Hexagones

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Hexagones 
Posté par 
Diablow  25-05-14 à 23:38
Bonjour à tous,

Ce problème tourne dans ma tête depuis longtemps, mais je n'ai jamais pu lui apporter une réponse... Peut-être quelqu'un pourra-t-il m'aider ici... Le problème est simple dans son énoncé. Soit les 7 hexagones ci-dessous. On affecte à chaque sommet (notés de "a" à "x") les nombres de 1 à 24 de telle façon que la somme des 6 nombres de chaque hexagone reste constante. Il faut tout simplement savoir combien il existe de solutions.

Je ne suis jamais allé bien loin : on peut montrer que la somme de chaque hexagone est forcement comprise entre 65 et 87. assez élémentaire... L'exploration systématique des solutions via un ordi me semble assez difficile vu le nombre de possibilités ...

Quelqu'un a-t-il une idée ?

 Posté par 
dpire : Hexagones   26-05-14 à 11:27
Bonjour,

C'est un exercice plaisant.

J'arrive à 75 comme total médian.

On voit 12 sommets uniques

6 sommets doublés et 6 sommets triplés.

Je pense qu'un seule solution est possible avec bien sûr les symétriques
 Posté par 
dpire : Hexagones   26-05-14 à 17:59
Bonjour suite

Je considérais que l'hexagone central faisait

partie du jeu.

Donc le maximum est 87 et le minimum 63

On peut chercher la médiane avec  75
 Posté par 
weierstrassre : Hexagones   26-05-14 à 20:54
Bonjour, j'ai crée un algorithme qui permet de résoudre le problème.

J'espère qu'il est assez efficace pour permettre de fournir la réponse.

Je fournirais les détails après l'avoir testé, je pense que dans tout les cas, ça peut te donner une idée...
 Posté par 
Diablowre : Hexagones   26-05-14 à 21:10
Quelques résultats par trop difficiles à établir:

La somme sur chaque hexagone

Dans mon esprit, l'hexagone central faisait partie des hexagones dont la somme est constante.

Si on désigne par Ei les douzes nombres externes (abfj...), Mi les six nombres médians(denv...), Ci les six nombres centraux (him...), alors on a:

Ei + Mi + Ci = 300

Ei + 2Mi + 3Ci = 7*somme

Ci = somme

Ce qui donne, tous les calculs réalises, que la somme est comprise entre 65 et 85 (bornes incluses)

Symétrie sur la somme sur les hexagones

Il existe une symétrie des solutions par rapport à 75. Supposons qu'il existe une solution dont la somme sur les hexagones égale 75+. Si on remplace chaque nombre de la solution par 25 moins ce nombre, alors, il est simple de constater que la figure résultante est aussi une solution mais dont la somme vaut 75-

Réduire le nombre de configuration à explorer

Concernant les différentes solutions, d'une façon simple, plusieurs facteurs qui permettent de réduire le problème:

Rotations : h est inférieur au min (i, m, q, p, l)

Symétrie : m < p

Sur les bords : a<b et c<g et o<s et w<x et r<v et f<j 

Si je compte bien, il existe 24! possibilités de disposer les nombres.

Si on tient compte des 3 points ci dessus on arrive à 24!/(2^8*3)=807875523090155520000 (ce qui fait beaucoup)

Le programme le plus rapide que j'ai réalisé évalue environ 10.000.000 de grilles par seconde. Soit environ 2.500.000 ans de calcul.
 Posté par 
Diablowre : Hexagones   26-05-14 à 21:13
Citation :

de weierstrass:

Bonjour, j'ai crée un algorithme qui permet de résoudre le problème. 

J'espère qu'il est assez efficace pour permettre de fournir la réponse. 

Je fournirais les détails après l'avoir testé, je pense que dans tout les cas, ça peut te donner une idée...

je suis impatient...
 Posté par 
weierstrassre : Hexagones   26-05-14 à 22:12
Comme tu t'y attendait, (et moi aussi un peu...), le programme semble un peu débordé par la tâche 

Néanmoins, voila mon idée d'algorithme, je ne sais pas si tu avais pensé à un similaire, il permet quand même de réduire pas mal le nombre annoncé de grille à tester.

Il est basé sur un principe récursif,n codé sur python.

Il à l'air de pas mal marcher, on peut visualiser les étapes en mettant des "print" aprés chaque boucle "for"

Mon programme est assez laid et on peut sans doute le simplifier, par exemple en utilisant des listes pour éviter les [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16... 

Il n'utilise pas le procédé de récursion visant à appeler la fonction dans la fonction, car je ne connais pas grand chose aux programmes récursifs, et que ce n'était pas pratique pour les conditions, mais le principe reste le même.

Je n'ai pas une très grande experience informatique, si vous voyez une erreur dans le raisonnement , dans l'écriture de l'algorithme, ou des améliorations possibles, n'hésite pas à le faire remarquer (surtout que comme le programme est long, on est pas à l'abri d'une faute vicieuse.)

A part ça, le programme n'est pas si compliqué à comprendre:

Le principe:

 Cliquez pour afficher
j'ai numéroté les sommets de 1 à 12 selon un ordre précis.

on associe à chaque sommet les lettres a1,a2,a3...

pour simplifier, on décide a1 = 1 et on cherchera la somme des hexagones égale à 75...

En changeant un peu le programme, on pourra faire les autres cas qui nous interesse?

En effet, grâce aux symétries, on peut décider de placer le 1 en 1, en 7, ou en 8, on aura parcouru tout les cas possibles.

(on verra que déjà avec ses conditions, le programme ne termine même pas sa tâche...)

Le principe est que a2 va prendre: la valeur 1 (comme ce n'est pas possible, le 1 est déjà pris, il va prendre la valeur 2, puis a3 va prendre la valeur 1 puis 2 puis 3, et ainsi de suite...

pendant le programme on vérifie d'autres conditions (la somme de l'hexagone fait 75, on prédit si avec les 4 ou 5 valeurs de l'hexagone déjà placée, on pourra trouver des nombres tels que la somme de l'hexagone fasse 75...)

