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:Z:j;<Z:>:1:" }{TEXT -1 0 "" }}{PARA 209 "" 0 "" {TEXT 203 17 "Solveur
 de Sudoku" }}{PARA 209 "" 0 "" {TEXT 287 43 "\251 Alain Le Stang - \"
Apprendre Maple\" - 2006" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 14 "Une grille de " }{TEXT 205 6 "Sudoku" }
{TEXT -1 26 " se compose de neuf  blocs" }{TEXT 206 1 " " }{TEXT -1 
197 "carr\351s 3x3 comportant chacun 9 cases, soit au total 81 cases. \+
\nLe but du jeu est de compl\351ter la grille (certaines des cases con
tenant initialement un chiffre) afin que chaque ligne,\nchaque colonne
" }{TEXT 207 1 " " }{TEXT -1 9 "et chaque" }{TEXT 208 1 " " }{TEXT -1 
166 "bloc 3x3 contienne une fois et une seule chacun des chiffres de 1
 \340 9. \nChaque case de la grille pourra \352tre caract\351ris\351e,
 comme on le voit dans la grille ci-dessous:" }}{PARA 0 "" 0 "" {TEXT 
-1 9 "- par sa " }{TEXT 209 5 "ligne" }{TEXT -1 29 ": nombre entier en
tre 0 et 8." }}{PARA 0 "" 0 "" {TEXT -1 9 "- par sa " }{TEXT 210 7 "co
lonne" }{TEXT -1 29 ": nombre entier entre 0 et 8." }}{PARA 0 "" 0 "" 
{TEXT -1 10 "- par son " }{TEXT 211 5 "index" }{TEXT -1 114 ":  nombre
 entier entre 0 et 80.\nAinsi, la case gris\351e de la grille est en l
igne 3, colonne 5 et a pour index 32. \n" }}}{PARA 201 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Une grille de " }{TEXT 
212 6 "Sudoku" }{TEXT -1 33 " sera repr\351sent\351e par un tableau " 
}{TEXT 213 7 "sudoku " }{TEXT -1 57 "\340 une dimension, dont les indi
ces \351volueront de 0 \340 80.\n\n" }{TEXT 214 7 "Exemple" }{TEXT -1 
1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "sudoku:=array(0..8
0,[\n 3, 0, 0, 1, 0, 0, 0, 0, 0,\n 0, 5, 7, 0, 0, 3, 1, 0, 0,\n 9, 0, \+
6, 0, 5, 0, 0, 0, 8,\n 4, 0, 0, 6, 0, 0, 9, 5, 1,\n 0, 0, 5, 8, 0, 0, \+
0, 0, 6, \n 0, 0, 0, 0, 0, 7, 0, 0, 0,\n 8, 0, 0, 0, 0, 0, 0, 0, 0,\n \+
0, 0, 0, 0, 0, 0, 0, 4, 2,\n 0, 0, 0, 0, 7, 1, 0, 0, 0\n]): " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "repr\351sentera la grille de " }
{TEXT 215 6 "Sudoku" }{TEXT -1 73 " ci-dessous.\nLes cases contenant i
nitialement un chiffre seront appel\351es " }{TEXT 216 12 "cases fix
\351es" }{TEXT -1 41 " et seront repr\351sent\351es dans le tableau \n
" }{TEXT 217 7 "sudoku " }{TEXT -1 79 "par ce chiffre. Tandis que les \+
cases vides seront repr\351sent\351es dans le tableau " }{TEXT 218 7 "
sudoku " }{TEXT -1 62 "par la valeur 0: ces \ncases seront appel\351es
 pour la suite des " }{TEXT 219 17 "cases modifiables" }{TEXT -1 12 " \+
du tableau." }}}{PARA 201 "" 0 "" {TEXT -1 0 "" }{GLPLOT2D 363 376 
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{TEXT -1 24 "1\260 Ecrire une proc\351dure " }{TEXT 220 8 "afficher" }
{TEXT -1 196 "(T::array) qui affiche une grille de Sudoku, \340 partir
 de sa repr\351sentation sous\n    forme de tableau. Pr\351voir de bie
n d\351limiter les blocs 3x3 en utilisant un trait plus \351pais, comm
e ci-dessous:   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "affiche
r(sudoku);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }{GLPLOT2D 361 359 359 
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{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "2\260 Ecrire les \+
proc\351dures suivantes, i \351tant de type nonnegint:\na) " }{TEXT 
221 17 "index_debut_ligne" }{TEXT -1 115 "(i) donne pour r\351sultat l
'index de la case situ\351e au d\351but de la ligne \340 laquelle appa
rtient la case d'index i.\nb) " }{TEXT 222 19 "index_debut_colonne" }
{TEXT -1 117 "(i) donne pour r\351sultat l'index de la case situ\351e \+
au d\351but de la colonne \340 laquelle appartient la case d'index i.
