Perfect Square Dissection

Author

Eric W. Weisstein
September 28, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Dissections/PerfectSquareDissection.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PerfectSquareDissection.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Smallest Perfect Square

s={{50,35,27},{8,19},{15,17,11},{6,24},{29,25,9,2},{7,18},{16},{42},{4,37},{33}};

BouwkampConstruction[s]

Show[Graphics[LabeledSquares[BouwkampConstruction[s]],AspectRatio->Automatic]]

g=MapIndexed[
Graphics[
LabeledPolygons[{#1},Hue[#2[[1]]/10]],AspectRatio->Automatic,PlotRange->{{0,112},{0,-112}}
]&,
BouwkampConstruction[s]
];

movie=Rest[FoldList[Show,{},g]]

Movie

<<Utilities`GIF`;

Export[gifs<>"perfsqr.gif",movie,ConversionOptions->{"Loop"->True,"AnimationDisplayTime"->1}]//Timing

70×70 (A square square pyramidal number)

{Sum[k^2,{k,24}],70^2}

Sum[k^2,{k,n}]

Simple Squares

ReleaseHold[#[[1]]&/@DownValues[SimpleSquare]/.SimpleSquare[x__]:>{x}]

Figures

Show[GraphicsArray[{{
		DrawSquare[SimpleSquare[21],0,DisplayFunction->Identity],
			DrawSquare[SimpleSquare[22,1],0,DisplayFunction->Identity]},
		{DrawSquare[SimpleSquare[22,2],0,DisplayFunction->Identity],
			DrawSquare[SimpleSquare[23],0,DisplayFunction->Identity]}}],
DisplayFunction->$DisplayFunction]

Ell Squares

Steinhaus 1983, p. 9

DrawEllSquare[24,1]

DrawEllSquare[25,1]

Gardner (1961, p. 203)

DrawSquare[CompoundSquare[39]]