Tout que ça marche, on continue de leur attribuer des valeurs, si ça ne marche pas, le programme change les valeurs des ak précédents.

L'avantage de cet algorithme est supprime de nombreux cas à tester:

si une configuration n'est pas bonne à un certain endroit, il ne testera pas toute les possibilités contenant cette configuration.

Voilà, j'espère que tu as a peu près compris...

Pour t'aider à comprendre l'algorithme:

ak in [a1,a2,a3...]==True

 on vérifie si la valeur que l'on viens d'attribuer n'aurait pas déjà été placée sur la grille

a1+a2+a6+a7<28

si cette somme est inférieure à 28, si on met derrière un 24 et un 23 pour compenser un si petit nombre, on aura pas 75, donc inutile de continuer avec ça...

continue

 on viens d'observer que si on continuait avec la dernière valeur renseignée, ça ne marcherait pas, donc ceci fait revenir au début de la dernière boucle "for" et attribue le numéro suivant.Sans ça, l'algorithme effectuerait tout les "for situés derrière pour rien (quel gâchis )

Avec ça tu devrais pouvoir t'en sortir sur la compréhension de l'algorithme...

LE voici...

bon courage:

 Cliquez pour afficher
def hexagones():

    L=[]

    a1=1

    for a2 in range(1,25):

        if a2 in [a1] == True:

            continue

        for a3 in range(1,25):

            if a3 in [a1,a2] == True:

                continue

            for a4 in range(1,25):

                if (a4 in [a1,a2,a3] == True) or (a1+a2+a3+a4<28) or (a1+a2+a3+a4>72):

                    continue

                for a5 in range(1,25):

                    a6=75-(a1+a2+a3+a4+a5)

                    if (a5 in [a1,a2,a3,a4]==True) or  (a1+a2+a3+a4+a5<51) or (a1+a2+a3+a4+a5>74) or (a6 in [a1,a2,a3,a4,a5]==True):

                        continue

                    for a7 in range(1,25):

                        if a7 in [a1,a2,a3,a4,a5,a6]==True:

                            continue

                        for a8 in range(1,25):

                            if (a8 in [a1,a2,a3,a4,a5,a6,a7]==True) or (a1+a2+a7+a8<28) or (a1+a2+a7+a8>72):

                                continue

                            for a9 in range(1,25):

                                a10=75-(a1+a2+a7+a8+a9)

                                if (a9 in [a1,a2,a3,a4,a5,a6,a7,a8]) or (a1+a2+a7+a8+a9<51) or (a1+a2+a7+a8+a9>74) or (a10 in [a1+a2+a3+a4+a5+a6+a7+a8+a9]==True):

                                    continue

                                for a11 in range(1,25):

                                    if (a11 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10]==True) or (a2+a3+a10+a11<28) or (a2+a3+a10+a11>72):

                                        continue

                                    for a12 in range(1,25):

                                        a13 = 75-(a2+a3+a10+a11+a12)

                                        if (a12 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11]==True) or (a2+a3+a10+a11+a12<51) or (a2+a3+a10+a11+a12>74) or (a13 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12]==True):

                                            continue

                                        for a14 in range(1,25):

                                            if (a14 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13]==True) or (a3+a4+a13+a14<28) or (a3+a4+a13+a14>72):

                                                continue

                                            for a15 in range(1,25):

                                                a16=75-(a3+a4++a13+a14+a15)

                                                if (a15 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14]==True) or (a3+a4+a13+a14+a15<51) or (a3+a4+a13+a14+a15>72) or (a16 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15]==True):

                                                    continue

                                                for a17 in range(1,25):

                                                    if (a17 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16]==True) or (a4+a5+a16+a17<28) or (a4+a5+a16+a17>72):

                                                        continue

                                                    for a18 in range(1,25):

                                                        a19=75-(a4+a5+a16+a17+a18)

                                                        if (a18 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17]==True) or (a4+a5+a16+a17+a18<51) or (a4+a5+a16+a17+a18>74) or (a19 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18]==True):

                                                            continue

                                                        for a20 in range(1,25):

                                                            if(a20 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19]==True) or (a5+a6+a19+a20<28) or (a5+a6+a19+a20>72):

                                                                continue

                                                            for a21 in range(1,25):

                                                                a22=75-(a5+a6+a19+a20+a21)

                                                                if (a21 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20]==True) or (a5+a6+a19+a20+a21<51) or (a5+a6+a19+a20+a21>74) or (a22 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21]==True) or (a7+a1+a6+a22<28) or (a7+a1+a6+a22>72):

                                                                    continue

                                                                for a23 in range(1,25):

                                                                    a24=75-(a7+a1+a6+a22+a23)

                                                                    if (a23 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22]==True) or (a7+a1+a6+a22+a23<51) or (a7+a1+a6+a22+a23>74) or (a24 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22,a23]==True):

                                                                        continue

                                                                    L.append([a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22,a23,a24])

    return(L)

cordialement.

 Posté par 
dpire : Hexagones   27-05-14 à 06:58
Bonjour,

On apprécie le travail de Weierstrass !

On attend une autre méthode.

Ou j'en suis:

1/somme de 1 à 24 (n*n+1)/2 =300

2/sommets uniques  u (abcfgjorsvwx)

3/sommets doubles  d(dehiklmnpqtu)

4/total logique 450 (150 u +300 d)

5/chaque hexagone a 2u+4d ---> 75

6/on décide d'un seul cas englobant toutes les symétries.

7/on teste sur un tableur pour obtenir 75

et en réduisant les différences par groupes.

de 2 si possible.

8/le hasard fera le reste..
 Posté par 
weierstrassre : Hexagones   27-05-14 à 07:06
Il y avait un problème, l'algorithme ne changeait pas de nombre si il était déjà présent dans la grille. je crois avoir réglé le problème, à checker...