\nc) " }{TEXT 223 5 "ligne" }{TEXT -1 90 "(i) donne pour r\351sultat l
e num\351ro de la ligne \340 laquelle appartient la case d'index i.\nd
) " }{TEXT 224 7 "colonne" }{TEXT -1 102 "(i) donne pour r\351sultat l
e num\351ro de la colonne \340 laquelle appartient la case d'index i  \+
(identique \340 " }{TEXT 225 19 "index_debut_colonne" }{TEXT -1 9 "(i)
).\ne) " }{TEXT 226 16 "index_debut_bloc" }{TEXT -1 102 "(i) donne pou
r r\351sultat l'index de la premi\350re case du bloc 3x3 auquel appart
ient la case d'index i.\n\n" }{TEXT 227 8 "Exemples" }{TEXT -1 1 ":" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "index_debut_ligne(32);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "index_debut_colonne(32);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lig
ne(32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "colonne(32);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "index_debut_blo
c(32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#I" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 97 "3\260 En utilisant les proc\351dures d\351finies dan
s la question 2\260, \351crire les proc\351dures suivantes:\na) " }
{TEXT 228 8 "OK_ligne" }{TEXT -1 75 "(T::array,v::posint,i::nonnegint)
 donne pour r\351sultat une valeur bool\351enne " }{TEXT 229 4 "true" 
}{TEXT -1 4 " ou " }{TEXT 230 5 "false" }{TEXT -1 92 " selon que le ch
iffre v figure\nd\351j\340 ou non sur la ligne de la case d'index i du
 tableau T. \n" }{TEXT 231 7 "Exemple" }{TEXT -1 133 ": on peut placer
 un 7 en ligne dans la case d'index 32, mais pas un 5 (car un 5 figure
 d\351j\340 sur cette ligne, dans la case d'index 34)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "OK_ligne(sudoku,7,32),OK_ligne(sudo
ku,5,32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueG%&falseG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "b) " }{TEXT 232 10 "OK_colonne" }
{TEXT -1 75 "(T::array,v::posint,i::nonnegint) donne pour r\351sultat \+
une valeur bool\351enne " }{TEXT 233 4 "true" }{TEXT -1 4 " ou " }
{TEXT 234 5 "false" }{TEXT -1 94 " selon que le chiffre v figure\nd
\351j\340 ou non sur la colonne de la case d'index i du tableau T. \n
" }{TEXT 235 7 "Exemple" }{TEXT -1 137 ": on peut placer un 5 en colon
ne dans la case d'index 32, mais pas un 1 (car un 1 figure d\351j\340 \+
sur cette colonne, dans la case d'index 77)" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 48 "OK_colonne(sudoku,5,32),OK_colonne(sudoku,1,32);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueG%&falseG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 3 "c) " }{TEXT 236 7 "OK_bloc" }{TEXT -1 75 "(T::array
,v::posint,i::nonnegint) donne pour r\351sultat une valeur bool\351enn
e " }{TEXT 237 4 "true" }{TEXT -1 4 " ou " }{TEXT 238 5 "false" }
{TEXT -1 92 " selon que le chiffre v figure\nd\351j\340 ou non dans le
 bloc de la case d'index i du tableau T. \n" }{TEXT 239 7 "Exemple" }
{TEXT -1 132 ": on peut placer un 3 dans le bloc de la case d'index 32
, mais pas un 8 (car un 8 figure d\351j\340 dans ce bloc, dans la case
 d'index 39)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "OK_bloc(sud
oku,3,32),OK_bloc(sudoku,8,32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%t
rueG%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "d) " }{TEXT 240 2 
"OK" }{TEXT -1 55 "(T::array,v::posint,i::nonnegint) donne pour r\351s
ultat  " }{TEXT 241 6 "false " }{TEXT -1 58 "si i=81 et sinon donne po
ur r\351sultat une valeur bool\351enne " }{TEXT 242 4 "true" }{TEXT 
-1 4 " ou " }{TEXT 243 6 "false\n" }{TEXT -1 117 "selon que le chiffre
 v peut \352tre plac\351 ou non dans la case d'index i du tableau T, c
onform\351ment \340 la r\350gle du jeu de " }{TEXT 244 6 "Sudoku" }
{TEXT -1 57 ". \nPour cela, on utilisera les 3 proc\351dures pr\351c
