{VERSION 2 3 "IBM INTEL NT" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 183 0 
0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 164 0 0 0 1 0 0 0 
0 0 0 }{CSTYLE "" -1 260 "" 0 1 184 0 108 0 0 0 1 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "System" 0 1 0 0 116 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 262 "" 0 1 109 101 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Syst
em" 0 1 120 111 114 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "System" 0 
1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "System" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 
-1 268 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "Syst
em" 0 1 0 0 4 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 
0 1 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 120 0 0 1 0 0 0 0 0 0 0 
}{CSTYLE "" -1 272 "" 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE 
"" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 
0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 
0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
256 41 "                               Gray codes" }}{PARA 0 "" 0 "" 
{TEXT -1 127 "                                                        \+
                      by David Joyner and  Jim McShea, Math Dept, USNA
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 " Bin
ary Gray codes have several applications:" }}{PARA 0 "" 0 "" {TEXT -1 
30 "* solving puzzles such as the " }{TEXT 271 13 "Tower of Hano" }
{TEXT -1 10 "i and the " }{TEXT 272 5 "Brain" }{TEXT -1 5 " [G]," }}
{PARA 0 "" 0 "" {TEXT -1 46 "* analog-digital-converters (goniometers)
 [S]," }}{PARA 0 "" 0 "" {TEXT -1 85 "* Hamiltonian circuits in hyperc
ubes [Gil] and Cayley graphs of Coxeter groups [CSW]," }}{PARA 0 "" 0 
"" {TEXT -1 45 "* capanology (the study of bell-ringing) [W]," }}
{PARA 0 "" 0 "" {TEXT -1 39 "* continuous space-filling curves [Gi]." 
}}{PARA 0 "" 0 "" {TEXT -1 33 "This worksheet and the text file " }
{TEXT 266 8 "gray.mpl" }{TEXT -1 609 " contains four algorithms for ge
nerating a binary Gray code (essentially the \"reflected\" binary Gray
 code) of length n. One is the binary-to-reflected Gray code conversio
n [G]. Another takes advantage of the periodicity of the reflected Gra
y code in each coordinate. The third uses a relationship between the i
-th codeword and the 2i-th codeword, which as far as we know is new. T
he third one is perhaps the fastest of these three but requires that t
he table of previous codewords be placed in memory. This package also \+
contains some plotting commands for visualizing the \"size\" of each c
odeword in the code. " }}{PARA 0 "" 0 "" {TEXT -1 270 " The fourth alg
orithm to produce Gray codes but this algorithm will actually produce \+
an m-ary (not just binary) Gray code of length n. It is compact and re
