Message: 3 Date: Sat, 18 Feb 2012 18:06:05 -0500 From: Warren Smith <warren.wds@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] cubed squares Message-ID:

The cubed square with the most water-volume is just to use the entire square without any tiles. [Maximizes  sum_j (s_j)^3.] Course, it is not so easy if you demand #tiles=27 (or whatever). Exactly, and tiles are always >21 for perfect (all squares in the dissection are different sizes) squared squares

Squared squares produced using Tuttes rotor-stator symmetry method seem to make big deep ponds. See the Federicos (PJF) order 50 squares in http://www.squaring.net/sq/ss/spss/o31+/SPSSo31-75.pdf

More interesting in my view is to MINIMIZE sum_j (s_j)^3.

The minimum size possible on the boundary of a perfect squared square is 5, Gambini proved this and produced some with 5 on the boundary using his packing program. Such squares would hold very little water, as small boundary squares need to be surrounded by other small squares creating an area with very low elevation and volume, and the water would just drain out from the boundary. There are many examples in the pdf above also in order 40s, 50's and 60's; I am trying to imagine other ways to minimise the water but cant think of another method of constructing perfect squares that would have this outcome. Then theres also the Imperfect squared squares, which exist in much greater numbers, which I havnt considered yet

Stuart