I had a look at James William's simple perfect squared squares (SPSSs) collection (15 million) and found some new record holder in higher orders. In order 41 and above several record breaking 'pairs' showed up. There is a pair in order 43 with a side of 35172, and largest/smallest elements of 10558 and 1312; 10558/1312=8.05. A pdf with 30 of the best is here; http://www.squaring.net/downloads/nice-spss-JBW.pdf I also had a look in simple perfect squared rectangles (SPSRs) all the way up to order 24. There are 63,301,415 in order 23 and 228,130,900 in order 24. In order 23 I found a SPSR 7526x5620 with largest/smallest elements of 2182 and 576; 2182/576 = 3.788. This was the only one with a ratio below 4, but there were many with 4 and above. A pdf is here; http://www.squaring.net/downloads/23-7526x5620-3.788.pdf So it looks like the ratio will approach 1 for SPSRs and I think this is probably the case for SPSSs too, though much higher orders would be required. A related problem is to find the maximum possible ratio of the LARGEST element in a squared square to the side of the squared square. In 2009 Brian Trial found 1xn SPSRs up to 1x18. The 1xn squared rectangle is a generalisation of squared squares and a harder problem, the squared domino (1x2) didnt appear until order 22, while the lowest order SPSS found by Duijvestijn was order 21 (112 side). Up until 2009 only one example of a 1x3 SPSR was known, due to Jasper Skinner, found in order 26, so Brian's finds were quite extraordinary. Two pdfs of Brian's 1xn SPSRs are here; (landscape version) http://www.squaring.net/sq/sr/B_T-18b.pdf and (portrait version) http://www.squaring.net/sq/sr/B_T-18a.pdf By scaling and fitting Brian's 1x17 and 1x18 SPSRs and adding one extra square, equal to the size of the long side of the scaled 1x17 rectangle, a compound perfect squared square (CPSS) is created with the largest element square having a side equal to 94% of the CPSS side. So it seems the maximum of largest element to squared square size also approaches 1, from below this time. A pdf of this CPSS is here http://www.squaring.net/downloads/560-338321736_BT.pdf , you need to zoom in to see the elements along the sides. Stuart Anderson

Message: 3 Date: Tue, 4 Jun 2013 20:30:11 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] squared squares Message-ID: <CAA-4O0FEeaXO0B3PRJ2Mi28Em5V7LzGTJ1my7MNo78gjgpwfTQ@mail.gmail.com> Content-Type: text/plain; charset=windows-1252

Stuart Anderson>Armin Singer has written a program to search for 'nice' squared squares http://www.squaring.net/downloads/squaredance-0.10.tgz He has a couple of 'nice' files in the tarball Also James Williams produced over 15 million new perfect squared squares earlier this year. Most of them havnt been added to the website yet. There are bound to be a whole lot of record breakers in that lot. There are some 'nice' squared rectangles here http://www.squaring.net/sq/sr/spsr/spsr_minmax.html

Stuart

While I don't see much nice in nice2.txt, the new record in nice1 appears to be 65/8:

ShowQuilt[{31, 1408, 1408, 448, 520, 440, 80, 360, 262, 186, 114, 239, 247, 175, 125, 163, 99, 260, 104, 96, 151, 118, 242, 64, 210, 200, 227, 145, 124, 81, 389, 366, 345, 308}, True]

where, from Ed Pegg's MondrianPuzzles.nb (*Code for generating positions of squares based on a Bouwkamp code.*)

BouwkampCoords[BouwkampFormat_List] := Module[{nums = BouwkampFormat}, {order, height, width } = Take[nums, 3]; squares = Drop[nums, 3]; active = {{{0, 0}, {0, height}}} ; coordinates = Table[{0}, {order}]; cornera = squares[[1]]; cornerb = cornerc = cornerd = 0; Do[ ss = squares[[n]]; coordinates[[n]] = Table[active[[1, 1]], {4}] + {{0, 0}, {0, ss}, {ss, ss}, {ss, 0}}; active = Sort[Switch[Sign[active[[1, 1, 2]] + ss - active[[1, 2, 2]]], 0, ReplacePart[active, 1 -> active[[1]] + {{ss, 0}, {ss, 0}}], -1, Flatten[{Drop[active, 1], {active[[1]] + {{0, ss}, {0, 0}}, {active[[1, 1]], active[[1, 1]]} + {{ss, 0}, {ss, ss}} }}, 1], _, Flatten[{active, {{"Error", n}}}, 1]]]; If[active[[1, 1]] == "Error", Break[]]; active = Partition[ Flatten[Select[Split[Flatten[active, 1]], Length[#] < 2 &], 1], 2]; If[coordinates[[n, 2]] == {0, height}, cornerb = squares[[n]]]; If[coordinates[[n, 3]] == {width, height}, cornerc = squares[[n]]]; If[coordinates[[n, 4]] == {width, 0}, cornerd = squares[[n]]], {n, 1, order}]; coordinates = If[active == {{{width, 0}, {width, height}}}, Append[ First /@ coordinates, {cornera, cornerb, cornerc, cornerd}], Append[ First /@ coordinates, {cornera, cornerb, cornerc, cornerd, active}]]]

(*Display squares of a Bouwkamp code.*)

ShowQuilt[BouwkampFormat_List, color_: False] := With[{ vals = Drop[BouwkampFormat, 3], order = BouwkampFormat[[1]], size = BouwkampFormat[[2]], coord = BouwkampCoords[BouwkampFormat]}, With[{colors = If[color, {Gray, White, Red, Yellow, Blue, Cyan, Magenta}[[ Mod[#, 7, 1]]] & /@ vals, Table[White, {Length[vals]}]]}, Graphics[{EdgeForm[{Black, Thick}], Opacity[.3], Table[{colors[[n]], Rectangle[coord[[n]], coord[[n]] + vals[[n]]]}, {n, 1, order}], Opacity[1], Black, Table[ If[vals[[n]] > size/15, Style[Text[vals[[n]], coord[[n]] + vals[[n]]/2], 18], Sequence @@ {}], {n, 1, order}]}, ImageSize -> {450, 370}]]]

I wonder if the min over all squared squares exists and is > 1. --rwg

On Tue, Jun 4, 2013 at 9:57 AM, Bill Gosper <billgosper@gmail.com> wrote: Ed> In the order 29 squares, a square of size 1702 has Smallest square =

...

62Largest Square = 567 That's a ratio of 9.1 --Ed Pegg Jr Wow.

--- On Tue, 6/4/13, Dan Asimov <dasimov@earthlink.net> wrote:

From: Dan Asimov <dasimov@earthlink.net> Subject: Re: [math-fun] squared squares To: "math-fun" <math-fun@mailman.xmission.com> Date: Tuesday, June 4, 2013, 8:44 AM

I see that Ian Gambini describes a squared square of side 110 in

Gambini, Ian A method for cutting squares into distinct squares. Discrete Appl. Math. 98 (1999), no. 1-2, 65?80,

with a total of 23 squares whose sides range from s=2 to s=44, for a ratio of 22.

--Dan No, there's a 1x1 in there. But a later example has 50/2.

I was just looking at a story where one Rodolfo Gambini uses quantum gravity to exonerate God from dividing by zero. Small legs must confer a selective advantage.

...

(There's an image of the squared square here: < http://www.sciencedirect.com/science/article/pii/S0166218X99001584 >, but it may require a subscription.)

On 2013-06-03, at 11:11 PM, Bill Gosper wrote:

http://www.squaring.net/sq/ss/spss/o22/spsso22.pdf has largest/smallest = 30. Is this minimal? What about squared rectangles? --rwg _______________________________________________ Tnx! --rwg