latively fast. Though discovered independently, this appears to be the
 same as the first algorithm of M. C. Er [E]." }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 18 "gray0.mws, 5-17-97" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "read(`d:/maplev4/share/games
/gray/gray.mpl`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "The binary G
ray code of length n has 2^n binary vectors. The algorithm in " }
{TEXT 261 4 "gray" }{TEXT -1 66 " uses a relationship between the i-th
 vector and the 2i-th vector." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "for i from 0 to 15 do gray(i,4); od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&\"\"!F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"
\"\"\"!F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"F$\"\"!F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!\"\"\"F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&\"\"!\"\"\"F%F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&
\"\"\"F$F$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"\"!F$F%" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$\"\"\"F$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7&\"\"!F$\"\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7&\"\"\"\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"F$F$F$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!\"\"\"F%F%" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7&\"\"!\"\"\"F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7&\"\"\"F$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"\"!F%F$
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 263 10 "gray_coord
" }{TEXT -1 104 " lists the Gray codewords coordinatewise. The first 3
0 Gray codewords of length 5 are obtained from the " }{TEXT 262 7 "col
umns" }{TEXT -1 25 " of the following output:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "for i from 1 to 5 do gray_coord(i,30); od;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7@\"\"!\"\"\"F%F$F$F%F%F$F$F%F%F$F$F%F
%F$F$F%F%F$F$F%F%F$F$F%F%F$F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7@
\"\"!F$\"\"\"F%F%F%F$F$F$F$F%F%F%F%F$F$F$F$F%F%F%F%F$F$F$F$F%F%F%F%" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7@\"\"!F$F$F$\"\"\"F%F%F%F%F%F%F%F$F$
F$F$F$F$F$F$F%F%F%F%F%F%F%F%F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7@
\"\"!F$F$F$F$F$F$F$\"\"\"F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F$F$F$F$F$F$" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7@\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$
\"\"\"F%F%F%F%F%F%F%F%F%F%F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 
10 "gray_2adic" }{TEXT -1 82 " takes advantage of the periodicity of t
he reflected Gray code in each coordinate:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "for i from 1 to 16 do gray_2adic(4,i); od;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&\"\"\"\"\"!F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&