\351dentes.\n" }{TEXT 245 7 "Exemple" }{TEXT -1 233 ": on peut placer \+
un 2 dans la case d'index 32, car 2 ne figure ni dans la ligne, ni dan
s la  colonne, ni dans le bloc auquel \nappartient cette case. Par con
tre, on ne peut y placer 1 qui figure d\351j\340 (\340 la fois en lign
e et en colonne)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "OK(sudo
ku,2,32),OK(sudoku,1,32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueG%&
falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "4\260 Ecrire une proc
\351dure " }{TEXT 246 7 "suivant" }{TEXT -1 47 "(T::array,i::nonnegint
) donnant pour r\351sultat :" }}{PARA 0 "" 0 "" {TEXT -1 141 " i si la
 case d'index i du tableau T est modifiable, ou sinon l'index de la pr
ochaine case modifiable suivant la case d'index i du tableau T." }}
{PARA 0 "" 0 "" {TEXT -1 138 "(pr\351voir dans ce dernier cas que le r
\351sultat pourra valoir 81, dans le cas o\371 les derni\350res cases \+
de la grille ne sont pas modifiables)  \n" }{TEXT 247 7 "Exemple" }
{TEXT -1 120 ": la case d'index 32 est vide, les cases d'index 33,34,3
5 contiennent respectivement 9,5,1, la case d'index 36 est vide." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "suivant(sudoku,32),suivant(s
udoku,33); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#K\"#O" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 2 "5\260" }{TEXT 248 1 " " }{TEXT -1 10 "Ecri
re une" }{TEXT 249 1 " " }{TEXT -1 10 "proc\351dure " }{TEXT 250 11 "i
ndex_fixes" }{TEXT -1 82 "(T::array) donnant pour r\351sultat la liste
 des index des cases fix\351es du tableau T." }}{PARA 0 "" 0 "" {TEXT 
251 7 "Exemple" }{TEXT -1 87 ": les cases contenant initialement un ch
iffre sont les cases d'index  0,3,10,...,76,77." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "index_fixes(sudoku);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7:\"\"!\"\"$\"#5\"#6\"#9\"#:\"#=\"#?\"#A\"#E\"#F\"#I\"#
L\"#M\"#N\"#Q\"#R\"#W\"#]\"#a\"#q\"#r\"#w\"#x" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 13 "6\260 Ecrire une" }{TEXT 252 1 " " }{TEXT -1 9 "proc
\351dure" }{TEXT 253 9 " solution" }{TEXT -1 57 "(T::array) donnant po
ur r\351sultat la solution du sudoku T." }}{PARA 0 "" 0 "" {TEXT -1 
63 "La proc\351dure fonctionnera sur un algorithme de retour arri\350r
e (" }{TEXT 254 12 "backtracking" }{TEXT -1 30 ") et utilisera les pro
c\351dures " }{TEXT 255 11 "index_fixes" }{TEXT -1 2 ",\n" }{TEXT 256 
7 "suivant" }{TEXT -1 5 ", et " }{TEXT 257 2 "OK" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 114 "On passera en revue toutes les possibili
t\351s offertes de chiffres candidats pour les diff\351rentes cases mo
difiables:" }}{PARA 0 "" 0 "" {TEXT -1 65 "- si le chiffre convient, o
n passe \340 la case modifiable suivante." }}{PARA 0 "" 0 "" {TEXT -1 
124 "- si on a \351puis\351 tous les chiffres candidats pour une case,
 on revient en arri\350re sur la case modifiable pr\351c\351dente et o
n y " }}{PARA 0 "" 0 "" {TEXT -1 34 "teste un nouveau chiffre candidat
." }}{PARA 0 "" 0 "" {TEXT -1 93 "Ainsi de suite ... jusqu'\340 attein
dre la fin du tableau T  (c'est \340 dire lorque index = 81 ). " }}}
{EXCHG {PARA 204 "" 1 "" {TEXT 258 7 "Exemple" }{TEXT -1 88 ": voici l
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le pr\351c\351dent" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6$/%'index
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%)chiffre~GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'index~G\"\")/%)chi
ffre~G\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~
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indexG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'index~G\"\"(/%)chiff
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~G\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~inde
