College Mathematics Journal

Vol48-N4

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Minimal Tilings of a Unit Square Iwan Praton Iwan Praton (ipraton@fandm.edu) traveled halfway around the world to study at Oberlin and the Massachusetts Institute of Technology. Because of his peripatetic youth, he finds it hard to believe that he has been at Franklin & Marshall College for nearly 20 years. Figure 1 shows two different tilings of the unit square, each using 10 square tiles. Figure 1. Two tilings of the unit square. In the left-hand tiling, there are two tiles of side length 1/2 and eight tiles of side length 1/4, so the total side length of the tiles is 2 · (1/2) + 8 · (1/4) = 3. In the second tiling on the right, there is one tile of side length 4/5 and nine tiles of side length 1/5, so the total side length is 4/5 + 9 · (1/5) = 13/5. Is it possible to find another tiling of the unit square with 10 square tiles so that the sum of the side lengths of the tiles is larger than 3? Also, is it possible to find a different tiling with 10 square tiles so that the sum of the side lengths is smaller than 13/5? You might want to try to answer these questions before reading on. Our first step, naturally, is to replace 10 with an arbitrary positive integer n. Let us be more precise and introduce some terminology and notation. Suppose T is a tiling of the unit square using n smaller squares (that is, we place n small squares—called tiles—inside the unit square, with no gaps or overlaps); we call T an n-tiling. Denote by σ (T ) the length of the tiling, that is, the sum of the side lengths of the n small squares. Define f M and f m by f M (n) = sup T σ (T ), f m (n) = inf T σ (T ) where the supremum and infimum are taken over all n-tilings of the unit square. The main problem is to describe these two functions as fully as possible. http://dx.doi.org/10.4169/college.math.j.48.4.242 MSC: 05B45, 52C20 242 © THE MATHEMATICAL ASSOCIATION OF AMERICA

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