Puzzlemaster

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The Independent Culture
LET'S WRAP up the tennis-court puzzle, fig a). This required us to devise a counting system ensuring that each rectangle was counted once and once only.

One strategy is to number each vertex with the number of rectangles having that point as the top left-hand corner. This is called splitting the problem into subtasks. The subtasks here are simple and all the same: how many rectangles have a given vertex as top left-hand corner? Then you need a system of going from vertex to vertex without missing any out. One way is to sweep from left to right and down the page as in reading.

The other thing about the tennis court is that it's not completely symmetrical. If you flip the court on its side, you get a different set of numbers. This gives a handy way of checking the total. One sort of checking is to retrace one's footsteps. The problem with this is that mistakes are habit-forming. But once an idea has been thought, it becomes infinitely easier to think it again. Psychologists call this "priming".

The other check is more exciting. You go by as different a route as possible and, voila, you end up with the same answer as before. This provides a sort of collateral confirmation.

The Garabaggio, fig b), last week has even less symmetry than the court: there are four ways of orienting the grid and so four ways of checking. We make it seven squares and 13 rectangles.

The racquet problem can of course be solved along similar lines, but it regularity permits a more general approach. The fact that any given rectangle in such an array is defined by a unique combination of a pair of vertical and a pair of horizontal lines means that we can count the rectangles by finding the total number of ways in which it is possible to combine the pairs of vertical with pairs of horizontal lines.

Points to Ponder

How many a) squares b) rectangles can be found in an 8x8 grid of squares (eg a chessboard)?

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