xG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~inde
xG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~inde
xG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'index~G\"\"'/%)chiffre~G
\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'index~G\"\"(/%)chiffre~G\"
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%'index~G\"\")/%)chiffre~G\"\"
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~indexG\"\"
)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~indexG\"\"
(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~indexG\"\"
'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8retour~`arri|cyre`:~indexG\"\"
&" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#%>----------~etc~--------
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%-%*GRIDSTYLEG6#%,RECTANGULARG" 1 2 2 1 10 1 2 9 1 1 2 1.000000 
45.000000 45.000000 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1
8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "
Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv
e 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37
" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "C
urve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve
 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56
" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "C
urve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve
 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75
" "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "C
urve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve
 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94
" "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" "
Curve 101" "Curve 102" "Curve 103" "Curve 104" "Curve 105" "Curve 106
" "Curve 107" "Curve 108" "Curve 109" "Curve 110" "Curve 111" "Curve 1
12" "Curve 113" "Curve 114" "Curve 115" "Curve 116" "Curve 117" "Curve
 118" "Curve 119" "Curve 120" "Curve 121" "Curve 122" "Curve 123" "Cur
ve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "C
urve 130" "Curve 131" "Curve 132" "Curve 133" "Curve 134" "Curve 135" 
"Curve 136" "Curve 137" "Curve 138" "Curve 139" "Curve 140" "Curve 141
" "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 1
47" "Curve 148" "Curve 149" "Curve 150" "Curve 151" "Curve 152" "Curve
 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Cur
ve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "C
urve 165" "Curve 166" "Curve 167" "Curve 168" "Curve 169" "Curve 170" 
"Curve 171" "Curve 172" "Curve 173" "Curve 174" "Curve 175" "Curve 176
" "Curve 177" "Curve 178" "Curve 179" "Curve 180" "Curve 181" "Curve 1
82" "Curve 183" "Curve 184" "Curve 185" "Curve 186" "Curve 187" "Curve
 188" "Curve 189" "Curve 190" "Curve 191" }}{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 42 "7\260 R\351soudre de m\352me les quatre g
rilles de " }{TEXT 259 7 "Sudoku " }{TEXT -1 10 "suivantes:" }}}
{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
279 "sudoku:=array(0..80,[\n 3, 0, 0, 1, 0, 0, 0, 0, 0,\n 0, 5, 7, 0, \+
0, 3, 1, 0, 0,\n 9, 0, 6, 0, 5, 0, 0, 0, 8,\n 4, 0, 0, 6, 0, 0, 9, 5, \+
1,\n 0, 0, 5, 8, 0, 0, 0, 0, 6, \n 0, 0, 0, 0, 0, 7, 0, 0, 0,\n 8, 0, \+
0, 0, 0, 0, 0, 0, 0,\n 0, 0, 0, 0, 0, 0, 0, 4, 2,\n 0, 0, 0, 0, 7, 1, \+
0, 0, 0\n]): \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 11 "1\260 affiche
r" }{TEXT -1 74 "() affiche la grille de Sudoku correspondant au table
au T des 81 nombres: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "wi
th(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoo
rds has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
502 "afficher:=proc(T::array)\nlocal L,e,texte,lig,col;\nL:=NULL: text
e:=NULL:\nfor lig from 0 to 9 do\n  if (lig mod 3=0) then e:=5 else e:
=1 end if;\n  L:=L,plot([t,9-lig,t=0..9],thickness=e);\n  for col from
