Catalogue of simple perfect squared squares of orders 21 through 25

Introduction A perfect squared square of order n is a square dissected into a finite number n of squares no two of which are of equal size. It is called compound if it contains a subset of component squares, more than one and less than all, arranged in a rectangle or a square. If it does not contain such subset it is called simple. The very first Simple Perfect Squared Square of this Catalogue was discovered by computer in March, 1978, by Duijvestijn. It has the lowest possible order, n = 21, and it is the only one of that order. In July of the same year, Duijvestijn found two simple perfect squared squares of order 22. After communicating these to P.J. Federico of Washington, D.C., U.S.A., who informed T.H. Willcocks of Bristol, U.K., the latter found a third simple perfect squared square of order 22 about August 1978. After Duijvestijn's proof in 1962 that there were no simple perfect squared squares of order less than 21, another computer attack, by J.C. Wilson of Waterloo, Canada, in his thesis of 1967, resulted in twenty-nine low-order simple perfect squared squares, five of order 25 and twenty-four of order 26. By certain transformation techniques, Federico and Willcocks were able to add three simple perfect squared squares of order 25 to the stock, not to mention their four and seven of orders 26 and 27, respectively. Then, in 1990, Duijvestijn decided to start all over again when computer power had tremendously grown compared to that of the sixties. He first found eight simple perfect squared squares of order 22, including his earlier two. Secondly, he found twelve of order 23 and twenty-six of order 24, making the total number of simple perfect squared squares of order less than 25 equal to forty-seven. From the order-22 results sent to J.D. Skinner, 11, of Lincoln, Nebraska, U.S.A., two simple perfect squared squares of order 23 were derived by Skinner even before these were detected in the output of Duijvestijn's computer(s). The same happened, when Bouwkamp found two simple perfect squared squares of order 24 from Duijvestijn's order-23 results even before the order-23 run was completed. In 1991 Duijvestijn started with order 25 and at the end of March, 1992, the unexpectedly large number of 160 simple perfect squared squares of order 25 were there, of which only very few had been discovered before by transform techniques by Federico, Willcocks, Skinner, and Bouwkamp. The Catalogue contains 207 simple perfect squared squares of which 182 were found by Duijvestijn, 13 by Bouwkamp, 5 by Wilson, 3 by Federico, 3 by Skinner, and 1 by Willcocks for the first time. Our thanks are due to F.H. Simons of the Department for providing us with the originals of the pages that follow. This Catalogue is rather an Album of Squared Squares, the creation of which was hoped for by Federico after we had demonstrated automatic drawing by computer and plotter of the squared squares just found by Wilson.