 Cliquez pour afficher
def hexagones():

    L=[]

    a1=1

    for a2 in range(1,25):

        print(a2,"a2")

        if (a2 in [a1])==True:

            continue

        for a3 in range(1,25):

            print(a3,"a3")

            if (a3 in [a1,a2])==True:

                continue

            for a4 in range(1,25):

                print(a4,"a4")

                if ((a4 in [a1,a2,a3])==True) or (a1+a2+a3+a4<28) or (a1+a2+a3+a4>72):

                    continue

                for a5 in range(1,25):

                    print(a5,"a5")

                    a6=75-(a1+a2+a3+a4+a5)

                    if ((a5 in [a1,a2,a3,a4])==True) or  (a1+a2+a3+a4+a5<51) or (a1+a2+a3+a4+a5>74) or ((a6 in [a1,a2,a3,a4,a5])==True):

                        continue

                    for a7 in range(1,25):

                        if (a7 in [a1,a2,a3,a4,a5,a6])==True:

                            continue

                        for a8 in range(1,25):

                            if ((a8 in [a1,a2,a3,a4,a5,a6,a7])==True) or (a1+a2+a7+a8<28) or (a1+a2+a7+a8>72):

                                continue

                            for a9 in range(1,25):

                                a10=75-(a1+a2+a7+a8+a9)

                                if ((a9 in [a1,a2,a3,a4,a5,a6,a7,a8])==True) or (a1+a2+a7+a8+a9<51) or (a1+a2+a7+a8+a9>74) or ((a10 in [a1+a2+a3+a4+a5+a6+a7+a8+a9])==True):

                                    continue

                                for a11 in range(1,25):

                                    if ((a11 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10])==True) or (a2+a3+a10+a11<28) or (a2+a3+a10+a11>72):

                                        continue

                                    for a12 in range(1,25):

                                        a13 = 75-(a2+a3+a10+a11+a12)

                                        if ((a12 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11])==True) or (a2+a3+a10+a11+a12<51) or (a2+a3+a10+a11+a12>74) or ((a13 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12])==True):

                                            continue

                                        for a14 in range(1,25):

                                            if ((a14 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13])==True) or (a3+a4+a13+a14<28) or (a3+a4+a13+a14>72):

                                                continue

                                            for a15 in range(1,25):

                                                a16=75-(a3+a4++a13+a14+a15)

                                                if ((a15 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14])==True) or (a3+a4+a13+a14+a15<51) or (a3+a4+a13+a14+a15>72) or ((a16 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15])==True):

                                                    continue

                                                for a17 in range(1,25):

                                                    if ((a17 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16])==True) or (a4+a5+a16+a17<28) or (a4+a5+a16+a17>72):

                                                        continue

                                                    for a18 in range(1,25):

                                                        a19=75-(a4+a5+a16+a17+a18)

                                                        if ((a18 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17])==True) or (a4+a5+a16+a17+a18<51) or (a4+a5+a16+a17+a18>74) or ((a19 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18])==True):

                                                            continue

                                                        for a20 in range(1,25):

                                                            if((a20 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19])==True) or (a5+a6+a19+a20<28) or (a5+a6+a19+a20>72):

                                                                continue

                                                            for a21 in range(1,25):

                                                                a22=75-(a5+a6+a19+a20+a21)

                                                                if ((a21 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20])==True) or (a5+a6+a19+a20+a21<51) or (a5+a6+a19+a20+a21>74) or ((a22 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21])==True) or (a7+a1+a6+a22<28) or (a7+a1+a6+a22>72):

                                                                    continue

                                                                for a23 in range(1,25):

                                                                    a24=75-(a7+a1+a6+a22+a23)

                                                                    if ((a23 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22])==True) or (a7+a1+a6+a22+a23<51) or (a7+a1+a6+a22+a23>74) or ((a24 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22,a23])==True):

                                                                        continue

                                                                    L.append([a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22,a23,a24])

    return(L)
 Posté par 
dpire : Hexagones   27-05-14 à 14:01
Bonjour,

je donne une solution avec une petite erreur...

abcdefghijklmnopqrstuvwxy

22 3 9 7 11 5 1 14 18 16 20 24 6 19 10 8 12 6 21 13 17 15 2 23
 Posté par 
weierstrassre : Hexagones   27-05-14 à 19:29
Si j'ai bien compris ta méthode dpi, elle permet de trouver facilement une solution au problème.

Malheureusement, le but est un peu plus dur, c'est trouver l'ensemble des solutions...
 Posté par 
dpire : Hexagones   28-05-14 à 06:50
Bonjour,

En en trouvant une ,on en déduit un

ensemble par permutations et autres symétries.

Mais je pense qu'il n'y a pas de méthode générale,

mais plutôt comme au sudoku des séries de sous-méthodes

pour arriver à réduire la grille.

Ici par exemple, une sous méthode qui ne marche pas...

c'est de coupler les sommets voisins de telle sorte

que leur total soit 25 (en cherchant le fameux 12.5 de moyenne)

C'est le genre de topic qui irait bien en "énigme" n'est-ce

pas Godefroy ?
 Posté par 
weierstrassre : Hexagones   28-05-14 à 18:09
 Posté par 
weierstrassre : Hexagones   28-05-14 à 18:10
une infime partie des solutions:

 Cliquez pour afficher
[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 15, 10, 8, 5, 7, 4, 11, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 15, 10, 8, 5, 7, 11, 4, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 15, 10, 8, 5, 7, 11, 4, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 17, 8, 4, 11, 7, 5, 6, 10, 15, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 17, 8, 4, 11, 7, 5, 6, 10, 16, 15]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 17, 8, 4, 11, 7, 5, 10, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 17, 8, 4, 11, 7, 5, 10, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 13, 17, 8, 4, 11, 7, 6, 5, 10, 15, 16]