 0 to 9 do\n    if (col mod 3=0) then e:=5 else e:=1 end if;\n    L:=L
,plot([col,t,t=-0.05..9.05],thickness=e):\n    if lig<9 and col<9 and \+
T[9*lig+col]>0 then texte:=texte,textplot([col+0.5,9-lig-0.5,T[9*lig+c
ol]],font=[HELVETICA,BOLD,32]) end if;\n  end do;\nend do:\ndisplay([L
,texte],axes=none)\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "afficher(sudoku);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }
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" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "C
urve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve
 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47
" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "C
urve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve
 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66
" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "C
urve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve
 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85
" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "C
urve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97" "Curve
 98" "Curve 99" "Curve 100" "Curve 101" "Curve 102" "Curve 103" "Curve
 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" "Curve 109" "Cur
ve 110" "Curve 111" "Curve 112" "Curve 113" "Curve 114" "Curve 115" "C
urve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 120" "Curve 121" 
"Curve 122" "Curve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127
" "Curve 128" "Curve 129" "Curve 130" "Curve 131" "Curve 132" "Curve 1
33" "Curve 134" }}{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 261 22 "2\260a) index_debut_ligne" }
{TEXT -1 111 "(i) donne pour r\351sultat l'index de la case situ\351e \+
au d\351but de la ligne \340 laquelle appartient la case d'index i:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "index_debut_ligne:=proc(i:
:nonnegint)\n  9*iquo(i,9)\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "seq(index_debut_ligne(k),k=0..80);" }}{PARA 200 "" 1 
"" {XPPMATH 20 "6]p\"\"!F#F#F#F#F#F#F#F#\"\"*F$F$F$F$F$F$F$F$\"#=F%F%F
%F%F%F%F%F%\"#FF&F&F&F&F&F&F&F&\"#OF'F'F'F'F'F'F'F'\"#XF(F(F(F(F(F(F(F
(\"#aF)F)F)F)F)F)F)F)\"#jF*F*F*F*F*F*F*F*\"#sF+F+F+F+F+F+F+F+" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 24 "2
\260b) index_debut_colonne" }{TEXT -1 113 "(i) donne pour r\351sultat \+
l'index de la case situ\351e au d\351but de la colonne \340 laquelle a
ppartient la case d'index i:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 62 "index_debut_colonne:=proc(i::nonnegint)\n  irem(i,9)\nend proc:
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(index_debut_colon
ne(k),k=0..80);" }}{PARA 200 "" 1 "" {XPPMATH 20 "6]p\"\"!\"\"\"\"\"#
\"\"$\"\"%\"\"&\"\"'\"\"(\"\")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#F$F%F&F'F(F)F*F+F#F$F%F&F'F(F)F*F+F#F$F%F&F'F(F)F*F+F#
F$F%F&F'F(F)F*F+F#F$F%F&F'F(F)F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "" 0 "" {TEXT 263 10 "2\260c) ligne" }{TEXT -1 86 "(i)
 donne pour r\351sultat le num\351ro de la ligne \340 laquelle apparti
ent la case d'index i:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "l
igne:=proc(i::nonnegint)\n  iquo(i,9)\nend proc:\n" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "seq(ligne(k),k=0..80);" }}{PARA 200 "" 1 "
" {XPPMATH 20 "6]p\"\"!F#F#F#F#F#F#F#F#\"\"\"F$F$F$F$F$F$F$F$\"\"#F%F%
F%F%F%F%F%F%\"\"$F&F&F&F&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)\"\"(F*F*F*F*F*F*F*F*\"\")F+F+F+F+F+F+F+F+" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 264 
12 "2\260d) colonne" }{TEXT -1 102 "(i) donne pour r\351sultat le num
\351ro de la colonne \340 laquelle appartient la case d'index i  (iden
tique \340 " }{TEXT 265 19 "index_debut_colonne" }{TEXT -1 5 "(i)):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "colonne:=proc(i::nonnegint
