A Note on the Purely Recursive Dissection for a Sequentially n-Divisible Square

Abstract

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 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 note, 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.