Counting The Number of Arrangements of Tatami Mats in a Rectangular Room of Vertical Length 2, 3 and 4

Graduate School of Mathematical Sciences, the University of Tokyo, 1538902 Japan

^* Corresponding author: susukeneko@gmail.com

Abstract

Japanese rooms are measured by the number of tatami mats that will fit inside. The size of a tatami mat can vary by region, but is generally around 180 cm by 90 cm, giving it a 2:1 ratio of length to width. In the following, for simplicity, we suppose that each tatami mat is a rectangle with two adjacent sides of lengths 1 and 2. A typical tea ceremony room is square-shaped and its area is the equivalent of 4 and a half tatami mats. Questions regarding 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 vertical length is fixed at four or less, by using the concept of compartments or indivisible factors. Since the area of each tatami mat is two, if the area of the room is odd, only one half-sized tatami mat is allowed to be used. Therefore, if the vertical length of the room is three, the results will be different depending on whether the horizontal length 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.