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[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 19, 11, 8, 4, 13, 5, 7, 6, 10, 16, 15]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 19, 11, 8, 13, 4, 5, 6, 7, 10, 15, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 19, 11, 8, 13, 4, 5, 6, 7, 10, 16, 15]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 19, 11, 8, 13, 4, 5, 7, 6, 10, 15, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 14, 20, 12, 19, 11, 8, 13, 4, 5, 7, 6, 10, 16, 15]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 6, 10, 8, 16, 17]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 6, 10, 8, 17, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 8, 10, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 8, 10, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 10, 6, 8, 16, 17]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 10, 6, 8, 17, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 10, 8, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 7, 14, 4, 10, 8, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 6, 10, 8, 16, 17]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 6, 10, 8, 17, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 8, 10, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 8, 10, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 10, 6, 8, 16, 17]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 10, 6, 8, 17, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 10, 8, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 13, 5, 14, 7, 4, 10, 8, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 8, 13, 5, 7, 10, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 8, 13, 5, 7, 10, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 8, 13, 5, 10, 7, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 8, 13, 5, 10, 7, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 13, 8, 5, 7, 10, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 13, 8, 5, 7, 10, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 13, 8, 5, 10, 7, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 12, 14, 4, 13, 8, 5, 10, 7, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 6, 10, 8, 16, 17]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 6, 10, 8, 17, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 8, 10, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 8, 10, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 10, 6, 8, 16, 17]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 10, 6, 8, 17, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 10, 8, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 7, 14, 4, 10, 8, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 14, 7, 4, 6, 10, 8, 16, 17]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 14, 7, 4, 6, 10, 8, 17, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 14, 7, 4, 8, 10, 6, 16, 19]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 14, 7, 4, 8, 10, 6, 19, 16]

[1, 2, 3, 22, 23, 24, 9, 18, 21, 24, 15, 11, 20, 13, 12, 5, 14, 7, 4, 10, 6, 8, 16, 17]

moi non plus je m'attendais pas à ça...
 Posté par 
dpire : Hexagones   28-05-14 à 18:56
>Weierstrass

J'ai regardé et mis sur mon tableur...

tu dois avoir une erreur dans ton algorithme car

1/sur mon tableur cela ne fait pas des totaux identiques

2/le 24 sort 2 fois
 Posté par 
weierstrassre : Hexagones   28-05-14 à 19:08
effectivement, je n'avais pas remarqué.

pour les totaux, je les trouves identiques, avec mon système de numérotation.

(enfin, j'en ai essayé deux et ça marchait...)
 Posté par 
weierstrassre : Hexagones   28-05-14 à 19:16
Quel boulet, j'ai mis des plus au lieu de virgules...

Il en reste néanmoins beaucoup:

 Cliquez pour afficher
[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 10, 14, 4, 7, 5, 6, 9, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 10, 14, 4, 7, 5, 9, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 10, 14, 4, 7, 5, 9, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 10, 14, 7, 4, 5, 6, 9, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 10, 14, 7, 4, 5, 6, 9, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 10, 14, 7, 4, 5, 9, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 10, 14, 7, 4, 5, 9, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 4, 7, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 4, 7, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 4, 7, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 4, 7, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 5, 6, 9, 4, 7, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 5, 6, 9, 4, 7, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 5, 6, 9, 7, 4, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 5, 6, 9, 7, 4, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 6, 5, 9, 4, 7, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 6, 5, 9, 4, 7, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 6, 5, 9, 7, 4, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 6, 5, 9, 7, 4, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 4, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 4, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 4, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 4, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 9, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 9, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 9, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 7, 9, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 9, 7, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 9, 7, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 9, 7, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 11, 14, 10, 9, 7, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 6, 7, 9, 4, 5, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 6, 7, 9, 4, 5, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 6, 7, 9, 5, 4, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 6, 7, 9, 5, 4, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 7, 6, 9, 4, 5, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 7, 6, 9, 4, 5, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 7, 6, 9, 5, 4, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 13, 14, 8, 7, 6, 9, 5, 4, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 6, 7, 4, 5, 9, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 6, 7, 4, 5, 9, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 6, 7, 4, 9, 5, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 6, 7, 4, 9, 5, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 7, 6, 4, 5, 9, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 7, 6, 4, 5, 9, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 7, 6, 4, 9, 5, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 8, 13, 7, 6, 4, 9, 5, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 4, 7, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 4, 7, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 4, 7, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 4, 7, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 5, 6, 9, 4, 7, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 5, 6, 9, 4, 7, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 5, 6, 9, 7, 4, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 5, 6, 9, 7, 4, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 6, 5, 9, 4, 7, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 6, 5, 9, 4, 7, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 6, 5, 9, 7, 4, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 6, 5, 9, 7, 4, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 4, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 4, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 4, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 4, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 9, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 9, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 9, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 7, 9, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 9, 7, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 9, 7, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 9, 7, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 11, 10, 9, 7, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 6, 7, 9, 4, 5, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 6, 7, 9, 4, 5, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 6, 7, 9, 5, 4, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 6, 7, 9, 5, 4, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 7, 6, 9, 4, 5, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 7, 6, 9, 4, 5, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 7, 6, 9, 5, 4, 10, 11, 17]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 14, 13, 8, 7, 6, 9, 5, 4, 10, 17, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 5, 9, 6, 4, 7, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 5, 9, 6, 4, 7, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 5, 9, 6, 7, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 5, 9, 6, 7, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 7, 9, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 7, 9, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 7, 9, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 7, 9, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 5, 6, 4, 7, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 5, 6, 4, 7, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 5, 6, 7, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 5, 6, 7, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 7, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 7, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 7, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 8, 10, 9, 7, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 7, 9, 6, 4, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 7, 9, 6, 4, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 7, 9, 6, 5, 4, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 7, 9, 6, 5, 4, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 9, 7, 6, 4, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 9, 7, 6, 4, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 9, 7, 6, 5, 4, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 16, 15, 17, 10, 8, 9, 7, 6, 5, 4, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 17, 14, 9, 16, 11, 4, 8, 7, 5, 6, 10, 13, 15]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 17, 14, 9, 16, 11, 4, 8, 7, 5, 6, 10, 15, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 17, 14, 9, 16, 11, 4, 8, 7, 6, 5, 10, 13, 15]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 17, 14, 9, 16, 11, 4, 8, 7, 6, 5, 10, 15, 13]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 17, 14, 9, 16, 11, 7, 8, 4, 5, 6, 13, 10, 15]

