%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: v7i1r46.dvi %%Pages: 9 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -V/dev/null v7i1r46 %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2000.10.01:1252 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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1 store /Prologue (dpost.ps) def /aspectratio 1 def /formsperpage 1 def /landscape false def /linewidth .3 def /magnification 1 def /margin 0 def /orientation 0 def /resolution 720 def /rotation 1 def /xoffset 0 def /yoffset 0 def /roundpage true def /useclippath true def /pagebbox [0 0 612 792] def /R /Times-Roman def /I /Times-Italic def /B /Times-Bold def /BI /Times-BoldItalic def /H /Helvetica def /HI /Helvetica-Oblique def /HB /Helvetica-Bold def /HX /Helvetica-BoldOblique def /CW /Courier def /CO /Courier def /CI /Courier-Oblique def /CB /Courier-Bold def /CX /Courier-BoldOblique def /PA /Palatino-Roman def /PI /Palatino-Italic def /PB /Palatino-Bold def /PX /Palatino-BoldItalic def /Hr /Helvetica-Narrow def /Hi /Helvetica-Narrow-Oblique def /Hb /Helvetica-Narrow-Bold def /Hx /Helvetica-Narrow-BoldOblique def /KR /Bookman-Light def /KI /Bookman-LightItalic def /KB /Bookman-Demi def /KX /Bookman-DemiItalic def /AR /AvantGarde-Book def /AI /AvantGarde-BookOblique def /AB /AvantGarde-Demi def /AX /AvantGarde-DemiOblique def /NR /NewCenturySchlbk-Roman def /NI /NewCenturySchlbk-Italic def /NB /NewCenturySchlbk-Bold def /NX /NewCenturySchlbk-BoldItalic def /ZD /ZapfDingbats def /ZI /ZapfChancery-MediumItalic def /S /S def /S1 /S1 def /GR /Symbol def /inch {72 mul} bind def /min {2 copy gt {exch} if pop} bind def /setup { counttomark 2 idiv {def} repeat pop landscape {/orientation 90 orientation add def} if /scaling 72 resolution div def linewidth setlinewidth 1 setlinecap pagedimensions xcenter ycenter translate orientation rotation mul rotate width 2 div neg height 2 div translate xoffset inch yoffset inch neg translate margin 2 div dup neg translate magnification dup aspectratio mul scale scaling scaling scale addmetrics 0 0 moveto } def /pagedimensions { % useclippath userdict /gotpagebbox known not and { % /pagebbox [clippath pathbbox newpath] def % roundpage currentdict /roundpagebbox known and {roundpagebbox} if % } if pagebbox aload pop 4 -1 roll exch 4 1 roll 4 copy landscape {4 2 roll} if sub /width exch def sub /height exch def add 2 div /xcenter exch def add 2 div /ycenter exch def userdict /gotpagebbox true put } def /landscapepage { landscape not { 0 height scaling div neg translate % not quite 90 rotate } if } bind def /portraitpage { landscape { width scaling div 0 translate % not quite -90 rotate } if } bind def /addmetrics { /Symbol /S null Sdefs cf /Times-Roman /S1 StandardEncoding dup length array copy S1defs cf } def /pagesetup { /page exch def currentdict /pagedict known currentdict page known and { page load pagedict exch get cvx exec } if } def /decodingdefs [ {counttomark 2 idiv {y moveto show} repeat} {neg /y exch def counttomark 2 idiv {y moveto show} repeat} {neg moveto {2 index stringwidth pop sub exch div 0 32 4 -1 roll widthshow} repeat} {neg moveto {spacewidth sub 0.0 32 4 -1 roll widthshow} repeat} {counttomark 2 idiv {y moveto show} repeat} {neg setfunnytext} ] def /setdecoding {/t decodingdefs 3 -1 roll get bind def} bind def /w {neg moveto show} bind def /m {neg dup /y exch def moveto} bind def /done {/lastpage where {pop lastpage} if} def /f { dup /font exch def findfont exch dup /ptsize exch def scaling div dup /size exch def scalefont setfont linewidth ptsize mul scaling 10 mul div setlinewidth /spacewidth ( ) stringwidth pop def } bind def /changefont { /fontheight exch def /fontslant exch def currentfont [ 1 0 fontheight ptsize div fontslant sin mul fontslant cos div fontheight ptsize div 0 0 ] makefont setfont } bind def /sf {f} bind def /cf { dup length 2 idiv /entries exch def /chtab exch def /newencoding exch def /newfont exch def findfont dup length 1 add dict /newdict exch def {1 index /FID ne {newdict 3 1 roll put}{pop pop} ifelse} forall newencoding type /arraytype eq {newdict /Encoding newencoding put} if newdict /Metrics entries dict put newdict /Metrics get begin chtab aload pop 1 1 entries {pop def} for newfont newdict definefont pop end } bind def % % A few arrays used to adjust reference points and character widths in some % of the printer resident fonts. If square roots are too high try changing % the lines describing /radical and /radicalex to, % % /radical [0 -75 550 0] % /radicalex [-50 -75 500 0] % % Move braceleftbt a bit - default PostScript character is off a bit. % /Sdefs [ /bracketlefttp [201 500] /bracketleftbt [201 500] /bracketrighttp [-81 380] /bracketrightbt [-83 380] /braceleftbt [203 490] /bracketrightex [220 -125 500 0] /radical [0 0 550 0] /radicalex [-50 0 500 0] /parenleftex [-20 -170 0 0] /integral [100 -50 500 0] /infinity [10 -75 730 0] ] def /S1defs [ /underscore [0 80 500 0] /endash [7 90 650 0] ] def end DpostDict begin mark /rotation 1 def /gotpagebbox true def /resolution 720 def setup 2 setdecoding % % Copyright (c) 1993 AT&T, All Rights Reserved % % Version 3.4 drawing procedures for dpost. Automatically pulled in when % needed. % /inpath false def /savematrix matrix def /Dl { inpath {pop pop neg lineto} {newpath neg moveto neg lineto stroke} ifelse } bind def /De { /y1 exch 2 div def /x1 exch 2 div def /savematrix savematrix currentmatrix def neg exch x1 add exch translate x1 y1 scale 0 0 1 0 360 inpath {1 0 moveto arc savematrix setmatrix} {newpath arc savematrix setmatrix stroke} ifelse } bind def /Da { /dy2 exch def /dx2 exch def /dy1 exch def /dx1 exch def dy1 add neg exch dx1 add exch dx1 dx1 mul dy1 dy1 mul add sqrt dy1 dx1 neg atan dy2 neg dx2 atan inpath {arc} {newpath arc stroke} ifelse } bind def /DA { /dy2 exch def /dx2 exch def /dy1 exch def /dx1 exch def dy1 add neg exch dx1 add exch dx1 dx1 mul dy1 dy1 mul add sqrt dy1 dx1 neg atan dy2 neg dx2 atan inpath {arcn} {newpath arcn stroke} ifelse } bind def /Ds { /y2 exch def /x2 exch def /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 5 x1 mul add 6 div y0 5 y1 mul add -6 div x2 5 x1 mul add 6 div y2 5 y1 mul add -6 div x1 x2 add 2 div y1 y2 add -2 div inpath {curveto} {newpath x0 x1 add 2 div y0 y1 add -2 div moveto curveto stroke} ifelse } bind def % % Copyright (c) 1993 AT&T, All Rights Reserved % % Color and reverse video support for dpost. Only defines a default % colordict dictionary if there's not already a definition. Means the % definitions in a private color dictionary included with -C always % win. % currentdict /colordict known not { /colordict 50 dict dup begin /red { 1 0 0 } def /green { 0 1 0 } def /blue { 0 0 1 } def /cyan { 0 1 1 } def /magenta { 1 0 1 } def /yellow { 1 1 0 } def /white { 1 1 1 } def /black { 0 0 0 } def end def } if /rgb {setrgbcolor} bind def /hsb {sethsbcolor} bind def /setcolor { counttomark 1 eq { dup colordict exch known not {pop /black} if colordict exch get exec setrgbcolor } if counttomark 2 eq { /backcolor exch def /textcolor exch def colordict backcolor known not colordict textcolor known not or { /backcolor colordict /black get def /textcolor colordict /white get def } if /backcolor colordict backcolor get def /textcolor colordict textcolor get def /dY1 0 def /dY2 0 def textcolor exec setrgbcolor } if } bind def /drawrvbox { /x2 exch def /x1 exch def currentpoint dup /y1 exch def /y2 exch def pop dY1 0 eq dY2 0 eq and { currentfont /FontBBox get aload pop currentfont /FontMatrix get dtransform /dY2 exch def pop currentfont /FontMatrix get dtransform /dY1 exch def pop } if /y1 y1 dY1 add def /y2 y2 dY2 add def backcolor exec setrgbcolor newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath fill textcolor exec setrgbcolor } bind def end DpostDict begin /saveobj save def mark 1 pagesetup 18 R f (32 disks)1 595 1 983 2328 t 14 R f ( bonds)1 370( 63)1 245(m = 0.21317456258988)2 1342 3 302 2490 t gsave newpath 1280 2470 m /inpath true def 0.7 setgray 10 R f 655 1546 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (1)829 1566 w gsave newpath 829 1546 m /inpath true def 0.1 setgray 632 1582 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1582 m /inpath true def 0.1 setgray 10 R f 802 1720 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1720 m /inpath true def 0.1 setgray 10 R f 802 1373 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1373 m /inpath true def 0.1 setgray 10 R f 976 1582 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1582 m /inpath true def 0.4899 setgray 10 R f 