Le GAP et Rubik's Cube

Difficulté: 1/20 facile-

24 Fevr 2024

GAP GAP est un logiciel de calcul formel, il permet de calculer un certain nombre de caractéristiques du Rubik's Cube, du Pocket, du Skewb, ...
comme le nombre d'états, le nombre de J-conjugaison-classes, ...

Commentaire

Calcul le nombre de J-conjugaison classes du Rubik's Cube


Autocollant numérotés

Voici un programme en GAP qui permet de calculer :
1) Le nombre de J-conjugaison-classes de Rubik's Cube
2) Le nombre de D-conjugaison-classes de Rubik's Cube
3) Le nombre de J-conjugaison-classes de Pocket
4) Le nombre de D-conjugaison-classes de Pocket

Programme 3 :

#gap_J-rubik.txt
# 5 6 7
# 4 H 8
# 3 2 1
#25 28 23|21 26 19|17 32 31|29 30 27
#38 G 36|12 A 10|34 D 40|16 P 14
#43 44 37|39 42 33|35 48 45|47 46 41
# 11 18 9
# 20 B 24
# 13 22 15

# Iso=le groupe isometrie du cube (48)
j1 := (6, 46, 18, 26)(8, 14, 24, 12)(38, 48, 36, 32)(2, 30, 22, 42)(16, 20, 10, 4)(28, 40, 44, 34)
(5, 45, 11, 17)(7, 13, 9, 3)(21, 31, 41, 35)(43, 33, 23, 29)(1, 25, 15, 37)(47, 39, 19, 27) ;

j2 := (6, 16, 22, 14)(8, 24, 20, 4)(38, 30, 40, 46)(2, 10, 18, 12)(28, 32, 48, 44)(34, 42, 36, 26)
(5, 31, 15, 43)(7, 45, 13, 25)(21, 19, 33, 39)(1, 35, 11, 23)(47, 41, 27, 29)(3, 17, 9, 37);

Iso := Group(j1,j2) ;

# Dep=le groupe isometrie+ du cube (24) ssg de Iso
d1 := (1,11)(2,18)(3,9)(4,24)(5,15)(6,22)(7,13)(8,20)(10,12)
(14,16)(17,37)(19,39)(21,33)(23,35)(25,45)(26,42)(27,47)(28,48)(29,41)
(30,46)(31,43)(32,44)(34,36)(38,40);

d2 := (1,15)(2,22)(3,13)(4,20)(5,11)(6,18)(7,9)(8,24)(10,16)(12,14)
(17,45)(19,47)(21,41)(23,43)(25,37)(26,46)(27,39)(28,44)(29,33)(30,42)
(31,35)(32,48)(34,40)(36,38);

d3 := (1,17,19)(2,32,10)(3,31,33)(4,40,42)
(5,45,39)(6,48,12)(7,35,21)(8,34,26)(9,23,29)(11,25,47)(13,43,41)
(14,22,44)(15,37,27)(16,18,28)(20,38,46)(24,36,30);

d4 := (1,35,11,23)(2,10,18,12)(3,17,9,37)(4,8,24,20)(5,31,15,43)
(6,16,22,14)(7,45,13,25)(19,33,39,21)(26,34,42,36)(27,29,47,41)
(28,32,48,44)(30,40,46,38) ;

Dep := Group(d1, d2,d3,d4) ;

# Rubik=le groupe du Rubik's Cube
pH := (2,4,6,8)(26,28,30,32) (1,3,5,7)(17,21,25,29)(19,23,27,31) ;
pB := (18,24,22,20)(42,48,46,44) (9,15,13,11)(33,45,41,37)(35,47,43,39);
pA := (2,34,18,36)(26,10,42,12) (1,35,11,23)(17,9,37,3)(19,33,39,21);
pP := (6,38,22,40)(30,14,46,16) (7,25,13,45)(29,27,41,47)(31,5,43 ,15);
pG := (4,12,20,14)(28,36,44,38) (3,39,13,27)(21,11,41,5)(23,37,43,25);
pD := (8,16,24,10)(32,40,48,34) (1,29,15,33)(17,31,45,35)(19,7,47,9);
Rubik := Group(pH,pB,pA,pP,pG,pD);
G := Rubik ;;
GG := "Rubik" ;;
J := Iso ;;
JJ := "J" ;;

# Pocket=le groupe Pocket (= Rubik sans arêtes)
#pH := (1,3,5,7)(17,21,25,29)(19,23,27,31) ;
#pB := (9,15,13,11)(33,45,41,37)(35,47,43,39);
#pA := (1,35,11,23)(17,9,37,3)(19,33,39,21);
#pP := (7,25,13,45)(29,27,41,47)(31,5,43 ,15);
#pG := (3,39,13,27)(21,11,41,5)(23,37,43,25);
#pD := (1,29,15,33)(17,31,45,35)(19,7,47,9);
#Pocket := Group(pH,pB,pA,pP,pG,pD);
#G := Pocket ;;
#GG := "Pocket" ;;
#J := Dep ;;
#JJ := "D" ;;

Jcjg := Sum(ConjugacyClasses(J),i -> (Size(i) * Size(Centralizer(G,Representative(i)))) / Size(J));;
Print("\n\n ",GG," = ", Size(G), "\n" );
Print("\n ",JJ,"-cjg = ", Jcjg, "\n" );

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DMJ: 22/02/2024









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