)\n  irem(i,9)  # identique \340 index_debut_colonne(i)\nend proc:\n" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seq(colonne(k),k=0..80);
" }}{PARA 200 "" 1 "" {XPPMATH 20 "6]p\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&
\"\"'\"\"(\"\")F#F$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
#F$F%F&F'F(F)F*F+F#F$F%F&F'F(F)F*F+F#F$F%F&F'F(F)F*F+F#F$F%F&F'F(F)F*F
+F#F$F%F&F'F(F)F*F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT 266 21 "2\260e) index_debut_bloc" }{TEXT -1 100 "(i) d
onne pour r\351sultat l'index de la premi\350re case du bloc 3x3 auque
l appartient la case d'index i:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 90 "index_debut_bloc:=proc(i::nonnegint)\n  27*iquo(ligne(i),3)+3*
iquo(colonne(i),3)\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "seq(index_debut_bloc(k),k=0..80);" }}{PARA 200 "" 1 "" 
{XPPMATH 20 "6]p\"\"!F#F#\"\"$F$F$\"\"'F%F%F#F#F#F$F$F$F%F%F%F#F#F#F$F
$F$F%F%F%\"#FF&F&\"#IF'F'\"#LF(F(F&F&F&F'F'F'F(F(F(F&F&F&F'F'F'F(F(F(
\"#aF)F)\"#dF*F*\"#gF+F+F)F)F)F*F*F*F+F+F+F)F)F)F*F*F*F+F+F+" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 267 5 "3
\260a) " }{TEXT -1 1 " " }{TEXT 268 8 "OK_ligne" }{TEXT -1 75 "(T::arr
ay,v::posint,i::nonnegint) donne pour r\351sultat une valeur bool\351e
nne " }{TEXT 269 4 "true" }{TEXT -1 4 " ou " }{TEXT 270 5 "false" }
{TEXT -1 91 " selon que le chiffre v figure\nd\351j\340 ou non sur la \+
ligne de la case d'index i du tableau T. " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 171 "OK_ligne:=proc(T::array,v::posint,i::nonnegint)\nl
ocal id,x;\nid:=index_debut_ligne(i):\nfor x from id to id+8 do\n  if \+
T[x]=v then return false end if\nend do:\ntrue\nend proc:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "OK_ligne(sudoku,7,32),OK_lig
ne(sudoku,5,32);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueG%&falseG" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 271 15 
"3\260b) OK_colonne" }{TEXT -1 75 "(T::array,v::posint,i::nonnegint) d
onne pour r\351sultat une valeur bool\351enne " }{TEXT 272 4 "true" }
{TEXT -1 4 " ou " }{TEXT 273 5 "false" }{TEXT -1 94 " selon que le chi
ffre v figure\nd\351j\340 ou non sur la colonne de la case d'index i d
u tableau T.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "OK_colon
ne:=proc(T::array,v::posint,i::nonnegint)\nlocal id,x;\nid:=index_debu
t_colonne(i):\nfor x from id to id+72 by 9 do\n  if T[x]=v then return
 false end if\nend do:\ntrue\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "OK_colonne(sudoku,5,32),OK_colonne(sudoku,1,32);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueG%&falseG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 274 12 "3\260c) OK_bloc" }{TEXT -1 75 "(T::array,v::posin
t,i::nonnegint) donne pour r\351sultat une valeur bool\351enne " }
{TEXT 275 4 "true" }{TEXT -1 4 " ou " }{TEXT 276 5 "false" }{TEXT -1 
91 " selon que le chiffre v figure\nd\351j\340 ou non dans le bloc de \+
la case d'index i du tableau T. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 210 "OK_bloc:=proc(T::array,v::posint,i::nonnegint)\nloca
l id,x,y;\nid:=index_debut_bloc(i):\nfor x from 0 to 2 do\n  for y fro
m 0 to 2 do \n     if T[id+9*x+y]=v then return false end if\n  end do
\nend do:\ntrue\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "OK_bloc(sudoku,3,32),OK_bloc(sudoku,8,32);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%%trueG%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 277 7 "3\260d) OK" }{TEXT -1 55 "(T::arra
y,v::posint,i::nonnegint) donne pour r\351sultat  " }{TEXT 278 6 "fals
e " }{TEXT -1 58 "si i=81 et sinon donne pour r\351sultat une valeur b
ool\351enne " }{TEXT 279 4 "true" }{TEXT -1 4 " ou " }{TEXT 280 6 "fal
se\n" }{TEXT -1 117 "selon que le chiffre v peut \352tre plac\351 ou n