[1, 2, 3, 22, 23, 24, 12, 19, 20, 21, 18, 17, 14, 9, 16, 11, 7, 8, 4, 5, 6, 13, 15, 10]

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[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 9, 10, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 9, 10, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 9, 10, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 9, 10, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 10, 9, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 10, 9, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 10, 9, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 8, 17, 7, 10, 9, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 5, 8, 4, 6, 11, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 5, 8, 4, 6, 11, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 5, 8, 4, 11, 6, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 5, 8, 4, 11, 6, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 8, 5, 4, 6, 11, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 8, 5, 4, 6, 11, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 8, 5, 4, 11, 6, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 9, 10, 13, 8, 5, 4, 11, 6, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 5, 8, 4, 6, 11, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 5, 8, 4, 6, 11, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 5, 8, 4, 11, 6, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 5, 8, 4, 11, 6, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 8, 5, 4, 6, 11, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 8, 5, 4, 6, 11, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 8, 5, 4, 11, 6, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 10, 9, 13, 8, 5, 4, 11, 6, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 4, 10, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 4, 10, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 4, 10, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 4, 10, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 9, 10, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 9, 10, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 9, 10, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 9, 10, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 4, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 4, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 4, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 4, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 9, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 9, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 9, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 11, 14, 7, 10, 9, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 4, 9, 6, 5, 10, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 4, 9, 6, 5, 10, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 4, 9, 6, 10, 5, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 4, 9, 6, 10, 5, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 9, 4, 6, 5, 10, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 9, 4, 6, 5, 10, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 9, 4, 6, 10, 5, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 13, 8, 11, 9, 4, 6, 10, 5, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 4, 10, 5, 6, 9, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 4, 10, 5, 6, 9, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 4, 10, 5, 9, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 4, 10, 5, 9, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 10, 4, 5, 6, 9, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 10, 4, 5, 6, 9, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 10, 4, 5, 9, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 7, 11, 10, 4, 5, 9, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 4, 10, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 4, 10, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 4, 10, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 4, 10, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 9, 10, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 9, 10, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 9, 10, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 9, 10, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 4, 9, 5, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 4, 9, 5, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 4, 9, 6, 5, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 4, 9, 6, 5, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 9, 4, 5, 6, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 9, 4, 5, 6, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 9, 4, 6, 5, 13, 8, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 14, 11, 7, 10, 9, 4, 6, 5, 13, 17, 8]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 7, 10, 4, 5, 8, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 7, 10, 4, 5, 8, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 7, 10, 4, 8, 5, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 7, 10, 4, 8, 5, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 10, 7, 4, 5, 8, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 10, 7, 4, 5, 8, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 10, 7, 4, 8, 5, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 6, 9, 10, 7, 4, 8, 5, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 9, 10, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 9, 10, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 9, 10, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 9, 10, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 10, 9, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 10, 9, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 10, 9, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 16, 18, 17, 8, 7, 10, 9, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 6, 9, 5, 4, 8, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 6, 9, 5, 4, 8, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 6, 9, 5, 8, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 6, 9, 5, 8, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 9, 6, 5, 4, 8, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 9, 6, 5, 4, 8, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 9, 6, 5, 8, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 7, 17, 10, 9, 6, 5, 8, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 5, 10, 6, 4, 7, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 5, 10, 6, 4, 7, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 5, 10, 6, 7, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 5, 10, 6, 7, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 7, 10, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 7, 10, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 7, 10, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 7, 10, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 5, 6, 4, 7, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 5, 6, 4, 7, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 5, 6, 7, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 5, 6, 7, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 7, 4, 5, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 7, 4, 5, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 7, 4, 6, 5, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 8, 17, 9, 10, 7, 4, 6, 5, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 5, 7, 4, 6, 10, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 5, 7, 4, 6, 10, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 5, 7, 4, 10, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 5, 7, 4, 10, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 7, 5, 4, 6, 10, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 7, 5, 4, 6, 10, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 7, 5, 4, 10, 6, 8, 13, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 11, 14, 7, 5, 4, 10, 6, 8, 17, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 5, 10, 7, 4, 6, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 5, 10, 7, 4, 6, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 5, 10, 7, 6, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 5, 10, 7, 6, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 7, 10, 5, 4, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 7, 10, 5, 4, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 7, 10, 5, 6, 4, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 7, 10, 5, 6, 4, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 5, 7, 4, 6, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 5, 7, 4, 6, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 5, 7, 6, 4, 11, 13, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 5, 7, 6, 4, 11, 14, 13]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 7, 5, 4, 6, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 7, 5, 4, 6, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 7, 5, 6, 4, 13, 11, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 9, 17, 8, 10, 7, 5, 6, 4, 13, 14, 11]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 5, 6, 8, 4, 9, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 5, 6, 8, 4, 9, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 5, 6, 8, 9, 4, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 5, 6, 8, 9, 4, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 6, 5, 8, 4, 9, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 6, 5, 8, 4, 9, 7, 17, 14]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 6, 5, 8, 9, 4, 7, 14, 17]

[1, 2, 3, 22, 23, 24, 12, 20, 19, 21, 15, 18, 16, 10, 13, 11, 6, 5, 8, 9, 4, 7, 17, 14]

j'espère qu'ils sont bons cela...

 Posté par 
weierstrassre : Hexagones   28-05-14 à 20:37
J'en suis à

Je vous fait la liste ou ça ira? 

Citation :
L'exploration systématique des solutions via un ordi me semble assez difficile vu le nombre de possibilités ... 

tu l'as dit (et surtout vu le nombre de solutions...)

Citation :
C'est le genre de topic qui irait bien en "énigme" n'est-ce

pas Godefroy ?

déjà que ca râle quand deux solutions sont possibles, je ne crois pas que certains en apprécieraient plus d'un million!

si la réponse de l'énigme se compte en Téraoctets, non merci...

Merci quand même pour cette énigme qui fut sympa à programmer, si tu en as d'autres dans le genre(avec moins de solutions, quand même ), je suis preneur...
 Posté par 
weierstrassre : Hexagones   28-05-14 à 20:48
pardon, j'en suis à 1500000 solutions en à peine 1/2heure (sans compter les symétries...)

un résultat remarquable qui aurait pu améliorer mon programme:

notons:

S1 la somme des nombres au sommets n'appartenant qu'à 1 hexagone

S2 ---------------------------------------------------2 hexagones

S3 ---------------------------------------------------3 hexagones

on à S1+S2+S3 = 24*25/3 = 300

7*75-S2-2S3 = 300

trivialement, S3 = 75

On en déduit:

S2 = 75

S1 = 150
 Posté par 
Diablowre : Hexagones   28-05-14 à 22:14
En ce qui me concerne, pour générer les différentes possibilités, en tenant compte des différentes symétries, j'utilise la numérotation des sommets ci-dessous.

ix represente le nombre affecté au sommet x

i1 est compris entre 1 et 11

i1 < i2

i1 < i3 et i3<>i2

i1 < i4 et i4<>i3 et i4<>i2

i1 < i5 et i5<>i4 et i5<>i3 et i5<>i2

i2 < i6 et i6<>i5 et i6<>i4 et i6<>i3

i1+i2+i3+i4+i5+i6 compris entre 65 et 85

On génère les noeuds intermédiaires (i7 à i12)

i7 de 1 à 24 et différent de  i1,i2,i3,i4,i5,i6

i8 de 1 à 24 et différent de  i1,i2,i3,i4,i5,i6,i7

i9 de 1 à 24 et différent de  i1,i2,i3,i4,i5,i6,I7,i8

i10 de 1 à 24 et différent de  i1,i2,i3,i4,i5,i6,i7,i8,i9

i11 de 1 à 24 et différent de  i1,i2,i3,i4,i5,i6,i7,i8,i9,i10

i12 est calculé, doit être compris entre 1 et 24 et différent de  i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11 

On calcule ensuite les noeuds externes...

Après avoir réalisé quelques tests sur l'ordre de générer les sommets, je pense que procéder de cette manière est est la façon la plus efficace... 

 Posté par 
weierstrassre : Hexagones   28-05-14 à 22:30
ta méthode à l'avantage de ne pas compter plusieurs fois les grilles où les sommets n'appartenant qu'à un sommet sont intervertis(enfin je pense)

on peut aller un peu plus vite en considérant les inégalités sur ou  sommets, et en utilisant le résultat montré dans mon précédent post.

Quoi qu'il en soit, ton algorithme ne te sera pas très utile car je fais tourner mon algo depuis plus de 2 heures, et je viens de dépasser les 5 millions de possibilités...

Pourtant, mon algo n'est pas près de s'arrêter, car les listes créées commencent par 1,2,5,21,22...

ce qui montre que ça en est pas loin dans l'exécution...

ajoutons à ça les symétries, les 3 positions possibles pour le 1, et la somme des sommets différente de 75, je pense que l'on n'est pas près de savoir le nombre de solutions... 

En tout cas, pour moi, le problème est classé, à moins qu'un rebondissement inattendu le rende un peu plus intéressant...
 Posté par 
Diablowre : Hexagones   28-05-14 à 22:40
Pour le calcul de i12, le calcul est simple puisque la somme des ix pour x de 7 à 12 est égal à 5*S-300 où S est la somme sur l'hexagone central (sommets de 1 à 6).

La clef de ce problème est bien sûr d'évaluer le moins de configurations possibles (La méthode débile consisterait à examiner les 24! possibilités). 

Pour cela, deux pistes:

- générer les solutions en affectant des valeurs aux sommets dans l'ordre le plus judicieux possible

- appliquer des relations qui permettent d'éliminer à priori des configurations. Par exemple toute configuration telle que la somme(ix avec x de 7 à 12) est différente de 5 * somme (ix avec x de 1 à 6) - 300 ne peut pas convenir. Cette simple relation permet d'éliminer très rapidement certaines solutions avant d'affecter les 24 sommets.

Pour moi, il faut arriver à trouver des relations entre les ix qui sont vérifiées uniquement quand la configuration est ok et appliquer ces relations pour éliminer des configurations le plus tôt possible dans la génération.
 Posté par 
weierstrassre : Hexagones   28-05-14 à 23:12
Une question:

A quoi ça va te servir de déterminer les milliards de solutions?
 Posté par 
Diablowre : Hexagones   28-05-14 à 23:36
Une réponse:

A rien. Mais comme dirait dpi, à quoi cela sert de faire un sudoku, ... Se prouver que l'on arrive à résoudre un problème que l'on juge au départ pas évident, pour lequel je n'ai trouvé aucune piste sue internet, à échanger quelques réflexions avec des personnes sur un fofo...

Pour ce problème, je ne me suis pas fixé comme but de lister les solutions mais juste de les compter... 
 Posté par 
weierstrassre : Hexagones   29-05-14 à 09:25
Ah d'accord, je croyais que tu voulais la liste de toute les solutions...

pour ma part, je chercherais plutôt d'un côté théorique, le nombre de solutions m'a l'air encore trop grand pour être traité par mon ordi...
 Posté par 
dpire : Hexagones   29-05-14 à 10:01
>Weierstrass

Tu as le record des lignes sur l'île et il faut

demander au gestionnaire de réduire tes octets

D'autant plus que j'ai passé la dernière à la moulinette

et je trouve pour les 6 hexagones

107 113 90 48 74 61 ...vous avez dit somme égale??

