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| <!doctype html public "-//IETF//DTD HTML//EN"><HTML><HEAD><TITLE>Squares in Squares</TITLE></HEAD><BODY BGCOLOR="#FFFFFF"><CENTER><font size=+4>Squares in Squares</font></CENTER><p><font size=+1>The following pictures show n unit squares packed inside the smallest known square (of side length s). For the n not pictured, the trivial packing (with no tilted squares) is the best known packing.<p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>1.<br><img src="s1.gif" align=abscenter><td> <TD><font size=+3>2.<br><img src="s2.gif" align=abscenter><td> <TD><font size=+3>3.<br><img src="s3.gif" align=abscenter><td> <TD><font size=+3>4.<br><img align=abscenter src="s4.gif"></tr><TR align=center><td align=center><font size=+1>s = 1<br>Trivial.<TD><td align=center><font size=+1>s = 2<br>Proved by Frits Göbel<br>in 1979.<TD><td align=center><font size=+1>s = 2<br>Proved by Frits Göbel<br>in 1979.<TD><td align=center><font size=+1>s = 2<br>Trivial.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>5.<br><img align=abscenter src="s5.gif"><td> <TD><font size=+3>6.<br><img align=abscenter src="s6.gif"><td> <TD><font size=+3>7.<br><img align=abscenter src="s7.gif"><td> <TD><font size=+3>8.<br><img align=abscenter src="s8.gif"></tr><TR align=center><td align=center><font size=+1>s = 2 + 1 / √2 = 2.707+<br>Proved by Frits Göbel<br>in 1979.<TD><td align=center><font size=+1>s = 3<br>Proved by Michael Kearney<br>and Peter Shiu in April 2002.<TD><td align=center><font size=+1>s = 3<br>Proved by Erich Friedman<br>in 1999.<TD><td align=center><font size=+1>s = 3<br>Proved by Erich Friedman<br>in 1999.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>9.<br><img align=abscenter src="s9.gif"><td> <TD><font size=+3>10.<br><img align=abscenter src="s10.gif"><td> <TD><font size=+3>11.<br><img align=abscenter src="s11.gif"><td> <TD><font size=+3>13.<br><img align=abscenter src="s13.gif"></tr><TR align=center><td align=center><font size=+1>s = 3<br>Trivial.<TD><td align=center><font size=+1>s = 3 + 1 / √2 = 3.707+<br>Proved by Walter Stromquist <br>in 2003.<TD><td align=center><font size=+1>s = 3.877+<br>Found by Walter Trump<br>in 1979.<TD><td align=center><font size=+1>s = 4<br>Proved by Wolfram Bentz<br>in 2010.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>14.<br><img align=abscenter src="s14.gif"><td> <TD><font size=+3>15.<br><img align=abscenter src="s15.gif"><td> <TD><font size=+3>17.<br><img align=abscenter src="s17.gif"><td> <TD><font size=+3>18.<br><img align=abscenter src="s18.gif"></tr><TR align=center><td align=center><font size=+1>s = 4<br>Proved by Erich Friedman<br>in 1999.<TD><td align=center><font size=+1>s = 4<br>Proved by Erich Friedman <br>in 1999.<TD><td align=center><font size=+1>s = 4.675+<br>Found by John Bidwell<br>in 1997.<TD><td align=center><font size=+1>s = (7 + √7) / 2 = 4.822+<br>Found by Pertti Hamalainen <br>in 1979.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>19.<br><img align=abscenter src="s19.gif"><td> <TD><font size=+3>24.<br><img align=abscenter src="s24.gif"><td> <TD><font size=+3>26.<br><img align=abscenter src="s26.gif"></tr><TR align=center><td align=center><font size=+1>s = 3 + 4 √2 / 3 = 4.885+<br>Found by Robert Wainwright<br>in 1979.<TD><td align=center><font size=+1>s = 5<br>Proved by Erich Friedman<br>in 1999.<TD><td align=center><font size=+1>s = 7 / 2 + 3 / √2 = 5.621+<br>Found by Erich Friedman<br>in 1997.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>27.<br><img align=abscenter src="s27.gif"><td> <TD><font size=+3>28.<br><img align=abscenter src="s28.gif"><td> <TD><font size=+3>29.<br><img align=abscenter src="s29.gif"></tr><TR align=center><td align=center><font size=+1>s = 5 + 1 / √2 = 5.707+<br>Found by Frits Göbel<br>in 1979.<TD><td align=center><font size=+1>s = 3 + 2 √2 = 5.828+<br>Found by Frits Göbel <br>in 1979.<TD><td align=center><font size=+1>s = 5.934+<br>Found by Thierry Gensane<br>and Philippe Ryckelynck<br>in April 2004.</TR></TABLE><p><hr><p> | ||
