The optimality of a certain purely recursive dissection for a sequentially n-divisible square

A dissection for a sequentially n-divisible square is a partition of a square into a number of polygons, not necessarily squares, which can be rearranged to form two squares, three squares, and so on, up to n squares successively. A dissection is called type-k iff k more pieces are needed to increase the maximum number n of composed squares by one. Ozawa found a general dissection of type-3, while Akiyama and Nakamura found a particular, “purely recursive” dissection of type-2. Nozaki has given a mixed procedure for a dissection of type-1. In this paper, we shall show that there is no type-1 purely recursive dissection for a sequentially n-divisible square. Therefore Akiyama and Nakamura’s dissection is optimal with respect to the type, among the purely recursive dissections. The results have been published in previous papers [1,2,6,7]. In this paper, we give a detailed proof.  2002 Elsevier Science B.V. All rights reserved.