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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 429 "Snow White distributed 21
 quarts of milk among the seven dwarfs. The first dwarf then distribut
ed the contents of his pail evenly to the pails of other six dwarfs. T
hen the second did the same, and so on. After the seventh dwarf distri
buted the contents of his pail evenly to the other six dwarfs, it was \+
found that each dwarf had exactly as much milk in his pail as at the s
tart.  What was the initial distribution of the milk? " }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(linalg
):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm a
nd trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 247 "v[1] is the initial vector.  We will find each subs
equent vector v[i+1] from v[i].  v[i+1] is the same as v[i], except th
at at the ith position the new value will be zero, while every other p
osition will have an equal proportion of that old value." }{MPLTEXT 1 
0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v[1] := vector([a,b,c,d,e,
f,g]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"\"-%'vectorG6#7
)%\"aG%\"bG%\"cG%\"dG%\"eG%\"fG%\"gG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "x := v[1][1] / 6: y := -v[1][1]:\nv[2] := evalm(v[1] \+
+ vector([y,x,x,x,x,x,x]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG
6#\"\"#-%'vectorG6#7)\"\"!,&%\"bG\"\"\"*&#F/\"\"'F/%\"aGF/F/,&%\"cGF/*
&F1F/F3F/F/,&%\"dGF/*&F1F/F3F/F/,&%\"eGF/*&F1F/F3F/F/,&%\"fGF/*&F1F/F3
F/F/,&%\"gGF/*&F1F/F3F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
79 "x := v[2][2] / 6: y := -v[2][2]:\nv[3] := evalm(v[2] + vector([x,y
,x,x,x,x,x]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"$-%'vec
torG6#7),&&&F%6#\"\"#F/#\"\"\"\"\"'F.F2,&F-!\"\"F.F2F,F,F,F,F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x := v[3][3] / 6: y := -v[3]
[3]:\nv[4] := evalm(v[3] + vector([x,x,y,x,x,x,x]));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%\"vG6#\"\"%-%'vectorG6#7),&&&F%6#\"\"#F/#\"\"(\"#
O*&#F2\"\"'\"\"\"F.F7F7,&F-#!#NF3*&F5F7F.F7F7\"\"!F,F,F,F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x := v[4][4] / 6: y := -v[4][4]:\nv
[5] := evalm(v[4] + vector([x,x,x,y,x,x,x]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"vG6#\"\"&-%'vectorG6#7),&&&F%6#\"\"#F/#\"#\\\"$;#*
&#F2\"#O\"\"\"F.F7F7,&F-#!$.#F3*&F5F7F.F7F7,&F-#\"\"(F3*&#F>F6F7F.F7F7
\"\"!F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x := v[5][5]
 / 6: y := -v[5][5]:\nv[6] := evalm(v[5] + vector([x,x,x,x,y,x,x]));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"'-%'vectorG6#7),&&&F%6#
\"\"#F/#\"$V$\"%'H\"*&#F2\"$;#\"\"\"F.F7F7,&F-#!%p6F3*&F5F7F.F7F7,&F-#
\"#\"*F3*&#F>F6F7F.F7F7,&F-#\"#\\F3*&#FCF6F7F.F7F7\"\"!F,F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x := v[6][6] / 6: y := -v[6]
[6]:\nv[7] := evalm(v[6] + vector([x,x,x,x,x,y,x]));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%\"vG6#\"\"(-%'vectorG6#7),&&&F%6#\"\"#F/#\"%,C\"%
wx*&#F2\"%'H\"\"\"\"F.F7F7,&F-#!%rmF3*&F5F7F.F7F7,&F-#\"$*))F3*&#F>F6F
7F.F7F7,&F-#\"$P'F3*&#FCF6F7F.F7F7,&F-#\"$V$F3*&#FHF6F7F.F7F7\"\"!F," 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x := v[7][7] / 6: y := -v
[7][7]:\nv[8] := evalm(v[7] + vector([x,x,x,x,x,x,y]));" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\")-%'vectorG6#7),&&&F%6#\"\"#F/#\"&2
o\"\"&cm%*&#F2\"%wx\"\"\"F.F7F7,&F-#!&Dw$F3*&F5F7F.F7F7,&F-#\"%NxF3*&#
F>F6F7F.F7F7,&F-#\"%BiF3*&#FCF6F7F.F7F7,&F-#\"%fWF3*&#FHF6F7F.F7F7,&F-
#\"%,CF3*&#FMF6F7F.F7F7\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "
Since the distribution is the same as it was in the beginning, we know
 that v[1] = v[8].  Solve for each value in v[8]." }{MPLTEXT 1 0 0 "" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "solve(\{v[1][1] = v[8][1], v[1][
2] = v[8][2], v[1][3] = v[8][3], v[1][4] = v[8][4], \n       v[1][5] =
 v[8][5], v[1][6] = v[8][6], v[1][7] = v[8][7]\}, \{a,b,c,d,e,f,g\});
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)/%\"aG,$%\"fG\"\"'/%\"dG,$F'\"\"
$/%\"cG,$F'\"\"%/%\"bG,$F'\"\"&/%\"eG,$F'\"\"#/%\"gG\"\"!/F'F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "And then knowing that this is a pr
obability vector, all of the values add up to one." }{MPLTEXT 1 0 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "solve(\{a=6*f, d=3*f, c=4*f, b=
5*f, e=2*f, g=0, f=f, a+b+c+d+e+f+g=1\}, \{a,b,c,d,e,f,g\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<)/%\"gG\"\"!/%\"eG#\"\"#\"#@/%\"bG#\"\"&F+/
%\"cG#\"\"%F+/%\"aG#F*\"\"(/%\"fG#\"\"\"F+/%\"dG#F;F7" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 58 "That is, for seven dwarves, the original \+
distribution is [" }{XPPEDIT 18 0 "6/21" "6#*&\"\"'\"\"\"\"#@!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "5/21" "6#*&\"\"&\"\"\"\"#@!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "4/21" "6#*&\"\"%\"\"\"\"#@!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "3/21" "6#*&\"\"$\"\"\"\"#@!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "2/21" "6#*&\"\"#\"\"\"\"#@!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "1/21" "6#*&\"\"\"F$\"#@!\"\"" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "0/21" "6#*&\"\"!\"\"\"\"#@!\"\"" }{TEXT -1 
33 "].\nTo generalize, for any number " }{XPPEDIT 18 0 "n;" "6#%\"nG" 
}{TEXT -1 7 ", let  " }{XPPEDIT 18 0 "r = n*(n-1)/2;" "6#/%\"rG*&*&%\"
nG\"\"\",&F'F(F(!\"\"F(F(\"\"#F*" }{TEXT -1 29 ".  The distribution th
en is [" }{XPPEDIT 18 0 "(n-1)/r;" "6#*&,&%\"nG\"\"\"F&!\"\"F&%\"rGF'
" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(n-2)/r;" "6#*&,&%\"nG\"\"\"\"\"#!
\"\"F&%\"rGF(" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "2/r;" "6#*&\"\"#
\"\"\"%\"rG!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "1/r;" "6#*&\"\"\"F$
%\"rG!\"\"" }{TEXT -1 5 ", 0]." }}}}{MARK "12 0 18" 23 }{VIEWOPTS 1 1 
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }