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Some Reminiscences of David Klarner |
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This essay is reprinted with the kind permission of David Singmaster and AK Peters, Ltd.. It originally appeared in the book Puzzler's Tribute: a feast for the mind, edited by David Wolfe and Tom Rodgers, AK Peters, 2001. ISBN 1-56881-121-7. 
Some Reminiscences of David Klarner I first met David in 1970 or 1971, when he was visiting Reading University. We discussed box-packing and I had found I had solved a problem that he'd been considering. In two dimensions, if a brick packs a box (larger than itself), then one can divide the box into two smaller boxes such that each smaller box can be packed (indeed with its bricks all in the same direction). The simplest illustration is filling a 6 x 5 box with 3 x 2 bricks, where the only packings exhibit this divisibility property. David had wondered if this still held in three dimensions and I had found that 25 1 x 3 x 4 bricks can pack into a 5 x 5 x 12 box but could not pack 5 x 5 x c, for c = 1, ..., 11, nor 1 x 5 x 12 nor 2 x 5 x 12 in any way. This is the smallest example of this behavior. David later mentioned this in his classic "Brick-packing puzzles" (J. Recreational Math. 6 (1973) 112-17), but he cited a different example: 2 x 3 x 7 in 8 x 11 x 21, apparently having forgotten the numbers in my example. In 1978, Dean Hoffman proposed the following. Can one fit 27 bricks, all a x b x c, into a cube of side a + b + c? The planar version is to use 4 bricks of size a x b to fit into a square of side a + b. This is easy to do and is a way of showing the arithmetic-geometric mean inequality The corresponding inequality for three variables gives us 27abc ≤ (a+b+c)3 so that a solution of Hoffman's problem gives a geometric proof of the arithmetic-geometric mean inequality for three variables. Hoffman tried to do this using pencil and paper and found it too hard to do, so he rang up David and asked if he could do it. David had inherited a fine table saw from his father and used it to make up a set of 27 blocks. As he made each one, he stacked it in the corner and found a solution as he went. In the early 1980s, I visited David at Binghamton. Dean Hoffman was present and David made me a set of 27 blocks from a lovely redwood. He also made a three-cornered frame to hold them. This set is one of the treasures of my collection. It was on this visit that I saw the bedspread made by Kara Lynn Klarner showing two orthogonal Latin squares of order 10. This is basically a 10 x 10 array of squares, using ten colors such that each color occurs once in each row and column. Then each square has a circle on it, using the same ten colors so that each color occurs once in each row and column, and further, so that each pair of colors occurs just once as a square-circle pair. They said the hardest part of making the spread was finding ten sufficiently contrasting colors. |