J'ai aussi testé celle de diablov

et je trouve:

45 53 73 61 77 169

Quant à "énigme" je dirai que le poseur devrait se contenter

d'une seule solution pour l'égalité hexagonale(tiens tiens !)
 Posté par 
weierstrassre : Hexagones   29-05-14 à 10:20
1+2+3+22+23+24 = 75

1+12+20+19+2 = 75

1+21+15+18+16+3 = 75

...

je fais avec le système de numérotation proposé dans mon algorithme

(la flemme de tout remettre dans l'ordre )
 Posté par 
weierstrassre : Hexagones   29-05-14 à 10:32
dpi

Je suis encore loin du record: Joute n°135 : Les nombres robustes 

 Posté par 
dpire : Hexagones   29-05-14 à 12:49
Bonjour,

Mon test est dans l'ordre alphabétique du dessin initial,je ne vois pas dans quel ordre vous répondez ??
 Posté par 
weierstrassre : Hexagones   29-05-14 à 13:05
celui de mon deuxième message 
 Posté par 
weierstrassre : Hexagones   29-05-14 à 13:20
J'ai modifié mon algo pour ne plus tenir compte des symétries, et des doubles sur les sommets n'appartenant qu'à un hexagone

 Cliquez pour afficher
def hexagones():

    T=0

    a1=1

    for a2 in range(1,25):

        if (a2 in [a1])==True:

            continue

        for a3 in range(1,25):

            if (a3 in [a1,a2])==True:

                continue

            for a4 in range(1,25):

                if ((a4 in [a1,a2,a3])==True) or (a1+a2+a3+a4<28) or (a1+a2+a3+a4>72):

                    continue

                for a5 in range(1,25):

                    a6=75-(a1+a2+a3+a4+a5)

                    if ((a5 in [a1,a2,a3,a4])==True) or  (a1+a2+a3+a4+a5<51) or (a1+a2+a3+a4+a5>74) or ((a6 in [a1,a2,a3,a4,a5])==True) or (a6>a2):

                        continue

                    for a7 in range(1,25):

                        if (a7 in [a1,a2,a3,a4,a5,a6])==True:

                            continue

                        for a8 in range(1,25):

                            if ((a8 in [a1,a2,a3,a4,a5,a6,a7])==True) or (a1+a2+a7+a8<28) or (a1+a2+a7+a8>72):

                                continue

                            for a9 in range(1,25):

                                a10=75-(a1+a2+a7+a8+a9)

                                if ((a9 in [a1,a2,a3,a4,a5,a6,a7,a8])==True) or (a1+a2+a7+a8+a9<51) or (a1+a2+a7+a8+a9>74) or ((a10 in [a1,a2,a3,a4,a5,a6,a7,a8,a9])==True) or (a9>a8):

                                    continue

                                for a11 in range(1,25):

                                    if ((a11 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10])==True) or (a2+a3+a10+a11<28) or (a2+a3+a10+a11>72):

                                        continue

                                    for a12 in range(1,25):

                                        a13 = 75-(a2+a3+a10+a11+a12)

                                        if ((a12 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11])==True) or (a2+a3+a10+a11+a12<51) or (a2+a3+a10+a11+a12>74) or ((a13 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12])==True) or (a12>a11):

                                            continue

                                        for a14 in range(1,25):

                                            if ((a14 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13])==True) or (a3+a4+a13+a14<28) or (a3+a4+a13+a14>72):

                                                continue

                                            for a15 in range(1,25):

                                                a16=75-(a3+a4++a13+a14+a15)

                                                if ((a15 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14])==True) or (a3+a4+a13+a14+a15<51) or (a3+a4+a13+a14+a15>72) or ((a16 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15])==True) or (a15>a14) or (a7+a10+a13+a16<28) or (a7+a10+a13+a16>72):

                                                    continue

                                                for a17 in range(1,25):

                                                    if ((a17 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16])==True) or (a4+a5+a16+a17<28) or (a4+a5+a16+a17>72):

                                                        continue

                                                    for a18 in range(1,25):

                                                        a19=75-(a4+a5+a16+a17+a18)

                                                        if ((a18 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17])==True) or (a4+a5+a16+a17+a18<51) or (a4+a5+a16+a17+a18>74) or ((a19 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18])==True) or (a18>a17) or (a7+a10+a13+a16+a19<51) or (a7+a10+a13+a16+a19>74):

                                                            continue

                                                        for a20 in range(1,25):

                                                            if((a20 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19])==True) or (a5+a6+a19+a20<28) or (a5+a6+a19+a20>72):

                                                                continue

                                                            for a21 in range(1,25):

                                                                a22=75-(a5+a6+a19+a20+a21)

                                                                if ((a21 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20])==True) or (a5+a6+a19+a20+a21<51) or (a5+a6+a19+a20+a21>74) or ((a22 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21])==True) or (a7+a1+a6+a22<28) or (a7+a1+a6+a22>72) or (a21>a20):

                                                                    continue

                                                                for a23 in range(1,25):

                                                                    a24=75-(a7+a1+a6+a22+a23)

                                                                    if ((a23 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22])==True) or (a7+a1+a6+a22+a23<51) or (a7+a1+a6+a22+a23>74) or ((a24 in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22,a23])==True) or (a24>a23):

                                                                        continue

                                                                    T=T+1

                                                                    if T%100==0:

                                                                        print(T)

                                                                        print([a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22,a23,a24])

ça trouve des solutions beaucoup moins vite(37000 en 8 min...), mais ça balaie beaucoup plus vite le nombre de configurations possibles...

Toutefois je suis encore loin de les compter toutes.