on dans la case d'index i du tableau T, conform\351ment \340 la r\350g
le du jeu de " }{TEXT 281 6 "Sudoku" }{TEXT -1 2 ". " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 148 "OK:=proc(T::array,v::posint,i::nonnegint
)\nif i=81 then return false \nelse OK_ligne(T,v,i) and OK_colonne(T,v
,i) and OK_bloc(T,v,i) end if\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "OK(sudoku,2,32),OK(sudoku,4,32);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$%%trueG%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 282 10 "4\260 suivant" }{TEXT -1 44 "(T::
array,i::nonnegint) donne pour r\351sultat:" }}{PARA 0 "" 0 "" {TEXT 
-1 58 " i si la case d'index i du tableau T contient 0, ou sinon " }}
{PARA 0 "" 0 "" {TEXT -1 80 " l'index de la prochaine case modifiable \+
suivant la case d'index i du tableau T:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 140 "suivant:=proc(T::array,i::nonnegint)\nlocal id;\nid:
=i;\nwhile T[id]<>0 do\n  if id<80 then id:=id+1 else return 81 end if
\nend do:\nid\nend proc:\n" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 39 "suivant(sudoku,32), suivant(sudoku,33);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$\"#K\"#O" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "" 0 "" {TEXT 283 14 "5\260 index_fixes" }{TEXT -1 73 
"(T) donne pour r\351sultat la liste des index des cases fix\351es du \+
tableau T:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "index_fixes:
=proc(T::array)\nlocal k,f;\nf:=NULL;\nfor k from 0 to 80 do\n  if T[k
]>0 then f:=f,k end if\nend do;\n[f]\nend proc:\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "index_fixes(sudoku);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7:\"\"!\"\"$\"#5\"#6\"#9\"#:\"#=\"#?\"#A\"#E\"#F\"#I\"#
L\"#M\"#N\"#Q\"#R\"#W\"#]\"#a\"#q\"#r\"#w\"#x" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 284 11 "6\260 solution" }
{TEXT -1 111 "(T) donne pour r\351sultat la solution du sudoku T ; la \+
proc\351dure fonctionne sur un algorithme de retour arri\350re (" }
{TEXT 285 12 "backtracking" }{TEXT -1 3 ") :" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 660 "solution:=proc(T::array)\nlocal index,valeur,fi
xes;\n\nindex:=0: valeur:=0: fixes:=index_fixes(T):   \nwhile index<81
 do  \n  index:=suivant(T,index): \n  valeur:=valeur+1:\n  if OK(T,val
eur,index) then \n     T[index]:=valeur: valeur:=0: \n  else \n    if \+
valeur=9 then \n     valeur:=0: \n     do\n       T[index]:=0: \n     \+
  index:=index-1:\n       while member(index,fixes) do\n         index
:=index-1\n       end do:\n       if index<0 then \n         error \"P
as de solution trouv\351e pour ce sudoku.\" \n       end if;\n       i
f T[index]<9 then \n         valeur:=T[index]: T[index]:=0:\n         \+
break \n       end if:\n     end do\n    end if\n  end if:\nend do:\ne
val(T)\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "affi
cher(solution(sudoku));" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }{GLPLOT2D 
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\"\"*F87$$\"3#******\\(Quh>!#=F97$$\"3S++DJnkoOF>F97$$\"3c++]7vB)e&F>F
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F[`lF]x7$F[`lF`x7$F[`lFcx7$F[`lFfx7$F[`lFix7$F[`lF\\y7$F[`lF_y7$F[`lFb
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$FcclFf^l7$FcclFi^l7$FcclF\\_l7$FcclF__l7$FcclFb_l7$FcclFe_lF]clFiu-F3
6%7S7$$\"\"$F8Fdv7$FiflFhv7$FiflF[w7$FiflF^w7$FiflFaw7$FiflFdw7$FiflFg
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l7$FiflFa]l7$FiflFd]l7$FiflFg]l7$FiflFj]l7$FiflF]^l7$FiflF`^l7$FiflFc^
l7$FiflFf^l7$FiflFi^l7$FiflF\\_l7$FiflF__l7$FiflFb_l7$FiflFe_lFeuFiu-F
36%7S7$$\"\"%F8Fdv7$F_jlFhv7$F_jlF[w7$F_jlF^w7$F_jlFaw7$F_jlFdw7$F_jlF
gw7$F_jlFjw7$F_jlF]x7$F_jlF`x7$F_jlFcx7$F_jlFfx7$F_jlFix7$F_jlF\\y7$F_
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