\"\"\"F$\"\"!F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!\"\"\"F$F$" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!\"\"\"F%F$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7&\"\"\"F$F$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7&\"\"\"\"\"!F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$\"\"\"F$
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$\"\"\"F%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7&\"\"\"\"\"!F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7&\"\"\"F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!\"\"\"F%F%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!\"\"\"F$F%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7&\"\"\"F$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7&\"\"\"\"\"!F%F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$\"
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "The coordinate (x,2^(y+1
)) of the plot below signifies that the y-th peg should be moved at th
e x-th step of the " }{TEXT 267 5 "Brain" }{TEXT -1 8 " puzzle." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "with(plots):\nwith(padic):\n
L:=[seq([n,inv_normp(n,2)],n=1..31)]:\npointplot(L);" }}{PARA 13 "" 1 
"" {INLPLOT "6#-%'POINTSG6A7$$\"\"\"\"\"!F'7$$\"\"#F)F+7$$\"\"$F)F'7$$
\"\"%F)F17$$\"\"&F)F'7$$\"\"'F)F+7$$\"\"(F)F'7$$\"\")F)F=7$$\"\"*F)F'7
$$\"#5F)F+7$$\"#6F)F'7$$\"#7F)F17$$\"#8F)F'7$$\"#9F)F+7$$\"#:F)F'7$$\"
#;F)FU7$$\"#<F)F'7$$\"#=F)F+7$$\"#>F)F'7$$\"#?F)F17$$\"#@F)F'7$$\"#AF)
F+7$$\"#BF)F'7$$\"#CF)F=7$$\"#DF)F'7$$\"#EF)F+7$$\"#FF)F'7$$\"#GF)F17$
$\"#HF)F'7$$\"#IF)F+7$$\"#JF)F'" 2 385 385 385 2 0 1 0 2 9 0 4 2 
1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 
0 0 0 0 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Another ver
sion of the same idea:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "l
istplot([seq(inv_normp(n,2),n=1..127)],color=blue);" }}{PARA 13 "" 1 "
" {INLPLOT "6$-%'CURVESG6#7[s7$$\"\"\"\"\"!F(7$$\"\"#F*F,7$$\"\"$F*F(7
$$\"\"%F*F27$$\"\"&F*F(7$$\"\"'F*F,7$$\"\"(F*F(7$$\"\")F*F>7$$\"\"*F*F
(7$$\"#5F*F,7$$\"#6F*F(7$$\"#7F*F27$$\"#8F*F(7$$\"#9F*F,7$$\"#:F*F(7$$
\"#;F*FV7$$\"#<F*F(7$$\"#=F*F,7$$\"#>F*F(7$$\"#?F*F27$$\"#@F*F(7$$\"#A
F*F,7$$\"#BF*F(7$$\"#CF*F>7$$\"#DF*F(7$$\"#EF*F,7$$\"#FF*F(7$$\"#GF*F2
7$$\"#HF*F(7$$\"#IF*F,7$$\"#JF*F(7$$\"#KF*F`q7$$\"#LF*F(7$$\"#MF*F,7$$
\"#NF*F(7$$\"#OF*F27$$\"#PF*F(7$$\"#QF*F,7$$\"#RF*F(7$$\"#SF*F>7$$\"#T
F*F(7$$\"#UF*F,7$$\"#VF*F(7$$\"#WF*F27$$\"#XF*F(7$$\"#YF*F,7$$\"#ZF*F(
7$$\"#[F*FV7$$\"#\\F*F(7$$\"#]F*F,7$$\"#^F*F(7$$\"#_F*F27$$\"#`F*F(7$$
\"#aF*F,7$$\"#bF*F(7$$\"#cF*F>7$$\"#dF*F(7$$\"#eF*F,7$$\"#fF*F(7$$\"#g
F*F27$$\"#hF*F(7$$\"#iF*F,7$$\"#jF*F(7$$\"#kF*F`w7$$\"#lF*F(7$$\"#mF*F
,7$$\"#nF*F(7$$\"#oF*F27$$\"#pF*F(7$$\"#qF*F,7$$\"#rF*F(7$$\"#sF*F>7$$
\"#tF*F(7$$\"#uF*F,7$$\"#vF*F(7$$\"#wF*F27$$\"#xF*F(7$$\"#yF*F,7$$\"#z
