What's special about 323 and squared rectangles?

Solution 1:

The three rectangles displayed are each composed of 3 sub-rectangles, and each of those sub-rectangles has it's tiling determined by the continued fraction from the Euclidean GCD algorithm. You could just use the Euclidean GCD to get a squaring of the 323x319 rectangle, but it wouldnt be minimal. The continued fraction of 323/319 = [1; 79,1,3], ie 323x319 = 1x319^2 + 79x4^2 + 1x3^2 + 3x1^2, and the sum of the c.f. values gives the order of the squared rectangle, ie 1+79+1+3 = 84, which is certainly not minimal.

However if the rectangle is dissected into several rectangles the order can be reduced.

For the three examples;

323x319 = 290x145 + 290x174 + 319x33

cont.frac [2] [1;1,2] [9,1,2] sum = 18

323x30 = 260x30 + 30x18 + 30x15

cont.frac [8;1,2] [1;1,2] [2] sum = 17

323x293 = 255x153 + 255x170 + 323x38 = 17

cont.frac [1;1,2] [1,2] [8,2] sum = 17

All the examples have three subrectangles and are minimal, so for these no better solutions exist with rectangles composed of just two subrectangles.

It would be worth checking to see if perhaps 323 and 352 are the first time a minimal solution requires 3 subrectangles. If all other smaller rectangles had only required dissection into 1 or 2 subrectangles, and if the average number of squares in a small rectangle was only half a dozen or so, then 3 rectangle solutions would jump from orders of 6 - 12 to 18 or thereabouts, and the next jump might occur when the first 4 subrectangle solutions appear.