1951 918 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (2)2125 938 w gsave newpath 2125 918 m /inpath true def 0.1 setgray 2098 1095 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1095 m /inpath true def 0.1 setgray 10 R f 2000 774 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 774 m /inpath true def 0.1 setgray 10 R f 2272 918 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 918 m /inpath true def 0.4901 setgray 10 R f 1346 1616 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (3)1519 1636 w gsave newpath 1519 1616 m /inpath true def 0.1 setgray 1319 1616 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1616 m /inpath true def 0.1 setgray 10 R f 1493 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 1493 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 1647 1703 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1703 m /inpath true def 0.1 setgray 10 R f 1647 1530 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1530 m /inpath true def 0.7 setgray 10 R f 1054 630 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (4)1228 650 w gsave newpath 1228 630 m /inpath true def 0.1 setgray 1117 784 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 784 m /inpath true def 0.1 setgray 10 R f 1113 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 1290 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.4901 setgray 10 R f 1346 1966 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (5)1519 1986 w gsave newpath 1519 1966 m /inpath true def 0.1 setgray 1319 1966 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 1493 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 1647 1877 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1877 m /inpath true def 0.1 setgray 10 R f 1493 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.4901 setgray 10 R f 1346 1269 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (6)1519 1289 w gsave newpath 1519 1269 m /inpath true def 0.1 setgray 1319 1269 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1269 m /inpath true def 0.1 setgray 10 R f 1493 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 1441 1102 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1102 m /inpath true def 0.1 setgray 10 R f 1647 1354 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1354 m /inpath true def 0.7 setgray 10 R f 655 1897 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (7)829 1917 w gsave newpath 829 1897 m /inpath true def 0.1 setgray 632 1930 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1930 m /inpath true def 0.1 setgray 10 R f 802 1720 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1720 m /inpath true def 0.1 setgray 10 R f 976 1930 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1930 m /inpath true def 0.4901 setgray 10 R f 1951 1616 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (8)2125 1636 w gsave newpath 2125 1616 m /inpath true def 0.1 setgray 1948 1703 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1703 m /inpath true def 0.1 setgray 10 R f 2098 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 1948 1530 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1530 m /inpath true def 0.1 setgray 10 R f 2098 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 2272 1616 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1616 m /inpath true def 0.4901 setgray 10 R f 1951 1269 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (9)2125 1289 w gsave newpath 2125 1269 m /inpath true def 0.1 setgray 2098 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 1948 1354 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1354 m /inpath true def 0.1 setgray 10 R f 2098 1095 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1095 m /inpath true def 0.1 setgray 10 R f 2272 1269 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1269 m /inpath true def 0.7 setgray 10 R f 887 938 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (10)1024 958 w gsave newpath 1061 938 m /inpath true def 0.1 setgray 920 1066 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1066 m /inpath true def 0.1 setgray 10 R f 1117 784 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 784 m /inpath true def 0.1 setgray 10 R f 1208 938 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 938 m /inpath true def 0.4901 setgray 10 R f 1231 329 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (11)1368 349 w gsave newpath 1405 329 m /inpath true def 0.1 setgray 1290 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 1467 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 1552 329 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 329 m /inpath true def 0.1 setgray 10 R f 1378 155 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 155 m /inpath true def 0.7 setgray 10 R f 999 1616 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (12)1135 1636 w gsave newpath 1172 1616 m /inpath true def 0.1 setgray 976 1582 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1582 m /inpath true def 0.1 setgray 10 R f 1146 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 1146 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 1319 1616 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1616 m /inpath true def 0.4899 setgray 10 R f 315 1616 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (13)451 1636 w gsave newpath 488 1616 m /inpath true def 0.1 setgray 462 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 462 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 632 1582 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1582 m /inpath true def 0.1 setgray 10 R f 289 1616 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1616 m /inpath true def 0.4901 setgray 10 R f 1405 630 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (14)1541 650 w gsave newpath 1578 630 m /inpath true def 0.1 setgray 1467 784 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 784 m /inpath true def 0.1 setgray 10 R f 1467 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 1725 630 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 630 m /inpath true def 0.1 setgray 10 R f 1640 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f 688 644 347 347 De (15)824 664 w gsave newpath 861 644 m /inpath true def 0.4899 setgray 999 1966 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (16)1135 1986 w gsave newpath 1172 1966 m /inpath true def 0.1 setgray 976 1930 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1930 m /inpath true def 0.1 setgray 10 R f 1146 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 1319 1966 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 1146 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.4901 setgray 10 R f 1951 1966 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (17)2088 1986 w gsave newpath 2125 1966 m /inpath true def 0.1 setgray 1948 1877 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1877 m /inpath true def 0.1 setgray 10 R f 2098 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 2272 1966 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 2098 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.4899 setgray 10 R f 878 329 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (18)1014 349 w gsave newpath 1051 329 m /inpath true def 0.1 setgray 848 329 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 329 m /inpath true def 0.1 setgray 10 R f 1113 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 1025 155 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 155 m /inpath true def 0.7 setgray 10 R f 999 1269 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (19)1135 1289 w gsave newpath 1172 1269 m /inpath true def 0.1 setgray 976 1233 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1233 m /inpath true def 0.1 setgray 10 R f 1146 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 1319 1269 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1269 m /inpath true def 0.7 setgray 10 R f 315 608 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (20)451 628 w gsave newpath 488 608 m /inpath true def 0.1 setgray 518 771 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 771 m /inpath true def 0.1 setgray 10 R f 567 466 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 466 m /inpath true def 0.1 setgray 10 R f 289 608 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 608 m /inpath true def 0.4901 setgray 10 R f 1650 1442 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (21)1786 1462 w gsave newpath 1823 1442 m /inpath true def 0.1 setgray 1647 1530 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1530 m /inpath true def 0.1 setgray 10 R f 1647 1354 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1354 m /inpath true def 0.1 setgray 10 R f 1797 1616 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1616 m /inpath true def 0.1 setgray 10 R f 1948 1530 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1530 m /inpath true def 0.1 setgray 10 R f 1948 1354 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1354 m /inpath true def 0.7 setgray 10 R f 1238 938 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (22)1374 958 w gsave newpath 1411 938 m /inpath true def 0.1 setgray 1208 938 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 938 m /inpath true def 0.1 setgray 10 R f 1441 1102 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1102 m /inpath true def 0.1 setgray 10 R f 1467 784 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 784 m /inpath true def 0.4901 setgray 10 R f 1578 329 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (23)1714 349 w gsave newpath 1751 329 m /inpath true def 0.1 setgray 1640 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 1552 329 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 329 m /inpath true def 0.1 setgray 10 R f 1814 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 1725 155 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 155 m /inpath true def 0.4899 setgray 10 R f 315 1966 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (24)451 1986 w gsave newpath 488 1966 m /inpath true def 0.1 setgray 462 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 632 1930 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1930 m /inpath true def 0.1 setgray 10 R f 289 1966 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 462 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.4901 setgray 10 R f 426 938 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (25)562 958 w gsave newpath 599 938 m /inpath true def 0.1 setgray 688 1069 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1069 m /inpath true def 0.1 setgray 10 R f 518 1102 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1102 m /inpath true def 0.1 setgray 10 R f 518 771 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 771 m /inpath true def 0.4901 setgray 10 R f 315 1269 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (26)451 1289 w gsave newpath 488 1269 m /inpath true def 0.1 setgray 462 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 632 1233 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1233 m /inpath true def 0.1 setgray 10 R f 518 1102 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1102 m /inpath true def 0.1 setgray 10 R f 289 1269 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f 1941 329 347 347 De (27)2078 349 w gsave newpath 2115 329 m /inpath true def 0.4899 setgray 527 329 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (28)664 349 w gsave newpath 701 329 m /inpath true def 0.1 setgray 567 466 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 466 m /inpath true def 0.1 setgray 10 R f 848 329 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 329 m /inpath true def 0.1 setgray 10 R f 675 155 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 155 m /inpath true def 0.4901 setgray 10 R f 1755 630 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (29)1891 650 w gsave newpath 1928 630 m /inpath true def 0.1 setgray 1725 630 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 630 m /inpath true def 0.1 setgray 10 R f 2000 774 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 774 m /inpath true def 0.1 setgray 10 R f 1814 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.4901 setgray 10 R f 1650 1792 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (30)1786 1812 w gsave newpath 1823 1792 m /inpath true def 0.1 setgray 1647 1877 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1877 m /inpath true def 0.1 setgray 10 R f 1647 1703 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1703 m /inpath true def 0.1 setgray 10 R f 1948 1877 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1877 m /inpath true def 0.1 setgray 10 R f 1948 1703 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1703 m /inpath true def 0.1 setgray 10 R f 1797 1616 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1616 m /inpath true def 0.4901 setgray 10 R f 655 1197 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (31)792 1217 w gsave newpath 829 1197 m /inpath true def 0.1 setgray 632 1233 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1233 m /inpath true def 0.1 setgray 10 R f 802 1373 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1373 m /inpath true def 0.1 setgray 10 R f 688 1069 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1069 m /inpath true def 0.1 setgray 10 R f 976 1233 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1233 m /inpath true def 0.1 setgray 10 R f 920 1066 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f 1594 981 347 347 De (32)1731 1001 w 315 153 315 2139 Dl 2301 153 315 153 Dl 2301 2139 2301 153 Dl 315 2139 2301 2139 Dl 18 R f (32 disks)1 595 1 3503 2328 t 14 R f ( bonds)1 370( 61)1 245(m = 0.21308235294443)2 1342 3 2822 2490 t gsave newpath 3800 2470 m /inpath true def 0.7 setgray 10 R f 3182 1582 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (1)3355 1602 w gsave newpath 3355 1582 m /inpath true def 0.1 setgray 3155 1599 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1599 m /inpath true def 0.1 setgray 10 R f 3329 1756 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1756 m /inpath true def 0.1 setgray 10 R f 3329 1406 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1406 m /inpath true def 0.1 setgray 10 R f 3502 1599 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1599 m /inpath true def 0.4899 setgray 10 R f 4471 886 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (2)4645 906 w gsave newpath 4645 886 m /inpath true def 0.1 setgray 4618 1062 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1062 m /inpath true def 0.1 setgray 10 R f 4501 761 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 761 m /inpath true def 0.1 setgray 10 R f 4792 886 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 886 m /inpath true def 0.7 setgray 10 R f 3875 1599 347 347 De gsave eofill grestore 0 setgray stroke 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781 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 781 m /inpath true def 0.1 setgray 10 R f 3767 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.4899 setgray 10 R f 3879 1966 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (5)4052 1986 w gsave newpath 4052 1966 m /inpath true def 0.1 setgray 3849 1966 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 4173 1874 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1874 m /inpath true def 0.1 setgray 10 R f 4026 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.7 setgray 10 R f 3875 1235 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (6)4049 1255 w 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stroke grestore /inpath false def gsave newpath 514 1949 m /inpath true def 0.7 setgray 10 R f 4471 1599 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (8)4645 1619 w gsave newpath 4645 1599 m /inpath true def 0.1 setgray 4471 1690 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1690 m /inpath true def 0.1 setgray 10 R f 4471 1508 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1508 m /inpath true def 0.1 setgray 10 R f 4792 1599 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1599 m /inpath true def 0.4899 setgray 10 R f 4471 1235 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (9)4645 1255 w gsave newpath 4645 1235 m /inpath true def 0.1 setgray 4471 1328 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1328 m /inpath true def 0.1 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setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 3849 1609 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1609 m /inpath true def 0.4899 setgray 10 R f 2835 1616 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (13)2971 1636 w gsave newpath 3008 1616 m /inpath true def 0.1 setgray 2982 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 3155 1599 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1599 m /inpath true def 0.1 setgray 10 R f 2982 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 2809 1616 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1616 m /inpath true def 0.4901 setgray 10 R f 3885 630 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (14)4022 650 