F*F(7$$\"#!)F*FV7$$\"#\")F*F(7$$\"##)F*F,7$$\"#$)F*F(7$$\"#%)F*F27$$\"
#&)F*F(7$$\"#')F*F,7$$\"#()F*F(7$$\"#))F*F>7$$\"#*)F*F(7$$\"#!*F*F,7$$
\"#\"*F*F(7$$\"##*F*F27$$\"#$*F*F(7$$\"#%*F*F,7$$\"#&*F*F(7$$\"#'*F*F`
q7$$\"#(*F*F(7$$\"#)*F*F,7$$\"#**F*F(7$$\"$+\"F*F27$$\"$,\"F*F(7$$\"$-
\"F*F,7$$\"$.\"F*F(7$$\"$/\"F*F>7$$\"$0\"F*F(7$$\"$1\"F*F,7$$\"$2\"F*F
(7$$\"$3\"F*F27$$\"$4\"F*F(7$$\"$5\"F*F,7$$\"$6\"F*F(7$$\"$7\"F*FV7$$
\"$8\"F*F(7$$\"$9\"F*F,7$$\"$:\"F*F(7$$\"$;\"F*F27$$\"$<\"F*F(7$$\"$=
\"F*F,7$$\"$>\"F*F(7$$\"$?\"F*F>7$$\"$@\"F*F(7$$\"$A\"F*F,7$$\"$B\"F*F
(7$$\"$C\"F*F27$$\"$D\"F*F(7$$\"$E\"F*F,7$$\"$F\"F*F(-%'COLOURG6&%$RGB
GF*F*$\"*++++\"!\")" 2 385 385 385 2 0 1 0 2 6 0 4 2 1.000000 
45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 
0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "And another:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "picture(7);" }}{PARA 13 "" 
1 "" {INLPLOT "6\\o-%)POLYGONSG6#7&7$$\"\"\"\"\"!F(7$$\"\"#F*F,7$$\"\"
$F*F(F+-F$6#7&7$$\"\"&F*F(7$$\"\"'F*F,7$$\"\"(F*F(F7-F$6#7&7$$\"\"*F*F
(7$$\"#5F*F,7$$\"#6F*F(FC-F$6#7&7$$\"#8F*F(7$$\"#9F*F,7$$\"#:F*F(FO-F$
6#7&7$$\"#<F*F(7$$\"#=F*F,7$$\"#>F*F(Fen-F$6#7&7$$\"#@F*F(7$$\"#AF*F,7
$$\"#BF*F(Fao-F$6#7&7$$\"#DF*F(7$$\"#EF*F,7$$\"#FF*F(F]p-F$6#7&7$$\"#H
F*F(7$$\"#IF*F,7$$\"#JF*F(Fip-F$6#7&7$$\"#LF*F(7$$\"#MF*F,7$$\"#NF*F(F
eq-F$6#7&7$$\"#PF*F(7$$\"#QF*F,7$$\"#RF*F(Far-F$6#7&7$$\"#TF*F(7$$\"#U
F*F,7$$\"#VF*F(F]s-F$6#7&7$$\"#XF*F(7$$\"#YF*F,7$$\"#ZF*F(Fis-F$6#7&7$
$\"#\\F*F(7$$\"#]F*F,7$$\"#^F*F(Fet-F$6#7&7$$\"#`F*F(7$$\"#aF*F,7$$\"#
bF*F(Fau-F$6#7&7$$\"#dF*F(7$$\"#eF*F,7$$\"#fF*F(F]v-F$6#7&7$$\"#hF*F(7
$$\"#iF*F,7$$\"#jF*F(Fiv-F$6#7&7$$\"#lF*F(7$$\"#mF*F,7$$\"#nF*F(Few-F$
6#7&7$$\"#pF*F(7$$\"#qF*F,7$$\"#rF*F(Fax-F$6#7&7$$\"#tF*F(7$$\"#uF*F,7
$$\"#vF*F(F]y-F$6#7&7$$\"#xF*F(7$$\"#yF*F,7$$\"#zF*F(Fiy-F$6#7&7$$\"#
\")F*F(7$$\"##)F*F,7$$\"#$)F*F(Fez-F$6#7&7$$\"#&)F*F(7$$\"#')F*F,7$$\"
#()F*F(Fa[l-F$6#7&7$$\"#*)F*F(7$$\"#!*F*F,7$$\"#\"*F*F(F]\\l-F$6#7&7$$
\"#$*F*F(7$$\"#%*F*F,7$$\"#&*F*F(Fi\\l-F$6#7&7$$\"#(*F*F(7$$\"#)*F*F,7
$$\"#**F*F(Fe]l-F$6#7&7$$\"$,\"F*F(7$$\"$-\"F*F,7$$\"$.\"F*F(Fa^l-F$6#
7&7$$\"$0\"F*F(7$$\"$1\"F*F,7$$\"$2\"F*F(F]_l-F$6#7&7$$\"$4\"F*F(7$$\"
$5\"F*F,7$$\"$6\"F*F(Fi_l-F$6#7&7$$\"$8\"F*F(7$$\"$9\"F*F,7$$\"$:\"F*F
(Fe`l-F$6#7&7$$\"$<\"F*F(7$$\"$=\"F*F,7$$\"$>\"F*F(Faal-F$6#7&7$$\"$@
\"F*F(7$$\"$A\"F*F,7$$\"$B\"F*F(F]bl-F$6#7&7$$\"$D\"F*F(7$$\"$E\"F*F,7
$$\"$F\"F*F(Fibl-F$6#7&F+7$$\"\"%F*FcclF7Fbcl-F$6#7&FC7$$\"#7F*FcclFOF
hcl-F$6#7&Fen7$$\"#?F*FcclFaoF^dl-F$6#7&F]p7$$\"#GF*FcclFipFddl-F$6#7&
Feq7$$\"#OF*FcclFarFjdl-F$6#7&F]s7$$\"#WF*FcclFisF`el-F$6#7&Fet7$$\"#_
F*FcclFauFfel-F$6#7&F]v7$$\"#gF*FcclFivF\\fl-F$6#7&Few7$$\"#oF*FcclFax
Fbfl-F$6#7&F]y7$$\"#wF*FcclFiyFhfl-F$6#7&Fez7$$\"#%)F*FcclFa[lF^gl-F$6
#7&F]\\l7$$\"##*F*FcclFi\\lFdgl-F$6#7&Fe]l7$$\"$+\"F*FcclFa^lFjgl-F$6#
