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