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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 16 "La course \340 z\351ro" }
}{PARA 19 "" 0 "" {TEXT -1 12 "Gilles Aldon" }}{PARA 3 "" 0 "" {TEXT 
-1 11 "Heuristique" }}{PARA 210 "" 0 "" {TEXT -1 0 "" }}{PARA 210 "" 
0 "" {TEXT -1 92 "La proc\351dure qui suit est \351crite r\351cursivem
ent  mais on pourrait l'\351crire dans une boucle..." }}{PARA 0 "" 0 "
" {TEXT -1 45 "si n est inf\351rieur \340 9 on soustrait n, sinon," }}
{PARA 0 "" 0 "" {TEXT -1 176 "si n est divisible par un \"grand\" nomb
re  (entre 5 et 9) on le divise par ce nombre et sinon, on le ram\350n
e par une soustraction ou une addition au multiple de 9 le plus proche
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 217 75 "ramene:=proc(n,compt,l)\noption remember;\nlocal x,
j,aux,k,aux2;\nx:=n;aux:=1;" }{MPLTEXT 1 217 98 "\nif x<=9 then return
 [compt+1,[op(l),x,\"-\",x,\"=0\"]]; fi;\nfor j from 5 to 8 do\nif (x \+
mod j=0) then " }{MPLTEXT 1 217 102 "\naux:=j fi;\nod;\nif  aux>=2 and
 x mod 9<>0  then ramene(x/aux,compt+1,[op(l),x,\"/\",aux,\"=\",x/aux]
) else" }{MPLTEXT 1 217 149 "\nif x mod 9=0 then ramene(x/9,compt+1,[o
p(l),x,\"/9=\",x/9]) else\n         \n          ramene(x-(x mod 9),com
pt+1,[op(l),x,\"-\",x mod 9,\"=\",x-(x mod 9)])" }{MPLTEXT 1 217 35 "
\n                       \nfi;fi;end:" }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 217 17 "ramene(990,0,[]);" }}{PARA 212 "" 1 "" {XPPMATH 20 
"6#7$\"\"&76\"$!**Q$/9=6\"\"$5\"F)Q\"/F(F$Q\"=F(\"#AF,Q\"-F(\"\"%F+\"#
=F/F'\"\"#F0F-F0Q#=0F(" }{TEXT 218 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 43 "On peut perfectionner un peu la proc\351dure :" }}}
{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 217 75 "ramene_plus_fin:=proc(n)
\nlocal i,k,x;\nx:=n;\nif op(1,ramene(x,0,[]))>4 then " }{MPLTEXT 1 
217 249 "\nfor k from 1 to 9 do\n   if op(1,ramene(x+k,0,[]))<5 then  \+
return [op(1,ramene(x+k,0,[]))+1,x,\"+\",k,\"=\",x+k,op(2,ramene(x+k,0
,[]))] fi;\n   if op(1,ramene(x-k,0,[]))<5 then  return [op(1,ramene(x
-k,0,[]))+1,x,\"-\",k,\"=\",x-k,op(2,ramene(x-k,0,[]))] fi;" }
{MPLTEXT 1 217 32 "\nod;\n fi;\nreturn ramene(x,0,[]);" }{MPLTEXT 1 
217 5 "\nend:" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 217 21 "ramene
_plus_fin(829);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#7)\"\"&\"$H)Q\"+6
\"\"\"$Q\"=F'\"$K)75F*Q\"/F'\"\")F)\"$/\"F.F,F-F)\"#8F/Q\"-F'\"\"%F)\"
\"*F2F0F2Q#=0F'" }{TEXT 218 1 " " }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 217 21 "ramene_plus_fin(851);" }}{PARA 212 "" 1 "" 
{XPPMATH 20 "6#7$\"\"'7;\"$^)Q\"-6\"\"\"&Q\"=F(\"$Y)F+Q$/9=F(\"#%*F-F'
\"\"%F*\"#!*F/F,\"#5F0Q\"/F(F)F*\"\"#F2F'F2Q#=0F(" }{TEXT 218 1 " " }}
}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 217 21 "ramene_plus_fin(990);" 
}}{PARA 212 "" 1 "" {XPPMATH 20 "6#7$\"\"&76\"$!**Q$/9=6\"\"$5\"F)Q\"/
F(F$Q\"=F(\"#AF,Q\"-F(\"\"%F+\"#=F/F'\"\"#F0F-F0Q#=0F(" }{TEXT 218 1 "
 " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 22 "En utilisant un graphe" }}
{PARA 0 "" 0 "" {TEXT -1 167 "On construit un graphe dont les sommets \+
sont les nombres de 1 \340 n+9 ; les ar\352tes relient deux nombres qu
i peuvent se d\351duire l'un de l'autre par une op\351ration permise.
" }}{PARA 0 "" 0 "" {TEXT -1 93 "On utilise ensuite les fonctionalit
\351s de Maple pour d\351terminer le plus court chemin de n \340 0." }
}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 217 15 "with(networks):" }}}
{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 217 76 "construis:=proc(n)\nloca
l vset,eset,G,i,j;\nvset:=\{seq(i,i=0..n+9)\};\neset:=\{\};" }
{MPLTEXT 1 217 69 "\nG:=graph(vset,eset);\nfor i from 0 to n+9 do\nfor
 j from i+1 to n+9 do" }{MPLTEXT 1 217 98 "\nif  j-i<=9 or (i>0 and j \+
mod i=0 and j/i<=9)  then addedge([j,i],G); addedge([i,j],G) fi;\nod;
\nod;" }{MPLTEXT 1 217 8 "\nG;\nend:" }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 217 51 "T:=shortpathtree(construis(829),0):path([0,829],T);
" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#7(\"\"!\"\"*\"#8\"$/\"\"$K)\"$H)
" }{TEXT 218 1 " " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 217 51 "T:
=shortpathtree(construis(851),0):path([0,851],T);" }{MPLTEXT 1 219 50 
"C'est un des deux cas \340 6 pas, l'autre \351tant 853 :" }}{PARA 
213 "" 1 "" {XPPMATH 20 "6#7)\"\"!\"\"*\"#:\"$?\"\"$S)\"$W)\"$^)" }
{TEXT 220 3 "   " }}}{EXCHG {PARA 210 "> " 0 "" {MPLTEXT 1 221 51 "T:=
shortpathtree(construis(853),0):path([0,853],T);" }}{PARA 212 "" 1 "" 
{XPPMATH 20 "6#7)\"\"!\"\"*\"#:\"$?\"\"$S)\"$W)\"$`)" }{TEXT 218 1 " \+
" }}}{EXCHG {PARA 211 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 69 "draw(construis(22));T:=shortpathtree(construis(22),
0):path([0,22],T);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 1313 829 829 
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