7&F]_l7$$\"$3\"F*FcclFi_lF`hl-F$6#7&Fe`l7$$\"$;\"F*FcclFaalFfhl-F$6#7&
F]bl7$$\"$C\"F*FcclFiblF\\il-F$6#7&Fbcl7$$\"\")F*FcilFhclFbil-F$6#7&F^
dl7$$\"#CF*FcilFddlFhil-F$6#7&Fjdl7$$\"#SF*FcilF`elF^jl-F$6#7&Ffel7$$
\"#cF*FcilF\\flFdjl-F$6#7&Fbfl7$$\"#sF*FcilFhflFjjl-F$6#7&F^gl7$$\"#))
F*FcilFdglF`[m-F$6#7&Fjgl7$$\"$/\"F*FcilF`hlFf[m-F$6#7&Ffhl7$$\"$?\"F*
FcilF\\ilF\\\\m-F$6#7&Fbil7$$\"#;F*Fc\\mFhilFb\\m-F$6#7&F^jl7$$\"#[F*F
c\\mFdjlFh\\m-F$6#7&Fjjl7$$\"#!)F*Fc\\mF`[mF^]m-F$6#7&Ff[m7$$\"$7\"F*F
c\\mF\\\\mFd]m-F$6#7&Fb\\m7$$\"#KF*F[^mFh\\mFj]m-F$6#7&F^]m7$$\"#'*F*F
[^mFd]mF`^m-F$6#7&Fj]m7$$\"#kF*Fg^mF`^mFf^m-%%VIEWG6$;F*$\"$G\"F*F\\_m
" 2 385 385 385 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 264 14 "plot
_gray(m,n)" }{TEXT -1 80 " command will plot the weights of the first \+
m binary Gray codewords of length n." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "plot_gray(15,4);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'C
URVESG6#727$$\"\"\"\"\"!F*7$$\"\"#F*F(7$$\"\"$F*F,7$$\"\"%F*F(7$$\"\"&
F*F,7$$\"\"'F*F/7$$\"\"(F*F,7$$\"\")F*F(7$$\"\"*F*F,7$$\"#5F*F/7$$\"#6
F*F27$$\"#7F*F/7$$\"#8F*F,7$$\"#9F*F/7$$\"#:F*F,7$$\"#;F*F(" 2 384 
384 384 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 -1 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 4 "The " }{TEXT 265 16 "plot_gray(m,n,a)" }{TEXT -1 110 " \+
command will plot the sum of the a^i, where i runs over the support of
 the binary Gray codewords of length n." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "wtd_plot_gray(15,4,1/2);" }}{PARA 13 "" 1 "" 
{INLPLOT "6#-%'CURVESG6#727$$\"\"\"\"\"!F*7$$\"\"#F*$\"+++++]!#57$$\"
\"$F*$\"+++++vF07$$\"\"%F*$\"+++++DF07$$\"\"&F*$\"++++]PF07$$\"\"'F*$
\"++++]()F07$$\"\"(F*$\"++++]iF07$$\"\")F*$\"++++]7F07$$\"\"*F*$\"++++
v=F07$$\"#5F*$\"++++voF07$$\"#6F*$\"++++v$*F07$$\"#7F*$\"++++vVF07$$\"
#8F*$\"++++DJF07$$\"#9F*$\"++++D\")F07$$\"#:F*$\"++++DcF07$$\"#;F*$FI!
#6" 2 384 384 384 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 184 76 0 0 0 0 0 1 }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "wtd_plot_gray(15,4,2);" }}{PARA 13 
"" 1 "" {INLPLOT "6#-%'CURVESG6#727$$\"\"\"\"\"!F*7$$\"\"#F*F,7$$\"\"$
F*$\"\"'F*7$$\"\"%F*F47$$\"\"&F*$\"#7F*7$F1$\"#9F*7$$\"\"(F*$\"#5F*7$$
\"\")F*FD7$$\"\"*F*$\"#CF*7$FA$\"#EF*7$$\"#6F*$\"#IF*7$F9$\"#GF*7$$\"#
8F*$\"#?F*7$F<$\"#AF*7$$\"#:F*$\"#=F*7$$\"#;F*F^o" 2 384 384 384 2 0 
1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
6 "  The " }{TEXT 269 21 "convert2base(m,b,n+1)" }{TEXT -1 80 " comman
d converts a positive integer n to a word of length m in base b notati
on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for i from 1 to 7 do
 convert2base(4,5,i); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$
F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$\"\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#7&\"\"!F$F$\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$F$
