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Counting the number of tatami mat arrangements in a rectangular room of fixed height 2, 3 and 4
Last modified: 2024-09-18
Abstract
Japanese rooms are measured by the number of tatami mats that will fit inside. In the following, we suppose that each tatami mat is a rectangle with two adjacent sides of length 1 and 2. For example, a typical tea room is square and its area is the equivalent of 4.5 tatami mats. Questions regarding the number of ways to lay tatami mats are not only fun for elementary school students, but also often included in entrance exams. In this paper, we derive recurrence formulae for determining the number of ways to lay tatami mats in a rectangular room whose width is fixed at 4 or less, by using the concept of indivisible factors. Since the area of each tatami mat is 2, if the area of the room is odd, only one half-sized tatami mat is allowed to be used. Therefore, if the width of the room is 3, the results will be different depending on whether the horizontal size of the room is even or odd. A generating function is used in this case, since it is difficult to derive the recurrence formula from direct consideration.
Keywords
Tatami tiling, pattern recognition, enumeration, combinatorics