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I thought I would share with our chess-loving readers the following interesting (and somewhat well-known) mathematical chess paradox , apparently proving that , and the accompanying explanation offered by Prof. Christian Hesse, University of Stuttgart (Germany). It shows a curious connection between the well-known Cassini’s identity (related to Fibonacci numbers) and the chessboard ( being a Fibonacci number!). The connection can be exploited further to come up with similar paradoxes wherein any -square can always be “rerranged” to form a -rectangle such that the difference between their areas is either or . Of course, for the curious reader there are plenty of such dissection problems listed in Prof David Eppstein’s Dissection page.

This being a mathematics blog, I’m sure a lot of readers out like to play Minesweeper. I’ve just obtained a personal best today (94 seconds on expert level) which, as Minesweeper buffs know, is nowhere close to world-class levels, but which made me happy anyway, as I’d never broken into the double digits before today!

It seems to me that world-class competitors must know some tricks with the mouse which I’ve never bothered to master, particularly because my laptop doesn’t have a mouse but rather a touchpad. This being the case, I keep my right index finger on the touchpad to guide the cursor, and the left index finger to click. I always left-click: that is, in my style of play, I [practically] never flag squares for bombs; I click only on non-bomb squares. For it’s well-known, or at least it should be, that the program doesn’t care if you identify where the bombs are — you get full credit for only identifying all the numbered squares.

To play in this style well, one needs to be fluent in a number of tactical tricks, which I don’t have good names for, but which in my personal argot I call things like “1-2-1”, “1-2-2-1”, “rule of three”, to name just a few. But that’s not what I set out to discuss, really. What I’d really like to hear from good players is: **what opening strategies do you use**?

The personal best I just set came after deciding on a new opening strategy. What I had been doing is clicking along border squares. Under that strategy, one *could* of course just keep clicking until one opens up an area, but often I would add to that the observation that if one clicked on a 1, thus leading to, e.g.,

x 1 x (–> border row)

x x x

then one could then click on the non-border square just adjacent to the 1, with only a 1 in 5 chance of setting off a bomb. If one then hits another 1:

x 1 x

x 1 x

x x x

then one can immediately open up the line of squares on the third rank, leading to a configuration such as

x 1 x

x 1 x

1 1 1

or better. This is often a cheap and quick way of opening up squares or otherwise getting a tactical toehold.

The new strategy I began using today is not to click along the border, but to click along the ranks or files adjacent to the border. Under this strategy, if one lands on a 1, leading to

x x x (–> border row)

x 1 x

x x x

then one can click on the border square adjacent to the 1, with only a 1 in 8 chance of setting off a bomb. If one does *not* set off a bomb, that square *has* to be a 1:

x 1 x

x 1 x

x x x

and then one can proceed as before. So I’ve just lowered my odds of hitting a bomb, plus a very small fractional gain in processing time that comes with the certain knowledge that it’s a 1 if not a bomb. So far the strategy has paid off well!

I’d like to hear other people’s opening strategies, and also I’d like to know some statistics. For example, considered over the space of expert-level games, what is the probability of getting a 1, a 2, and so on? Does anyone know? (It seems this would be very difficult computing analytically — one is probably better off using a Monte Carlo simulation. But I don’t have the wherewithal to set that kind of thing up.)

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