Diablow, ou en est ton algo, que l'on compare quelle numérotation est la plus efficace?
 Posté par 
castoriginalre : Hexagones   29-05-14 à 14:18
Bonjour à tous,

considérons tout d'abord le dessin de départ amélioré pour scinder les hexagones ( la numérotation est la même que celle de l'énoncé.)

On ne considère pas  les 6 symétries obtenues par la rotation de la figure ni celles par retournement.

on voit qu'il y 6 points qui apparaissent 3 fois, 6 points qui apparaissent 2 fois et 12 points qui n'apparaissent qu'une fois.

Si l'on prend les hexagones 1,2,3, on a 13 points (nombres) qui sont définis sur les 24.

Le nombre de combinaisons est donc de C2413= 24!/(13!*11!) = 2496144 solutions

Si l'on examine maintenant l'hexagone 4, on peut encore définir 3 points f,j,n. Comme il reste 24-13=11 points non encore attribués

le nombre de combinaisons pour f,j,n est ce C113= 11!/(8!*3!) = 66 solutions

Passons à l'hexagone 5, on peut encore de finir les points r,u,v.

Il reste 24-16 = 8 points non attribués. Le nombre de combinaisons pour r,u,v est C83= 8!/(5!*3!) = 56 solutions

Il reste 3 points à définir pour l'hexagone 6: x,w,t.

On a déjà défini 19 points, il reste 5 points. Donc on a pour x,w,t un nombre de combinaisons égal à  C53=5!/(2!*3!) = 10 solutions.

Les deux derniers points de l'hexagone 7,  o et s sont alors déterminés.

Finalement, le nombre de solutions total est de :

2496144 * 66 * 56 * 10 *1 = 92.257.482.240  solutions

Amitiés
 Posté par 
castoriginalre : Hexagones   29-05-14 à 14:23
J'ai oublié d'ajouter qu'il s'agit d'un nombre maximal sans tenir compte de l'égalité des sommes de chaque hexagone.

La méthode pour ce calcul suivra !
 Posté par 
weierstrassre : Hexagones   29-05-14 à 20:07
pourquoi faire ces détours pour en arriver là?

Un peu de souci avec le dénombrement? 

imaginons que tu veuilles placer tes 24 chiffres dans 13 emplacements,

plaçons le 1 dans le premier, le 2 dans le deuxième, ..., le douzième dans le douzième.

Pour le treizième emplacement, il reste 24-12 = 12 possibilités.

Imaginos maintenant que tu n'est fixé que les 11 premiers, le douzième peut prendre 13 valeurs possibles. On a alors 1312 possibilités.

Et ainsi de suite...

On arrive à 2423...1312 = 24!/11!

Eh oui, c'est un arrangement...

Faisons le calcul:

24!/(24-13)!11!/(11-3)!8!/(8-3)!5!/(5-3)!2 = 6.204481023

mais pourquoi ne pas calculer directement le nombre de possibilités?

24!/1! = 6.204881023 

(notons qu'il était logique que ça ne pouvait être la combinaison. Pour l'ensemble, on aurait eu 24!/(24!1!) = 1, ce qui était peu probable )

en tout cas, merci quand même de ta contribution.

cordialement.
 Posté par 
Diablowre : Hexagones   31-05-14 à 08:56
Merci de toutes ces contributions.

Voici quelques résultats donnés par un programme en C qui évalue systématiquement configuration en partant de l'hexagone central, puis de l'intermédiaire.Le prog a tourné une heure environ, les résultats sont pour des hexagones de somme 75 uniquement. Toute interpolation des résultats me semble assez aléatoire vu la variation du nombre de solution en fonction du triplet (1, 24, X) choisi.

Je donne aussi ces résultats pour comparer, histoire de voir si les programmes sont buggués. 

Dans le tableau ci-dessous on peut lire que pour le triplet initial (AWR)=(1,23,18) il existe 481891 solutions en tenant compte des régles de symétries deja définies plus haut:

i1 < i2, i1 < i3, i1 < i4, i1 < i5, i2 < i6, 

i13 < i14, i15 < i16, i17 < i18, i19 < i20, i21 < i22, i23 < i24

Théoriquement, je peux lancer sur chaque processeur de mon ordi un calcul avec des triplets différents (1xx), (2xx), etc... et additionner les résultats... mais bon...

 Posté par 
dpire : Hexagones   31-05-14 à 19:06
Bonjour,

Dommage qu'il n'y ait pas eu plus de participants..

Je pense que 75 est la seule somme qui fonctionne,

mais de là à le démontrer...
 Posté par 
weierstrassre : Hexagones   31-05-14 à 19:09
tu as testé les autres sommes?

parce que vu le nombre de solutions pour 75, ça ne m'étonnerait pas que l'on en trouve quelques unes pour 74 ou 76.

Après, pour 67,68, je pense effectivement que l'on en trouvera pas.
 Posté par 
Diablowre : Hexagones   31-05-14 à 22:30
Non, il existe des solutions pour tous les nombre entre 65 et 85.

Ci dessous quelques exemples pour 

 Posté par 
Diablowre : Hexagones   31-05-14 à 22:34
Avec des notations identiques, la suite ci-dessous

 Posté par 
dpire : Hexagones   01-06-14 à 06:59
Bonjour,

Ceci élimine définitivement ma comparaison avec

le sudoku qui en japonais signifie solution unique
  Posté par 
weierstrassre : Hexagones   01-06-14 à 15:39
Je sens que l'on ne s'en sortira pas...

J'ai utilisé toute les symétries possibles, toutes les résultats établis,j'ai fixé un point, je n'ai considéré que les sommes égales à 75, j'ai fait toutner mon programme plusieurs heures et trouver des millions de solutions (bien qu'en tenant compte des symétries, et j'ai l'impression de toujours en être au même point. (mon algo n'est pas près d'avoir passé en revue toutes les solutions...)