w gsave newpath 4059 630 m /inpath true def 0.1 setgray 3944 781 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 781 m /inpath true def 0.1 setgray 10 R f 4121 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 3944 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 4206 630 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 630 m /inpath true def 0.4901 setgray 10 R f 3185 634 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (15)3321 654 w gsave newpath 3358 634 m /inpath true def 0.1 setgray 3155 634 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave 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stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 3676 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.4899 setgray 10 R f 4471 1966 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (17)4608 1986 w gsave newpath 4645 1966 m /inpath true def 0.1 setgray 4471 1874 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1874 m /inpath true def 0.1 setgray 10 R f 4792 1966 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 4618 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.4901 setgray 10 R f 3355 329 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (18)3492 349 w gsave newpath 3529 329 m /inpath true def 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De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 781 m /inpath true def 0.1 setgray 10 R f 3938 1085 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1085 m /inpath true def 0.1 setgray 10 R f 3944 781 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 781 m /inpath true def 0.4901 setgray 10 R f 4062 329 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (23)4198 349 w gsave newpath 4235 329 m /inpath true def 0.1 setgray 4121 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 4294 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 4209 155 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 155 m /inpath true def 0.4899 setgray 10 R f 2835 1966 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (24)2971 1986 w gsave newpath 3008 1966 m /inpath true def 0.1 setgray 2982 1792 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1792 m /inpath true def 0.1 setgray 10 R f 3155 1949 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1949 m /inpath true def 0.1 setgray 10 R f 2809 1966 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1966 m /inpath true def 0.1 setgray 10 R f 2982 2139 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 2139 m /inpath true def 0.4901 setgray 10 R f 2979 951 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (25)3115 971 w gsave newpath 3152 951 m /inpath true def 0.1 setgray 3227 1091 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1091 m /inpath true def 0.1 setgray 10 R f 3054 1108 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1108 m /inpath true def 0.1 setgray 10 R f 3054 790 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 790 m /inpath true def 0.4901 setgray 10 R f 2835 1269 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (26)2971 1289 w gsave newpath 3008 1269 m /inpath true def 0.1 setgray 2982 1442 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1442 m /inpath true def 0.1 setgray 10 R f 3155 1252 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1252 m /inpath true def 0.1 setgray 10 R f 3054 1108 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1108 m /inpath true def 0.1 setgray 10 R f 2809 1269 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f 4468 352 347 347 De (27)4604 372 w gsave newpath 4641 352 m /inpath true def 0.4899 setgray 3008 329 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (28)3145 349 w gsave newpath 3182 329 m /inpath true def 0.1 setgray 3067 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 3329 329 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 329 m /inpath true def 0.1 setgray 10 R f 3155 155 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 155 m /inpath true def 0.4901 setgray 10 R f 4232 634 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (29)4369 654 w gsave newpath 4406 634 m /inpath true def 0.1 setgray 4294 479 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 479 m /inpath true def 0.1 setgray 10 R f 4206 630 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 630 m /inpath true def 0.1 setgray 10 R f 4501 761 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 761 m /inpath true def 0.7 setgray 10 R f 4173 1782 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (30)4310 1802 w gsave newpath 4347 1782 m /inpath true def 0.1 setgray 4173 1874 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1874 m /inpath true def 0.1 setgray 10 R f 4173 1690 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1690 m /inpath true def 0.1 setgray 10 R f 4471 1874 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1874 m /inpath true def 0.1 setgray 10 R f 4471 1690 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1690 m /inpath true def 0.4901 setgray 10 R f 3182 1233 347 347 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (31)3318 1253 w gsave newpath 3355 1233 m /inpath true def 0.1 setgray 3329 1406 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1406 m /inpath true def 0.1 setgray 10 R f 3155 1252 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1252 m /inpath true def 0.1 setgray 10 R f 3227 1091 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1091 m /inpath true def 0.1 setgray 10 R f 3502 1252 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 514 1252 m /inpath true def 0.1 setgray 10 R f 3417 1082 52 52 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f 4098 968 347 347 De (32)4234 988 w 2835 153 2835 2139 Dl 4821 153 2835 153 Dl 4821 2139 4821 153 Dl 2835 2139 4821 2139 Dl 18 R f (37 disks)1 595 1 973 5208 t 14 R f ( bonds)1 370( 77)1 280(m = 0.19642918462957)2 1342 3 274 5370 t gsave newpath 1270 5350 m /inpath true def 0.4901 setgray 10 R f 1311 3722 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (1)1474 3742 w gsave newpath 1474 3722 m /inpath true def 0.1 setgray 1284 3722 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3722 m /inpath true def 0.1 setgray 10 R f 1368 3862 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3862 m /inpath true def 0.1 setgray 10 R f 1531 3862 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3862 m /inpath true def 0.1 setgray 10 R f 1368 3579 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3579 m /inpath true def 0.1 setgray 10 R f 1548 3592 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3592 m /inpath true def 0.7 setgray 10 R f 804 4562 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (2)967 4582 w gsave newpath 967 4562 m /inpath true def 0.1 setgray 861 4702 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4702 m /inpath true def 0.1 setgray 10 R f 861 4419 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4419 m /inpath true def 0.1 setgray 10 R f 1031 4426 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4426 m /inpath true def 0.4901 setgray 10 R f 1931 3949 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (3)2094 3969 w gsave newpath 2094 3949 m /inpath true def 0.1 setgray 2094 4112 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 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4835 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4835 m /inpath true def 0.1 setgray 10 R f 3964 5009 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 5009 m /inpath true def 0.4901 setgray 10 R f 4154 4825 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (16)4280 4845 w gsave newpath 4317 4825 m /inpath true def 0.1 setgray 4127 4835 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4835 m /inpath true def 0.1 setgray 10 R f 4200 4689 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4689 m /inpath true def 0.1 setgray 10 R f 4453 4835 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4835 m /inpath true def 0.4901 setgray 10 R f 2827 4166 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (17)2953 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0 setgray stroke grestore /inpath false def gsave newpath 515 3755 m /inpath true def 0.1 setgray 10 R f 2800 3839 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3839 m /inpath true def 0.4901 setgray 10 R f 3354 4555 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (19)3480 4575 w gsave newpath 3517 4555 m /inpath true def 0.1 setgray 3400 4692 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4692 m /inpath true def 0.1 setgray 10 R f 3363 4449 