares, you could simulate other shapes of cell. For example, using 1×2 areas, you could create a pattern that matches a hexagonal game board by offsetting every second column of areas downwards one cell. A hexagonal board gives you a possible mine count in the range 0..6, so although the board construction seems less trivial, the gaming possibilities are more constrained.

We could also create shapes that have more neighbours, like the one pictured. Rather than the 8 neighbours of a square grid, this "Scroll Lock 2" variant has up to 9 neighbours. Other variants go further, or introduce patterns where there are easy and difficult cells. Large cells can make the smaller cells more difficult to solve, and so on.

Here's the question: is there potential for interesting mathematical exploration here? What would the intrinsic and derived properties be? Can this be applied usefully to anything else, or can other studies be applied here?

Refs:

**[1]**Here's the app, made for Windows 95–10 Desktop:

http://johnvalentine.co.uk/crossmines.php

Note that there's a bug, where, is you choose a 'pattern' game, you have to go in and choose the pattern each time.

**[2]**"How does one choose which bits of maths will be the ones to make the huge impact down the track?" https://plus.google.com/+DavidRoberts/posts/duMz4SU32ST

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