\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$\"\"\"F%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&\"\"!F$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 232 "The following table is also given in M. Gardner's S
cientific American article \"The binary Gray code\" [G]. It gives the \+
binary representation and then the reflected binary Gray codeword asso
ciated to the decimal numbers from 1 to 42." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 70 "for i from 1 to 42 do \nprint(i,convert2binary(i),
convert2gray(i));\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"7#F#F$
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#7$\"\"\"\"\"!7$F%F%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"\"$7$\"\"\"F%7$F%\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"\"%7%\"\"\"\"\"!F&7%F%F%F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"&7%\"\"\"\"\"!F%7%F%F%F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"'7%\"\"\"F%\"\"!7%F%F&F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"(7%\"\"\"F%F%7%F%\"\"!F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\")7&\"\"\"\"\"!F&F&7&F%F%F&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"\"*7&\"\"\"\"\"!F&F%7&F%F%F&F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#57&\"\"\"\"\"!F%F&7&F%F%F%F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#67&\"\"\"\"\"!F%F%7&F%F%F%F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#77&\"\"\"F%\"\"!F&7&F%F&F%F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#87&\"\"\"F%\"\"!F%7&F%F&F%F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#97&\"\"\"F%F%\"\"!7&F%F&F&F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#:7&\"\"\"F%F%F%7&F%\"\"!F'F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#;7'\"\"\"\"\"!F&F&F&7'F%F%F&F&F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"#<7'\"\"\"\"\"!F&F&F%7'F%F%F&F&F%" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6%\"#=7'\"\"\"\"\"!F&F%F&7'F%F%F&F%F%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%\"#>7'\"\"\"\"\"!F&F%F%7'F%F%F&F%F&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%\"#?7'\"\"\"\"\"!F%F&F&7'F%F%F%F%F&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6%\"#@7'\"\"\"\"\"!F%F&F%7'F%F%F%F%F%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"#A7'\"\"\"\"\"!F%F%F&7'F%F%F%F&F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#B7'\"\"\"\"\"!F%F%F%7'F%F%F%F&F&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#C7'\"\"\"F%\"\"!F&F&7'F%F&F%F&F&" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#D7'\"\"\"F%\"\"!F&F%7'F%F&F%F&F%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#E7'\"\"\"F%\"\"!F%F&7'F%F&F%F%F
%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#F7'\"\"\"F%\"\"!F%F%7'F%F&F%F%
F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#G7'\"\"\"F%F%\"\"!F&7'F%F&F&F