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4449 m /inpath true def 0.1 setgray 10 R f 3564 4699 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4699 m /inpath true def 0.1 setgray 10 R f 3517 4392 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4392 m /inpath true def 0.7 setgray 10 R f 3927 3832 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (20)4054 3852 w gsave newpath 4091 3832 m /inpath true def 0.1 setgray 3930 3739 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3739 m /inpath true def 0.1 setgray 10 R f 3930 3929 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3929 m /inpath true def 0.1 setgray 10 R f 4200 3922 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3922 m /inpath true def 0.1 setgray 10 R f 4207 3752 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3752 m /inpath true def 0.4901 setgray 10 R f 3377 3482 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (21)3503 3502 w gsave newpath 3540 3482 m /inpath true def 0.1 setgray 3377 3579 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3579 m /inpath true def 0.1 setgray 10 R f 3410 3356 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3356 m /inpath true def 0.1 setgray 10 R f 3654 3565 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3565 m /inpath true def 0.1 setgray 10 R f 3570 3332 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3332 m /inpath true def 0.7 setgray 10 R f 2847 4846 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (22)2974 4866 w gsave newpath 3011 4846 m /inpath true def 0.1 setgray 3147 4835 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4835 m /inpath true def 0.1 setgray 10 R f 2974 4682 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4682 m /inpath true def 0.1 setgray 10 R f 2984 5009 53 53 De gsave eofill grestore 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53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3586 m /inpath true def 0.1 setgray 10 R f 4617 3342 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3342 m /inpath true def 0.1 setgray 10 R f 4794 3506 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3506 m /inpath true def 0.4901 setgray 10 R f 4220 4339 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (28)4347 4359 w gsave newpath 4384 4339 m /inpath true def 0.1 setgray 4234 4445 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4445 m /inpath true def 0.1 setgray 10 R f 4204 4285 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4285 m /inpath true def 0.1 setgray 10 R f 4494 4429 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4429 m 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/inpath true def 0.1 setgray 10 R f 4631 3995 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3995 m /inpath true def 0.1 setgray 10 R f 4794 4159 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4159 m /inpath true def 0.4901 setgray 10 R f 4494 3832 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (33)4620 3852 w gsave newpath 4657 3832 m /inpath true def 0.1 setgray 4631 3995 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3995 m /inpath true def 0.1 setgray 10 R f 4490 3749 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3749 m /inpath true def 0.1 setgray 10 R f 4631 3669 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3669 m /inpath true def 0.1 setgray 10 R f 4794 3832 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3832 m /inpath true def 0.4901 setgray 10 R f 3494 3179 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (34)3620 3199 w gsave newpath 3657 3179 m /inpath true def 0.1 setgray 3471 3202 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3202 m /inpath true def 0.1 setgray 10 R f 3570 3332 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3332 m /inpath true def 0.1 setgray 10 R f 3794 3179 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3179 m /inpath true def 0.1 setgray 10 R f 3630 3015 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3015 m /inpath true def 0.7 setgray 10 R f 3660 4022 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (35)3786 4042 w gsave newpath 3823 4022 m /inpath true def 0.1 setgray 3674 4126 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4126 m /inpath true def 0.1 setgray 10 R f 3664 3929 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3929 m /inpath true def 0.1 setgray 10 R f 3924 4126 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4126 m /inpath true def 0.1 setgray 10 R f 3930 3929 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3929 m /inpath true def 0.4901 setgray 10 R f 4144 3229 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (36)4270 3249 w gsave newpath 4307 3229 m /inpath true def 0.1 setgray 4120 3202 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3202 m /inpath true def 0.1 setgray 10 R f 4181 3356 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3356 m /inpath true def 0.1 setgray 10 R f 4443 3202 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 3202 m /inpath true def 0.7 setgray 10 R f 4494 4519 326 326 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (37)4620 4539 w gsave newpath 4657 4519 m /inpath true def 0.1 setgray 4623 4682 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4682 m /inpath true def 0.1 setgray 10 R f 4494 4429 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 515 4429 m /inpath true def 0.1 setgray 10 R f 4794 4519 53 53 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f 2827 3016 2827 5009 Dl 4820 3015 2827 3015 Dl 4821 5008 4821 3015 Dl 2828 5009 4821 5009 Dl cleartomark showpage saveobj restore end DpostDict begin done end %%EndDocument @endspecial 203 x(Figure)29 b(2.1:)42 b(Impro)m(v)m(ed)31 b(pac)m(kings)g(\(left\))e(and)h(previous)h(record)f(pac)m(kings)h (\(righ)m(t\))e(of)h(32)f(and)i(37)e(disks)-60 5499 y(in)j(a)g(square)p eop %%Page: 3 3 3 2 bop -60 250 a Fe(the)37 b(electr)n(onic)g(journal)i(of)f(combina)-7 b(torics)39 b(\(2000\),#R46)1084 b Fk(3)-60 5176 y @beginspecial 10 @llx 241 @lly 489 @urx 789 @ury 4680 @rwi 5400 @rhi @clip @setspecial %%BeginDocument: bdgl2.ps /DpostDict 200 dict def DpostDict begin % % Copyright (c) 1993 AT&T, All Rights Reserved % % Version 3.4 prologue for troff files. % /#copies 1 store /Prologue (dpost.ps) def /aspectratio 1 def /formsperpage 1 def /landscape false def /linewidth .3 def /magnification 1 def /margin 0 def /orientation 0 def /resolution 720 def /rotation 1 def /xoffset 0 def /yoffset 0 def /roundpage true def /useclippath true def /pagebbox [0 0 612 792] def /R /Times-Roman def /I /Times-Italic def /B /Times-Bold def /BI /Times-BoldItalic def /H /Helvetica def /HI /Helvetica-Oblique def /HB /Helvetica-Bold def /HX /Helvetica-BoldOblique def /CW /Courier def /CO /Courier def /CI /Courier-Oblique def /CB /Courier-Bold def /CX /Courier-BoldOblique def /PA /Palatino-Roman def /PI /Palatino-Italic def /PB /Palatino-Bold def /PX /Palatino-BoldItalic def /Hr /Helvetica-Narrow def /Hi /Helvetica-Narrow-Oblique def /Hb /Helvetica-Narrow-Bold def /Hx /Helvetica-Narrow-BoldOblique def /KR /Bookman-Light def /KI /Bookman-LightItalic def /KB /Bookman-Demi def /KX /Bookman-DemiItalic def /AR /AvantGarde-Book def /AI /AvantGarde-BookOblique def /AB /AvantGarde-Demi def /AX /AvantGarde-DemiOblique def /NR /NewCenturySchlbk-Roman def /NI /NewCenturySchlbk-Italic def /NB /NewCenturySchlbk-Bold def /NX /NewCenturySchlbk-BoldItalic def /ZD /ZapfDingbats def /ZI /ZapfChancery-MediumItalic def /S /S def /S1 /S1 def /GR /Symbol def /inch {72 mul} bind def /min {2 copy gt {exch} if pop} bind def /setup { counttomark 2 idiv {def} repeat pop landscape {/orientation 90 orientation add def} if /scaling 72 resolution div def linewidth setlinewidth 1 setlinecap pagedimensions xcenter ycenter translate orientation rotation mul rotate width 2 div neg height 2 div translate xoffset inch yoffset inch neg translate margin 2 div dup neg translate magnification dup aspectratio mul scale scaling scaling scale addmetrics 0 0 moveto } def /pagedimensions { % useclippath userdict /gotpagebbox known not and { % /pagebbox [clippath pathbbox newpath] def % roundpage currentdict /roundpagebbox known and {roundpagebbox} if % } if pagebbox aload pop 4 -1 roll exch 4 1 roll 4 copy landscape {4 2 roll} if sub /width exch def sub /height exch def add 2 div /xcenter exch def add 2 div /ycenter exch def userdict /gotpagebbox true put } def /landscapepage { landscape not { 0 height scaling div neg translate % not quite 90 rotate } if } bind def /portraitpage { landscape { width scaling div 0 translate % not quite -90 rotate } if } bind def /addmetrics { /Symbol /S null Sdefs cf /Times-Roman /S1 StandardEncoding dup length array copy S1defs cf } def /pagesetup { /page exch def currentdict /pagedict known currentdict page known and { page load pagedict exch get cvx exec } if } def /decodingdefs [ {counttomark 2 idiv {y moveto show} repeat} {neg /y exch def counttomark 2 idiv {y moveto show} repeat} {neg moveto {2 index stringwidth pop sub exch div 0 32 4 -1 roll widthshow} repeat} {neg moveto {spacewidth sub 0.0 32 4 -1 roll widthshow} repeat} {counttomark 2 idiv {y moveto show} repeat} {neg setfunnytext} ] def /setdecoding {/t decodingdefs 3 -1 roll get bind def} bind def /w {neg moveto show} bind def /m {neg dup /y exch def moveto} bind def /done {/lastpage where {pop lastpage} if} def /f { dup /font exch def findfont exch dup /ptsize exch def scaling div dup /size exch def scalefont setfont linewidth ptsize mul scaling 10 mul div setlinewidth /spacewidth ( ) stringwidth pop def } bind def /changefont { /fontheight exch def /fontslant exch def currentfont [ 1 0 fontheight ptsize div fontslant sin mul fontslant cos div fontheight ptsize div 0 0 ] makefont setfont } bind def /sf {f} bind def /cf { dup length 2 idiv /entries exch def /chtab exch def /newencoding exch def /newfont exch def findfont dup length 1 add dict /newdict exch def {1 index /FID ne {newdict 3 1 roll put}{pop pop} ifelse} forall newencoding type /arraytype eq {newdict /Encoding newencoding put} if newdict /Metrics entries dict put newdict /Metrics get begin chtab aload pop 1 1 entries {pop def} for newfont newdict definefont pop end } bind def % % A few arrays used to adjust reference points and character widths in some % of the printer resident fonts. If square roots are too high try changing % the lines describing /radical and /radicalex to, % % /radical [0 -75 550 0] % /radicalex [-50 -75 500 0] % % Move braceleftbt a bit - default PostScript character is off a bit. % /Sdefs [ /bracketlefttp [201 500] /bracketleftbt [201 500] /bracketrighttp [-81 380] /bracketrightbt [-83 380] /braceleftbt [203 490] /bracketrightex [220 -125 500 0] /radical [0 0 550 0] /radicalex [-50 0 500 0] /parenleftex [-20 -170 0 0] /integral [100 -50 500 0] /infinity [10 -75 730 0] ] def /S1defs [ /underscore [0 80 500 0] /endash [7 90 650 0] ] def end DpostDict begin mark /rotation 1 def /gotpagebbox true def /resolution 720 def setup 2 setdecoding % % Copyright (c) 1993 AT&T, All Rights Reserved % % Version 3.4 drawing procedures for dpost. Automatically pulled in when % needed. % /inpath false def /savematrix matrix def /Dl { inpath {pop pop neg lineto} {newpath neg moveto neg lineto stroke} ifelse } bind def /De { /y1 exch 2 div def /x1 exch 2 div def /savematrix savematrix currentmatrix def neg exch x1 add exch translate x1 y1 scale 0 0 1 0 360 inpath {1 0 moveto arc savematrix setmatrix} {newpath arc savematrix setmatrix stroke} ifelse } bind def /Da { /dy2 exch def /dx2 exch def /dy1 exch def /dx1 exch def dy1 add neg exch dx1 add exch dx1 dx1 mul dy1 dy1 mul add sqrt dy1 dx1 neg atan dy2 neg dx2 atan inpath {arc} {newpath arc stroke} ifelse } bind def /DA { /dy2 exch def /dx2 exch def /dy1 exch def /dx1 exch def dy1 add neg exch dx1 add exch dx1 dx1 mul dy1 dy1 mul add sqrt dy1 dx1 neg atan dy2 neg dx2 atan inpath {arcn} {newpath arcn stroke} ifelse } bind def /Ds { /y2 exch def /x2 exch def /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 5 x1 mul add 6 div y0 5 y1 mul add -6 div x2 5 x1 mul add 6 div y2 5 y1 mul add -6 div x1 x2 add 2 div y1 y2 add -2 div inpath {curveto} {newpath x0 x1 add 2 div y0 y1 add -2 div moveto curveto stroke} ifelse } bind def % % Copyright (c) 1993 AT&T, All Rights Reserved % % Color and reverse video support for dpost. Only defines a default % colordict dictionary if there's not already a definition. Means the % definitions in a private color dictionary included with -C always % win. % currentdict /colordict known not { /colordict 50 dict dup begin /red { 1 0 0 } def /green { 0 1 0 } def /blue { 0 0 1 } def /cyan { 0 1 1 } def /magenta { 1 0 1 } def /yellow { 1 1 0 } def /white { 1 1 1 } def /black { 0 0 0 } def end def } if /rgb {setrgbcolor} bind def /hsb {sethsbcolor} bind def /setcolor { counttomark 1 eq { dup colordict exch known not {pop /black} if colordict exch get exec setrgbcolor } if counttomark 2 eq { /backcolor exch def /textcolor exch def colordict backcolor known not colordict textcolor known not or { /backcolor colordict /black get def /textcolor colordict /white get def } if /backcolor colordict backcolor get def /textcolor colordict textcolor get def /dY1 0 def /dY2 0 def textcolor exec setrgbcolor } if } bind def /drawrvbox { /x2 exch def /x1 exch def currentpoint dup /y1 exch def /y2 exch def pop dY1 0 eq dY2 0 eq and { currentfont /FontBBox get aload pop currentfont /FontMatrix get dtransform /dY2 exch def pop currentfont /FontMatrix get dtransform /dY1 exch def pop } if /y1 y1 dY1 add def /y2 y2 dY2 add def backcolor exec setrgbcolor newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath fill textcolor exec setrgbcolor } bind def end DpostDict begin /saveobj save def mark 1 pagesetup 18 R f (48 disks)1 595 1 957 2288 t 14 R f ( bonds)1 370( 111)1 315(m = 0.16940542937029)2 1342 3 241 2450 t gsave newpath 1254 2430 m /inpath true def 0.4901 setgray 10 R f 1008 1462 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (1)1146 1482 w gsave newpath 1146 1462 m /inpath true def 0.1 setgray 1021 1557 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1557 m /inpath true def 0.1 setgray 10 R f 1002 1393 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1393 m /inpath true def 0.1 setgray 10 R f 1244 1530 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1530 m /inpath true def 0.1 setgray 10 R f 1244 1393 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1393 m /inpath true def 0.1 setgray 10 R f 1123 1321 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1321 m /inpath true def 0.4901 setgray 10 R f 1751 1654 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (2)1889 1674 w gsave newpath 1889 1654 m /inpath true def 0.1 setgray 1735 1697 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1697 m /inpath true def 0.1 setgray 10 R f 1836 1792 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1792 m /inpath true def 0.1 setgray 10 R f 1970 1563 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1563 m /inpath true def 0.4901 setgray 10 R f 1247 1321 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (3)1385 1341 w gsave newpath 1385 1321 m /inpath true def 0.1 setgray 1362 1462 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1462 m /inpath true def 0.1 setgray 10 R f 1244 1393 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1393 m /inpath true def 0.1 setgray 10 R f 1362 1184 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1184 m /inpath true def 0.1 setgray 10 R f 1244 1252 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1252 m /inpath true def 0.1 setgray 10 R f 1483 1393 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1393 m /inpath true def 0.1 setgray 10 R f 1483 1252 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1252 m /inpath true def 0.4901 setgray 10 R f 324 441 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (4)462 461 w gsave newpath 462 441 m /inpath true def 0.1 setgray 439 578 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 578 m /inpath true def 0.1 setgray 10 R f 560 509 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 509 m /inpath true def 0.1 setgray 10 R f 557 365 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 365 m /inpath true def 0.1 setgray 10 R f 301 441 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 441 m /inpath true def 0.4901 setgray 10 R f 1728 1321 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (5)1866 1341 w gsave newpath 1866 1321 m /inpath true def 0.1 setgray 1722 1393 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1393 m /inpath true def 0.1 setgray 10 R f 1722 1252 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1252 m /inpath true def 0.1 setgray 10 R f 1957 1396 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1396 m /inpath true def 0.1 setgray 10 R f 1957 1246 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1246 m /inpath true def 0.4901 setgray 10 R f 766 1046 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (6)903 1066 w gsave newpath 903 1046 m /inpath true def 0.1 setgray 763 1114 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1114 m /inpath true def 0.1 setgray 10 R f 779 