Paul Leeuw

For his Bachelor's thesis, Paul Leeuw chose to research the problem of dissecting the square into squares. Duijvestijn was one of Leeuw's thesis advisers and they both knew the lowest order perfect squared square had been found in 1978 by Duijvestijn. Finding and proving the lowest order compound perfect square square was still an open problem so this is the subject of the thesis. Leeuw finds that Willcocks 24:175 is the lowest order CPSS and only CPSS of order 24. The research involved a process of generating c-net graphs, and from them tables of squared rectangles. Also created were 'Deficient squares', squared squares, composed of squares and one subrectangle. If a squared rectangle fits the subrectangle of the Deficient square, and no two squares in the whole dissection are the same size, the resulting dissection is a compound perfect squared square. Of course not all CPSSs are of this form, and depending on the order, a number of other compound perfect squared square constructions are possible, particularly where two or more rectangles are involved. The thesis examines the possibilities for these constructions also.

Leeuw states; "The idea of this way of solving the problem comes from P. J. Federico, the mapping into the computer, the development of the necessary algorithms is performed by A.J. W. Duijvestijn and P. Leeuw"[1].

The research resulted in several thousand compound perfect squared squares being created, with only several published, but the main result was establishing Willcocks 24:175 as lowest order compound perfect squared square.

The thesis result was republished in a journal article with co-authors P.J. Federico and A.J.W. Duijvestijn in 1982[3].

  1. P. Leeuw SQUARED SQUARES, A Bachelor's Thesis, Technological University Twente, August 1979,
  2. P. Leeuw, "Compound squared squares : programs", Technische Hogeschool Twente, Onderafdeling der Toegepaste Wiskunde, 1980,
  3. A. J. W. Duijvestijn, P. J. Federico, P. Leeuw, Compound Perfect Squares, Source: The American Mathematical Monthly, Vol. 89, No. 1 (Jan., 1982), pp. 15-32