%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#H7'\"\"\"F%F%\"\"!F%7'F%F&F&
F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#I7'\"\"\"F%F%F%\"\"!7'F%F&F
&F&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#J7'\"\"\"F%F%F%F%7'F%\"\"!
F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#K7(\"\"\"\"\"!F&F&F&F&7(F
%F%F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#L7(\"\"\"\"\"!F&F&F&
F%7(F%F%F&F&F&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#M7(\"\"\"\"\"!F
&F&F%F&7(F%F%F&F&F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#N7(\"\"\"
\"\"!F&F&F%F%7(F%F%F&F&F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#O7(
\"\"\"\"\"!F&F%F&F&7(F%F%F&F%F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%
\"#P7(\"\"\"\"\"!F&F%F&F%7(F%F%F&F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%\"#Q7(\"\"\"\"\"!F&F%F%F&7(F%F%F&F%F&F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#R7(\"\"\"\"\"!F&F%F%F%7(F%F%F&F%F&F&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%\"#S7(\"\"\"\"\"!F%F&F&F&7(F%F%F%F%F&F&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"#T7(\"\"\"\"\"!F%F&F&F%7(F%F%F%F%F&F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#U7(\"\"\"\"\"!F%F&F%F&7(F%F%F%F%F%F
%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "A binary Gray code of length
 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "graycode(4,2);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#727&\"\"!F%F%F%7&\"\"\"F%F%F%7&F'F'F%F
%7&F%F'F%F%7&F%F'F'F%7&F'F'F'F%7&F'F%F'F%7&F%F%F'F%7&F%F%F'F'7&F'F%F'F
'7&F'F'F'F'7&F%F'F'F'7&F%F'F%F'7&F'F'F%F'7&F'F%F%F'7&F%F%F%F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "A 4-ary Gray code of length 4:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "graycode(4,4);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#7\\[l7&\"\"!F%F%F%7&\"\"\"F%F%F%7&\"\"#F%F%F%7&
\"\"$F%F%F%7&F+F'F%F%7&F)F'F%F%7&F'F'F%F%7&F%F'F%F%7&F%F)F%F%7&F'F)F%F
%7&F)F)F%F%7&F+F)F%F%7&F+F+F%F%7&F)F+F%F%7&F'F+F%F%7&F%F+F%F%7&F%F+F'F
%7&F'F+F'F%7&F)F+F'F%7&F+F+F'F%7&F+F)F'F%7&F)F)F'F%7&F'F)F'F%7&F%F)F'F
%7&F%F'F'F%7&F'F'F'F%7&F)F'F'F%7&F+F'F'F%7&F+F%F'F%7&F)F%F'F%7&F'F%F'F
%7&F%F%F'F%7&F%F%F)F%7&F'F%F)F%7&F)F%F)F%7&F+F%F)F%7&F+F'F)F%7&F)F'F)F
%7&F'F'F)F%7&F%F'F)F%7&F%F)F)F%7&F'F)F)F%7&F)F)F)F%7&F+F)F)F%7&F+F+F)F
%7&F)F+F)F%7&F'F+F)F%7&F%F+F)F%7&F%F+F+F%7&F'F+F+F%7&F)F+F+F%7&F+F+F+F
%7&F+F)F+F%7&F)F)F+F%7&F'F)F+F%7&F%F)F+F%7&F%F'F+F%7&F'F'F+F%7&F)F'F+F
%7&F+F'F+F%7&F+F%F+F%7&F)F%F+F%7&F'F%F+F%7&F%F%F+F%7&F%F%F+F'7&F'F%F+F
'7&F)F%F+F'7&F+F%F+F'7&F+F'F+F'7&F)F'F+F'7&F'F'F+F'7&F%F'F+F'7&F%F)F+F
'7&F'F)F+F'7&F)F)F+F'7&F+F)F+F'7&F+F+F+F'7&F)F+F+F'7&F'F+F+F'7&F%F+F+F
'7&F%F+F)F'7&F'F+F)F'7&F)F+F)F'7&F+F+F)F'7&F+F)F)F'7&F)F)F)F'7&F'F)F)F
'7&F%F)F)F'7&F%F'F)F'7&F'F'F)F'7&F)F'F)F'7&F+F'F)F'7&F+F%F)F'7&F)F%F)F
'7&F'F%F)F'7&F%F%F)F'7&F%F%F'F'7&F'F%F'F'7&F)F%F'F'7&F+F%F'F'7&F+F'F'F