951 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 951 m /inpath true def 0.1 setgray 10 R f 881 1184 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1184 m /inpath true def 0.1 setgray 10 R f 1002 1114 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1114 m /inpath true def 0.1 setgray 10 R f 1002 974 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 974 m /inpath true def 0.4901 setgray 10 R f 527 1184 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (7)665 1204 w gsave newpath 665 1184 m /inpath true def 0.1 setgray 540 1278 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1278 m /inpath true def 0.1 setgray 10 R f 540 1089 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1089 m /inpath true def 0.1 setgray 10 R f 763 1252 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1252 m /inpath true def 0.1 setgray 10 R f 763 1114 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1114 m /inpath true def 0.4901 setgray 10 R f 1008 905 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (8)1146 925 w gsave newpath 1146 905 m /inpath true def 0.1 setgray 1002 974 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 974 m /inpath true def 0.1 setgray 10 R f 1021 814 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 814 m /inpath true def 0.1 setgray 10 R f 1244 974 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 974 m /inpath true def 0.1 setgray 10 R f 1123 1046 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1046 m /inpath true def 0.1 setgray 10 R f 1237 830 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 830 m /inpath true def 0.4901 setgray 10 R f 563 1510 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (9)701 1530 w gsave newpath 701 1510 m /inpath true def 0.1 setgray 678 1648 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1648 m /inpath true def 0.1 setgray 10 R f 560 1580 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1580 m /inpath true def 0.1 setgray 10 R f 560 1442 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1442 m /inpath true def 0.1 setgray 10 R f 799 1580 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1580 m /inpath true def 0.1 setgray 10 R f 779 1415 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1415 m /inpath true def 0.7 setgray 10 R f 557 290 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (10)657 310 w gsave newpath 694 290 m /inpath true def 0.1 setgray 557 365 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 365 m /inpath true def 0.1 setgray 10 R f 792 358 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 358 m /inpath true def 0.1 setgray 10 R f 671 153 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 153 m /inpath true def 0.4901 setgray 10 R f 324 1373 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (11)425 1393 w gsave newpath 462 1373 m /inpath true def 0.1 setgray 439 1510 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1510 m /inpath true def 0.1 setgray 10 R f 560 1442 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1442 m /inpath true def 0.1 setgray 10 R f 540 1278 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1278 m /inpath true def 0.1 setgray 10 R f 301 1373 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1373 m /inpath true def 0.4901 setgray 10 R f 805 1926 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (12)906 1946 w gsave newpath 943 1926 m /inpath true def 0.1 setgray 799 1858 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1858 m /inpath true def 0.1 setgray 10 R f 920 1789 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1789 m /inpath true def 0.1 setgray 10 R f 1038 1858 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1858 m /inpath true def 0.1 setgray 10 R f 920 2063 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f 1934 297 275 275 De (13)2035 317 w gsave newpath 2072 297 m /inpath true def 0.4901 setgray 563 856 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (14)664 876 w gsave newpath 701 856 m /inpath true def 0.1 setgray 560 925 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 925 m /inpath true def 0.1 setgray 10 R f 560 788 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 788 m /inpath true def 0.1 setgray 10 R f 678 718 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 718 m /inpath true def 0.1 setgray 10 R f 779 951 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 951 m /inpath true def 0.1 setgray 10 R f 799 788 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 788 m /inpath true def 0.4901 setgray 10 R f 1277 428 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (15)1377 448 w gsave newpath 1414 428 m /inpath true def 0.1 setgray 1270 500 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 500 m /inpath true def 0.1 setgray 10 R f 1270 358 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 358 m /inpath true def 0.1 setgray 10 R f 1493 522 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 522 m /inpath true def 0.1 setgray 10 R f 1512 358 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 358 m /inpath true def 0.4901 setgray 10 R f 324 1926 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (16)425 1946 w gsave newpath 462 1926 m /inpath true def 0.1 setgray 560 1858 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1858 m /inpath true def 0.1 setgray 10 R f 439 1789 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1789 m /inpath true def 0.1 setgray 10 R f 301 1926 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1926 m /inpath true def 0.1 setgray 10 R f 439 2063 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 2063 m /inpath true def 0.4901 setgray 10 R f 1008 1184 275 275 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (17)1109 1204 w gsave newpath 1146 1184 m /inpath true def 0.1 setgray 1002 1252 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1252 m /inpath true def 0.1 setgray 10 R f 1002 1114 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 1114 m /inpath true def 0.1 setgray 10 R f 1244 1252 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 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914 4206 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4206 m /inpath true def 0.1 setgray 10 R f 1051 4446 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4446 m /inpath true def 0.1 setgray 10 R f 1051 4206 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4206 m /inpath true def 0.4899 setgray 10 R f 1977 3533 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (15)2081 3553 w gsave newpath 2118 3533 m /inpath true def 0.1 setgray 2094 3670 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3670 m /inpath true def 0.1 setgray 10 R f 1998 3433 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3433 m /inpath true def 0.1 setgray 10 R f 2234 3533 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 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0 setgray stroke grestore /inpath false def gsave newpath 485 3963 m /inpath true def 0.1 setgray 10 R f 1051 3723 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3723 m /inpath true def 0.7 setgray 10 R f 1977 4323 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (18)2081 4343 w gsave newpath 2118 4323 m /inpath true def 0.1 setgray 2028 4443 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4443 m /inpath true def 0.1 setgray 10 R f 2041 4193 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4193 m /inpath true def 0.1 setgray 10 R f 2234 4323 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4323 m /inpath true def 0.7 setgray 10 R f 1781 3336 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (19)1884 3356 w gsave newpath 1921 3336 m /inpath 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def gsave newpath 485 4206 m /inpath true def 0.1 setgray 10 R f 1607 4446 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4446 m /inpath true def 0.1 setgray 10 R f 1607 4206 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4206 m /inpath true def 0.7 setgray 10 R f 727 4566 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (21)831 4586 w gsave newpath 868 4566 m /inpath true def 0.1 setgray 774 4686 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4686 m /inpath true def 0.1 setgray 10 R f 774 4446 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4446 m /inpath true def 0.1 setgray 10 R f 914 4686 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4686 m /inpath true def 0.1 setgray 10 R f 914 4446 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4446 m /inpath true def 0.7 setgray 10 R f 1641 3576 