| <TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>37.<br><img align=abscenter src="s37.gif"><td> <TD><font size=+3>38.<br><img align=abscenter src="s38.gif"><td> <TD><font size=+3>39.<br><img align=abscenter src="s39.gif"></tr><TR align=center><td align=center><font size=+1>s = 6.598+<br>Found by David W. Cantrell<br>in September 2002.<TD><td align=center><font size=+1>s = 6 + 1 / √2 = 6.707+<br>Found by Frits Göbel<br>in 1979.<TD><td align=center><font size=+1>s = 6.818+<br>Found by David W. Cantrell<br>in August 2002.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>40.<br><img align=abscenter src="s40.gif"><td> <TD><font size=+3>41.<br><img align=abscenter src="s41.gif"><td> <TD><font size=+3>50.<br><img align=abscenter src="s50.gif"></tr><TR align=center><td align=center><font size=+1>s = 4 + 2 √2 = 6.828+<br>Found by Frits Göbel<br>in 1979.<TD><td align=center><font size=+1>s = 6.937+<br>Found by Joe DeVincentis<br>in April 2014.<TD><td align=center><font size=+1>s = 7.598+<br>Found by David W. Cantrell<br>in September 2002.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>51.<br><img align=abscenter src="s51.gif"><td> <TD><font size=+3>52.<br><img align=abscenter src="s52.gif"><td> <TD><font size=+3>53.<br><img align=abscenter src="s53.gif"></tr><TR align=center><td align=center><font size=+1>s = 7.704+<br>Found by Károly Hajba<br>in July 2009.<TD><td align=center><font size=+1>s = 7 + 1 / √2 = 7.707+<br>Found by Frits Göbel<br>in 1979.<TD><td align=center><font size=+1>s = 7.823+<br>Found by David W. Cantrell<br>in September 2002.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>54.<br><img align=abscenter src="s54.gif"><td> <TD><font size=+3>55.<br><img align=abscenter src="s55.gif"><td> <TD><font size=+3>65.<br><img align=abscenter src="s65.gif"></tr><TR align=center><td align=center><font size=+1>s = 7.846+<br>Found by Joe DeVincentis<br>in April 2014.<TD><td align=center><font size=+1>s = 7.954+<br>Found by Joe DeVincentis<br>in April 2014.<TD><td align=center><font size=+1>s = 5 + 5 / √2 = 8.535+<br>Found by Frits Göbel<br>in 1979.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>66.<br><img align=abscenter src="s66.gif"><td> <TD><font size=+3>67.<br><img align=abscenter src="s67.gif"><td> <TD><font size=+3>68.<br><img align=abscenter src="s68.gif"></tr><TR align=center><td align=center><font size=+1>s = 3 + 4 √2 = 8.657+<br>Found by Evert Stenlund<br>in 1980.<TD><td align=center><font size=+1>s = 8 + 1 / √2 = 8.707+<br>Found by Frits Göbel<br>in 1979. <TD><td align=center><font size=+1>s = 15/2 + √7/2 = 8.822+<br>Found by David W. Cantrell<br>in September 2002.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>69.<br><img align=abscenter src="s69.jpg"><td> <TD><font size=+3>70.<br><img align=abscenter src="s70.gif"><td> <TD><font size=+3>71.<br><img align=abscenter src="s71.gif"></tr><TR align=center><td align=center><font size=+1>s = 8.828+<br>Found by Maurizio Morandi<br>in June 2010.<TD><td align=center><font size=+1>s = 8.881+<br>Found by Joe DeVincentis<br>in April 2014.<TD><td align=center><font size=+1>s = 8.960+<br>Found by Joe DeVincentis<br>in April 2014.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>82.<br><img align=abscenter src="s82.gif"><td> <TD><font size=+3>83.<br><img align=abscenter src="s83.gif"><td> <TD><font size=+3>84.<br><img align=abscenter src="s84.gif"></tr><TR align=center><td align=center><font size=+1>s = 6 + 5 / √2 = 9.535+<br>Found by Frits Göbel<br>in 1979. | ||
| <TD><td align=center><font size=+1>s = 4 + 4 √2 = 9.657+<br>Found by Evert Stenlund<br>in 1980.<TD><td align=center><font size=+1>s = 9 + 1 / √2 = 9.707+<br>Found by Frits Göbel<br>in 1979.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=100% cellspacing=0><TR align=center><TD><font size=+3>85.<br><img align=abscenter src="s85.gif"><td> <TD><font size=+3>86.<br><img align=abscenter src="s86.gif"><td> <TD><font size=+3>87.<br><img align=abscenter src="s87.gif"></tr><TR align=center><td align=center><font size=+1>s = 11 / 2 + 3 √2 = 9.742+<br>Found by Erich Friedman<br>in 1997. | ||
| <TD><td align=center><font size=+1>s = 17 / 2 + √7 / 2 = 9.822+<br>Found by Erich Friedman <br>in 1997.<TD><td align=center><font size=+1>s = 9.851+<br>Found by David W. Cantrell<br>in August 2002.</TR></TABLE><p><hr><p><TABLE BORDER=0 width=67% cellspacing=0><TR align=center><TD><font size=+3>88.<br><img align=abscenter src="s88.gif"><td> <TD><font size=+3>89.<br><img align=abscenter src="s89.gif"></tr><TR align=center><td align=center><font size=+1>s = 9.901+<br>Found by David W. Cantrell<br>in August 2002.<TD><td align=center><font size=+1>s = 5 + 7 / √2 = 9.950+<br>Found by Evert Stenlund <br>in 1980.</TR></TABLE><p><hr><p><font size=+1>For more details, see my paper on the subject:<A HREF="https://erich-friedman.github.io/papers/squares/squares.html" >Packing Unit Squares in Squares: A Survey and New Results</A>.<P></BODY></HTML> | ||
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| The <a href="https://kingbird.myphotos.cc/packing/squares_in_squares.html">squares in squares page</a> is now being maintained by David Ellsworth. |
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