Take a chessboard. Take a set of dominoes, each big enough to cover exactly two adjacent squares on the board. Take a pair of scissors and—quietly, so the owner doesn’t realize your nefarious intent—cut out two diagonally opposite corner squares of the board. That is, either the top-right and bottom-left corners, or the top-left and bottom-right corners.

Note that the squares you have snipped off are either both black or both white.

While you ponder over that puzzle, I’ll tell you why I’ve been thinking about it, then sling another one at you.

December is, of course, NRI migratory season. Niloufer and Steve Mackey, two favourite friends, winged into town from Kalamazoo in the US, where they are mathematicians (and hobbyist tomato- and peach-canners, and avid hummingbird-feeders, and who knows what else, but of all that another time). We talked about a lot of things—as friends do who meet only every December—among which were some things mathematical. Given that they are professionals and academics, and given that I’m a strictly amateur dabbler, there’s a level of competence our conversations quickly reach. Unsurprisingly, that level is dictated firmly by my incompetence. But still, it is fun.

And this time, we spoke about puzzles, or, more interestingly, how mathematicians think about puzzles.

The dominoes, for example. At first glance, you might think like I did when I first came upon this puzzle: each piece covers two squares, there are 62 squares on the mutilated board, therefore 31 domino pieces will indeed cover them all. Done. Where’s the problem?

Though not quite done, actually. Just a little reflection leads you to this. Whenever you lay a piece on the board, you cover two adjacent squares, one black and one white. Thus at any time, the number of black squares hidden is the same as the number of white ones. Now remember that the mutilated board has, say, two less black squares than white. Progressing one black-white pair at a time, it’s impossible to cover those 62 squares, because you can never cover the two extra white squares. Done.

Though not quite, again. One reason I like mathematics is that what really interests its practitioners, like the Mackeys, is not so much the solution to a problem. It’s instead the new questions the solution opens up.

Here, the first question might be: okay, with the diagonally opposite squares, this reasoning is clear. But suppose you remove the top-right and bottom-right squares, i.e., one black and one white. Since the black and white squares now pair up, can we cover them with 31 domino pieces? The answer is yes, and you’ll quickly figure out one easy way to do it: place three pieces vertically along the extreme-right column, four pieces vertically along each of the other seven columns.

On to the next question then: what if you remove one black and one white square from anywhere on the board? That’s a little bit more of a challenge to answer, which is why I’ll leave you to figure it out (I’m just the dabbler). Once you do, maybe you’ll have even more interesting questions to attempt.

A simple example? Perhaps. But it is still an instructive way to understand what makes mathematicians tick, the way of mathematics itself. The challenge and satisfaction lie less in destinations reached than in the paths explored along the way, the vistas that yawn beyond. What new prospects lie along those paths? What new ways of thinking do they encourage? Do they help in tackling other problems?

And in that vein, the Mackeys and I chatted about another puzzle, the one I promised to sling at you.

The Maharaja of Gaipajama’s palace has a really long room with a row of lights on the ceiling: let’s say 25,000 of them. Each light switches on or off with a hanging cord. Pull once, it switches on, pull again, it’s off, like that. During the day, all 25,000 are off. At dusk, an attendant walks down the length of the room, pulling each cord and thus turning on all the lights. Behind him walks a second attendant, starting at the second light and pulling every second cord (#2, #4, #6, etc). Attendant #3 starts at light #3 and pulls every third cord (#3, #6, #9, etc). Attendant #4: #4, #8, #12, etc. And so we go until attendant #25,000 walks to light #25,000 and pulls its cord.

Never mind the Maharaja’s quirky ways and his 25,000 attendants stuck in dead-end jobs. Question: at the end of this (yes, futile) exercise, which lights are on, if any?

Far more interesting question: will anything we learnt while solving the chessboard puzzle help us here?

Fun stuff, really. Which is why even dabbling has its rewards. One pair at a time.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. Comments are welcome at dilip@livemint.com

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