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (22)1744 3596 w gsave newpath 1781 3576 m /inpath true def 0.1 setgray 1661 3476 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3476 m /inpath true def 0.1 setgray 10 R f 1787 3713 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3713 m /inpath true def 0.1 setgray 10 R f 1828 3456 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3456 m /inpath true def 0.7 setgray 10 R f 1561 4566 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (23)1664 4586 w gsave newpath 1701 4566 m /inpath true def 0.1 setgray 1607 4686 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4686 m /inpath true def 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gsave newpath 485 4443 m /inpath true def 0.4899 setgray 10 R f 311 4806 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (25)414 4826 w gsave newpath 451 4806 m /inpath true def 0.1 setgray 568 4806 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4806 m /inpath true def 0.1 setgray 10 R f 494 4686 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4686 m /inpath true def 0.1 setgray 10 R f 288 4806 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4806 m /inpath true def 0.1 setgray 10 R f 427 4946 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 4946 m /inpath true def 0.7 setgray 10 R f 1144 3846 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (26)1247 3866 w gsave newpath 1284 3846 m /inpath true def 0.1 setgray 1331 3966 46 46 De gsave 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f 844 3380 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (40)947 3400 w gsave newpath 984 3380 m /inpath true def 0.1 setgray 861 3476 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3476 m /inpath true def 0.1 setgray 10 R f 891 3259 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3259 m /inpath true def 0.1 setgray 10 R f 1041 3493 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3493 m /inpath true def 0.1 setgray 10 R f 1028 3259 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave newpath 485 3259 m /inpath true def 0.4901 setgray 10 R f 584 3846 280 280 De gsave eofill grestore 0 setgray stroke grestore /inpath false def 10 R f (41)687 3866 w gsave newpath 724 3846 m /inpath true def 0.1 setgray 617 3956 46 46 De gsave eofill grestore 0 setgray stroke grestore /inpath false def gsave 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y(and)33 b(the)g(examination)d(of)i (an)h(exp)s(onen)m(tially)e(increasing)h(n)m(um)m(b)s(er)h(of)f(cases.) 86 798 y(Before)26 b(pro)m(ving)f(a)f(displa)m(y)m(ed)i(pac)m(king)f (is)g(optimal,)f(it)g(is)h(\014rst)g(necessary)j(to)d(pro)m(v)m(e)h (that)f(it)f(actually)-60 918 y(exists!)66 b(That)40 b(is,)h(that)e(there)i(is)e(a)g(geometrical)f(con\014guration)g (corresp)s(onding)i(to)f(the)i(p)s(oin)m(ts)e(and)-60 1038 y(edges)g(of)e(a)h(giv)m(en)g(diagram)e(in)h(whic)m(h)h(all)e(in)m (ter-cen)m(ter)i(distances)h(are)f(at)f(least)h(one)g(disk)g(diameter) -60 1159 y(and)45 b(cen)m(ter-b)s(oundary)i(distances)f(are)f(at)f (least)h(a)g(disk)g(radius.)81 b(F)-8 b(or)44 b(example,)k(existence)f (of)e(the)-60 1279 y(pac)m(kings)37 b(in)e(Figures)h(2.1)g(and)g(2.2,)h (is)e(not)h(ob)m(vious)h(\(to)f(us!\))55 b(from)35 b(just)h(lo)s(oking) e(at)i(the)h(diagrams.)-60 1399 y(\(F)-8 b(or)35 b(simplicit)m(y)f(of)i (discussion,)i(w)m(e)f(will)d(assume)j(for)f(no)m(w)h(that)f(all)f 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2137 y Ff(i)1755 2122 y Fk(and)g Fg(y)1992 2137 y Ff(i)2052 2122 y Fk(expressing)g(the)h(v)-5 b(arious)31 b(b)s(onds)h(\(or)f(edges\))i(in)-60 2242 y(the)d(diagram.)40 b(W)-8 b(e)30 b(can)g(then)g(eliminate)d(all)g(the)j(v)-5 b(ariables)28 b(except)j(one)f(\(sa)m(y)g Fg(x)2936 2257 y Fc(1)2976 2242 y Fk(\))g(from)e(the)i(equations)-60 2363 y(\(using)f(resultan)m(ts\),)h(to)f(obtain)f(a)h(single)f(p)s (olynomial)e(equation)j Fg(P)14 b Fk(\()p Fg(x)2568 2378 y Fc(1)2607 2363 y Fk(\))28 b(=)f(0)i(whic)m(h)h(m)m(ust)f(b)s(e)g (satis\014ed)-60 2483 y(b)m(y)34 b(an)m(y)f(solution)e(\()p Fg(x)722 2498 y Fc(1)762 2483 y Fg(;)17 b(y)854 2498 y Fc(1)893 2483 y Fg(;)g(:::;)g(x)1117 2498 y Ff(n)1164 2483 y Fg(;)g(y)1256 2498 y Ff(n)1302 2483 y Fk(\))33 b(of)f(the)h(original)d(set)j(of)f(equations.)45 b(W)-8 b(e)33 b(can)g(then)g(express)i(eac)m(h)-60 2603 y Fg(z)-15 2618 y Ff(i)46 2603 y Fk(in)c(terms)h(of)g Fg(x)596 2618 y Fc(1)635 2603 y Fk(,)h(and)f(\014nally)f(b)s(ound)h(the)h(distances)g 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y([GL]\))e(ha)m(v)m(e)i(also)e(b)s(een)h(found)g(b)m(y)g(the)g(new)h (com)m(bination)d(algorithm,)e(whic)m(h)k(also)f(ran)h(more)f(quic)m (kly)-60 4778 y(in)h(these)i(cases)g(than)e(the)h(\\billiards")c (algorithm)h(alone.)-60 5049 y Fh(5.)84 b(The)38 b(new)g(pac)m(king)f (algorithm)127 5268 y Fk(W)-8 b(e)42 b(describ)s(e)f(the)h(algorithm)c (of)i(Phase)j(1)d(in)h(a)f(form)g(applicable)f(for)i(pac)m(king)g (equal)g(spheres)-60 5388 y(in)32 b(v)-5 b(ariously-shap)s(ed)33 b(b)s(ounded)h(regions)e(of)h(euclidean)g Fg(n)p Fk(-space.)46 b(Our)33 b(computer)g(realization)e(of)h(this)-60 5509 y(pro)s(cedure)37 b(pac)m(ks)g(equal)f(disks)g(in)g(rectangles,)h (equilateral)d(triangles,)h(or)g(circles.)53 b(W)-8 b(e)37 b(ha)m(v)m(e)g(used)g(it)-60 5629 y(primarily)29 b(for)j(pac)m(king)h (disks)g(in)f(squares.)p eop %%Page: 7 7 7 6 bop -60 250 a Fe(the)37 b(electr)n(onic)g(journal)i(of)f(combina)-7 b(torics)39 b(\(2000\),#R46)1084 b Fk(7)59 557 y(1.)49 b(Arbitrarily)44 b(initialize)f(the)k(cen)m(ter)h(p)s(osition)d(of)i (eac)m(h)h(sphere)g(so)f(that)f(the)i(spheres)g(do)f(not)184 677 y(in)m(tersect)h(while)f(eac)m(h)i(has)f(a)f(unit)g(radius.)88 b(Arbitrarily)45 b(initialize)f(a)j(direction)g(of)g(motion)184 798 y(for)37 b(eac)m(h)i(sphere.)60 b(\(In)38 b(our)f(pac)m (king-disks-in-a-square)g(exp)s(erimen)m(ts,)j(w)m(e)e(usually)f (randomly)184 918 y(scatter)f(the)h(disks)f(in)f(a)h(su\016cien)m(tly)g (large)f(square)i(and)f(for)f(a)h(disk)g(cen)m(tered)i(at)d Fg(a)p Fk(,)i(w)m(e)g(set)f(its)184 1038 y(initial)29 b(motion)h(direction)i(to)g(b)s(e)h Fd(\000)p Fg(a)p Fk(.\))59 1235 y(2.)49 b(Determine)40 b(the)i(smallest)d(size)j(b)s (ounding)e(shap)s(e)i(that)f(con)m(tains)h(all)d(spheres.)71 b(\(Sp)s(eci\014cally)-8 b(,)184 1355 y(w)m(e)34 b(determine)g(the)g (smallest)e(b)s(ounding)h(square)h(for)f(all)f(disks.)47 b(Without)33 b(losing)f(pac)m(king)h(p)s(er-)184 1476 y(formance,)e(the)i(square)g(size)f(minimization)27 b(is)k(done)i(more) 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b(the)g(spheres)i(sequen)m(tially)e(without)f (rep)s(etition)g(in)g(some)h(order)g(and)g(attempt)f(to)h(mo)m(v)m(e) 184 2864 y(eac)m(h)k(sphere)h(a)e(distance)h Fg(s)f Fk(from)f(its)h (curren)m(t)i(p)s(osition)c(using)j(pro)s(cedure)g Fg(mov)t(e)p 3308 2864 30 4 v 35 w(attempt)h Fk(de-)184 2984 y(tailed)c(b)s(elo)m (w.)54 b(Eac)m(h)37 b Fg(mov)t(e)p 1263 2984 V 35 w(attempt)g Fk(for)f(a)g(sphere)h(returns)g(either)f Fg(success)f Fk(or)h Fg(f)11 b(ail)r(ur)s(e)p Fk(.)54 b(\(In)184 3105 y(our)39 b(exp)s(erimen)m(ts,)j(the)e(order)f(of)g(sphere)i(selection)e (app)s(ears)g(non-essen)m(tial.)64 b(W)-8 b(e)40 b(set)g(the)g(se-)184 3225 y(lection)33 b(order)j(randomly)d(at)i(the)g(start)g(of)f(the)h (pro)s(cedure)h(and)f(randomly)f(reset)i(it)e(eac)m(h)h(time)184 3345 y(when)f(resetting)e Fg(s)h Fk(in)e(Step)j(8)e(b)s(elo)m(w.\))59 3542 y(7.)49 b(Determine)33 b(the)h(smallest)e(size)i(b)s(ounding)g (shap)s(e)g(that)g(con)m(tains)g(all)d(spheres.)50 b(If)34 b(the)g(new)h(size)184 3662 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Fk(.)85 4742 y Fd(\017)49 b Fk(A)m(ttempt)36 b(to)g(mo)m(v)m(e)g(the)h(sphere)g(a)f (distance)g Fg(s)g Fk(along)f(its)g(previously)i(c)m(hosen)g (direction.)53 b(\(The)184 4863 y(initial)29 b(direction)i(has)i(b)s (een)g(c)m(hosen)i(in)c(Step)j(1)e(of)g(the)h(main)e(pro)s(cedure.\))85 5059 y Fd(\017)49 b Fk(If)31 b(in)g(the)g(new)i(p)s(osition)c(the)j (sphere)h(do)s(es)e(not)g(in)m(tersect)i(other)e(spheres)i(and)f(do)s (es)f(not)g(extend)184 5179 y(b)s(ey)m(ond)40 b(the)f(region)e(b)s (oundary)-8 b(,)41 b(then)e Fg(mov)t(e)p 1929 5179 V 35 w(attempt)h Fk(terminates)e(after)g(accepting)h(the)g(new)184 5300 y(sphere)34 b(p)s(osition)d(and)h(rep)s(orting)g Fg(success)g Fk(to)g(the)h(calling)d(pro)s(cedure.)85 5496 y Fd(\017)49 b Fk(Otherwise)38 b(start)g(with)f(the)h(old)f (sphere)i(p)s(osition)d(again)g(and)i(reset)g(the)g(direction)f(of)g (motion)184 5617 y(using)32 b(the)i(pro)s(cedure)f Fg(r)s(eset)p 1284 5617 V 36 w(dir)s(ection)g Fk(detailed)f(b)s(elo)m(w)g(and)h 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