minesweeper generalizes unusually well

As a person of a particular neurotype, I always find it very fun to consider generalized versions of simple¹¹ For some definition of simple. games. For example, I think infinite chess²² Infinite chess enjoyers should consider reading the paper about how infinite chess can embed any open Gale-Stewart game, and therefore every ordinal less than the Church-Kleene ordinal can be attained as the game value of some infinite chess position. is very cool. A chess variant I designed several years ago, Chess 9b60, is also what I'd call a natural “generalization” of chess, in some ways — it keeps the board the same, but extends the set of possible pieces to include fairy chess pieces that seem (to me) to fit in reasonably well with the existing chess pieces.

One obvious way you might try to generalize chess is by making the board multidimensional — indeed, a common turn of phrase refers to 4D chess, which Wiktionary defines as

A sophisticated strategy that is far beyond the comprehension of others, especially one in which apparent blundersare simply indicators of yet-to-be-understood brilliance.

Famously, 4D chess is a game that actually exists, sold as a video game under the title 5D Chess with Multiverse Time Travel. 5D Chess always went over my head, but people do play it seriously. However, notice the with Multiverse Time Travel part. This game isn’t the most natural or immediate generalization of chess into higher dimensions. If you naïvely plop down even an 8x8x8 board, nevermind 4D, you get something pretty much totally unplayable. People try it, and I love when they do, but 5D Chess really relies on its bespoke Multiverse Time Travel nonsense to play as well as it does, I think.

The issue is one of topology: boards just open up a hell of a lot when you add an extra dimension! The things that made it balance and play well in 2D vanish. This isn't just a chess thing, it happens in a lot of simple games. Perhaps³³ Actually, probably not. Standard tic-tac-toe already sucks. one particularly good example of this is tic-tac-toe, which becomes pathological in high dimensions.

But there’s one game⁴⁴ Although I suppose one can observe that it is a singleplayer game, rather than a multiplayer one. which I can give as an example of something that cleanly avoids this issue: Minesweeper. There is 4D Minesweeper, there is Non-Euclidean Minesweeper, these are both among the most natural Minesweeper generalizations you can make without requiring any galaxy-brain multiverse time travel mechanic, and they are in my opinion pretty solid as games.

Why does Minesweeper work well where other games fail? I think the basic observation I'd make here, as an immediate guess, is that Minesweeper just outright works pretty okay on any graph. You don't care about any geometric information beyond just file adjacencies, basically, and it’s all mostly about making slightly local deductions (and about doing so reliably on a large board without ever misstepping and blowing up the mines). So you could really just take some vertices connected by edges in whatever way, put bombs on some of the vertices, and then play Minesweeper on it. The connectedness and openness of the board or whatever changes the texture of the game a little but it fundamentally works out to play roughly the same way for the same reasons even if you play it on your favourite weird choice of graph.

I’m sure Minesweeper isn’t the only example of a game that generalizes to other similarly well-balanced games⁵⁵ Finding fully diagonalized Latin hypercubes is generalized Sudoku., but I thought it was neat how well it works.