'7&F)F'F'F'7&F'F'F'F'7&F%F'F'F'7&F%F)F'F'7&F'F)F'F'7&F)F)F'F'7&F+F)F'F
'7&F+F+F'F'7&F)F+F'F'7&F'F+F'F'7&F%F+F'F'7&F%F+F%F'7&F'F+F%F'7&F)F+F%F
'7&F+F+F%F'7&F+F)F%F'7&F)F)F%F'7&F'F)F%F'7&F%F)F%F'7&F%F'F%F'7&F'F'F%F
'7&F)F'F%F'7&F+F'F%F'7&F+F%F%F'7&F)F%F%F'7&F'F%F%F'7&F%F%F%F'7&F%F%F%F
)7&F'F%F%F)7&F)F%F%F)7&F+F%F%F)7&F+F'F%F)7&F)F'F%F)7&F'F'F%F)7&F%F'F%F
)7&F%F)F%F)7&F'F)F%F)7&F)F)F%F)7&F+F)F%F)7&F+F+F%F)7&F)F+F%F)7&F'F+F%F
)7&F%F+F%F)7&F%F+F'F)7&F'F+F'F)7&F)F+F'F)7&F+F+F'F)7&F+F)F'F)7&F)F)F'F
)7&F'F)F'F)7&F%F)F'F)7&F%F'F'F)7&F'F'F'F)7&F)F'F'F)7&F+F'F'F)7&F+F%F'F
)7&F)F%F'F)7&F'F%F'F)7&F%F%F'F)7&F%F%F)F)7&F'F%F)F)7&F)F%F)F)7&F+F%F)F
)7&F+F'F)F)7&F)F'F)F)7&F'F'F)F)7&F%F'F)F)7&F%F)F)F)7&F'F)F)F)7&F)F)F)F
)7&F+F)F)F)7&F+F+F)F)7&F)F+F)F)7&F'F+F)F)7&F%F+F)F)7&F%F+F+F)7&F'F+F+F
)7&F)F+F+F)7&F+F+F+F)7&F+F)F+F)7&F)F)F+F)7&F'F)F+F)7&F%F)F+F)7&F%F'F+F
)7&F'F'F+F)7&F)F'F+F)7&F+F'F+F)7&F+F%F+F)7&F)F%F+F)7&F'F%F+F)7&F%F%F+F
)7&F%F%F+F+7&F'F%F+F+7&F)F%F+F+7&F+F%F+F+7&F+F'F+F+7&F)F'F+F+7&F'F'F+F
+7&F%F'F+F+7&F%F)F+F+7&F'F)F+F+7&F)F)F+F+7&F+F)F+F+7&F+F+F+F+7&F)F+F+F
+7&F'F+F+F+7&F%F+F+F+7&F%F+F)F+7&F'F+F)F+7&F)F+F)F+7&F+F+F)F+7&F+F)F)F
+7&F)F)F)F+7&F'F)F)F+7&F%F)F)F+7&F%F'F)F+7&F'F'F)F+7&F)F'F)F+7&F+F'F)F
+7&F+F%F)F+7&F)F%F)F+7&F'F%F)F+7&F%F%F)F+7&F%F%F'F+7&F'F%F'F+7&F)F%F'F
+7&F+F%F'F+7&F+F'F'F+7&F)F'F'F+7&F'F'F'F+7&F%F'F'F+7&F%F)F'F+7&F'F)F'F
+7&F)F)F'F+7&F+F)F'F+7&F+F+F'F+7&F)F+F'F+7&F'F+F'F+7&F%F+F'F+7&F%F+F%F
+7&F'F+F%F+7&F)F+F%F+7&F+F+F%F+7&F+F)F%F+7&F)F)F%F+7&F'F)F%F+7&F%F)F%F
+7&F%F'F%F+7&F'F'F%F+7&F)F'F%F+7&F+F'F%F+7&F+F%F%F+7&F)F%F%F+7&F'F%F%F
+7&F%F%F%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "                                         \+
        " }{TEXT 257 18 "        References" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "[CSW] J. Conway, N. Sloane, an
d A. Wilks, \"Gray codes and reflection groups\", Graphs and combinato
rics " }{TEXT 259 1 "5" }{TEXT -1 13 "(1989)315-325" }}{PARA 0 "" 0 "
" {TEXT -1 94 "[E] M. C. Er, \"On generating the N-ary reflected Gray \+
codes\", IEEE transactions on computers, " }{TEXT 273 2 "33" }{TEXT 
-1 13 "(1984)739-741" }}{PARA 0 "" 0 "" {TEXT -1 43 "[G] M. Gardner, \+
\"The binary Gray code\", in " }{TEXT 258 52 "Knotted donuts and other
 mathematical entertainments" }{TEXT -1 33 ", F. H. Freeman and Co., N
Y, 1986" }}{PARA 0 "" 0 "" {TEXT -1 61 "[Gi] W. Gilbert, \"A cube-fill
ing Hilbert curve\", Math Intell " }{TEXT 270 1 "6" }{TEXT -1 9 " (198
4)78" }}{PARA 0 "" 0 "" {TEXT -1 102 "[Gil] E. Gilbert, \"Gray codes a
nd paths on the n-cube\", Bell System Technical Journal 37 (1958)815-8
26" }}{PARA 0 "" 0 "" {TEXT -1 88 "[S] Web page of T. Sillke at http:/
/www.mathematik.uni-bielefeld.de/~sillke/RESULTS/gray" }}{PARA 0 "" 0 
"" {TEXT -1 56 "[W] A. White, \"Ringing the cosets\", Amer. Math. Mont
hly " }{TEXT 260 2 "94" }{TEXT -1 13 "(1987)721-746" }}}}{MARK "0 11 0
" 1 }{VIEWOPTS 1 1 0 1 1 1803 }