Merge remote-tracking branch 'origin/master'

MPSI/obligatoire/TP_GoL/Pipfile

@@ -0,0 +1,12 @@
+[[source]]
+name = "pypi"
+url = "https://pypi.org/simple"
+verify_ssl = true
+
+[dev-packages]
+
+[packages]
+pygame = "*"
+
+[requires]
+python_version = "3.7"

MPSI/obligatoire/TP_GoL/Pipfile.lock

@@ -0,0 +1,53 @@
+{
+    "_meta": {
+        "hash": {
+            "sha256": "e172a3e30534a9a0b74742497b40f2ab196a0890d1cb082eb481fb89e384456e"
+        },
+        "pipfile-spec": 6,
+        "requires": {
+            "python_version": "3.7"
+        },
+        "sources": [
+            {
+                "name": "pypi",
+                "url": "https://pypi.org/simple",
+                "verify_ssl": true
+            }
+        ]
+    },
+    "default": {
+        "pygame": {
+            "hashes": [
+                "sha256:0480fe82cd41a43e3eea497fa2c059c72ac54cb5d003d5aa2ed06a04541c384e",
+                "sha256:136a3b5711d9ec369a0407e4e08ffced3ba61aa41059e9280ffffa79c8614f65",
+                "sha256:2622b9dd95f445c887a36a57eade42c672598589f69a8052ccdb8eeeffa4dbb1",
+                "sha256:301c6428c0880ecd4a9e3951b80e539c33863b6ff356a443db1758de4f297957",
+                "sha256:396320aa29a925feed0b64639f77ce1418722ea7f536b4e4936083dd8d4c4535",
+                "sha256:398c42b605ecc514e62f68f1944a2d21e247938309f598de6cb0ad3c207324a8",
+                "sha256:4aaff572a273a32e70ec3593d213e59ab11c183a9916616562247930f17a5447",
+                "sha256:4e1065577f1b29111113be5deb2ea88553551a5e1cf33e0c08fa32768f285809",
+                "sha256:5f052dc2975a399aa1830c1f04c5f72856aa416bf3cd4b31375a058015a5c620",
+                "sha256:68ea43e51150316b9fb08e251209d4e2b4e76a340b5b6fc8cdf1a898c78f7e5b",
+                "sha256:698433a9fcefca0527244dc44dff9503eb26157494730b1cc80e6e4dbb246e92",
+                "sha256:6f714986f7987f10cb94f1be0753318e341a7ea6b12d66f37a4d5d6dd4695023",
+                "sha256:73cd9df328c7e72638dbcc1d18e7155225faed880a53db6bad90d1d7c0a71dfd",
+                "sha256:7876d1f29f66d3d7cac46479503891ee1ef409b0fbce54b0d74f3a6b33a46dba",
+                "sha256:854e87b8b2b76e3ed11d64985fcfdd7af919659503de99fc2b0a717b314c3cf0",
+                "sha256:8da13704ad45b7d5de8a8cca135a7f44c7fc6aa9f691abe7b0392468a34a8013",
+                "sha256:9ce22fb72298ea33dbb3a1c6c60a4a4e19d9698df6f3f5782eba4dada7b7736d",
+                "sha256:a37b6c59e7b8feadc51db5197052b86ceb6443f9fb2a6f7d6527620e707c558c",
+                "sha256:a6e8d2f99dbe1dfe72d0c019693c14d93c410f702d0b04ec9a81b36dacd55a23",
+                "sha256:a9ac862dd7159861f2c6443b0029089e1c0c4ec762a8074022914ec52fe4dfac",
+                "sha256:ae1bc3e78ed28f20878e7ca2c98663a6634e9c00d7746d39413fc18e907dc162",
+                "sha256:be7e70f91bd4eb35ae081062f16bf434619b3292358d9b061f8159ddc570c7f0",
+                "sha256:c895cf9c1b6d1cbba8cb8cc3f5427febcf8aa41a9333697741abeea1c537a350",
+                "sha256:e3e7e4a09dfd8b03663222d6bcadec9fef021404f4d9eecf56825342e039dfc1",
+                "sha256:f1f5714d2c23f6a64ef2ac4fcd36a2dd2689da85978d951a99a6ae5dfdf9bdbc",
+                "sha256:fa788f775680fc5d268ab00a2da29c9a22830032cfab732730298a2952cd87f3"
+            ],
+            "index": "pypi",
+            "version": "==1.9.6"
+        }
+    },
+    "develop": {}
+}

MPSI/obligatoire/TP_GoL/README

@@ -0,0 +1,26 @@
+Le simulateur utilise pygame, pour l'installer :
+    pip install pygame
+une fois fait, pour lancer le simulateur, executez le main.py en tant que script (Ca permet à Pyzo de lancer le script dans le bon dossier)
+
+Comment l'utiliser :
+
+    Les touches :
+        PARTOUT :
+            ESPACE : Met en pause le jeu
+            C : Diminuer la fréquence de mise à jour des cellules
+            V : Augmenter la fréquence de mise à jour des cellules
+            Q : Quitter
+            Suppr : Effacer la grille
+            D : Activer le mode débogage (Affiche les compteurs des cellules à l'itération précédente)
+            M : Basculer en mode Clavier / Basculer en mode Souris
+        Mode Souris :
+            Clic gauche : Changer l'état de la cellule selectionnée (rester appuyé pour en changer plusieurs)
+            Clic droit / Clic molette (glisser) : Changer la zone visible de la grille
+            Molette vers le haut : Zoomer
+            Molette vers le bas : Dézoomer
+        Mode Clavier :
+            Touches directionnelles (Flèches) : Déplacer le pointeur (cadre jaune)
+            T : changer l'état de la cellule selectionnée
+
+À propos :
+    La grille est infinie, et par défaut, le mode souris est activé. Le mode clavier est incomplet mais permet de faire des actions basique.

MPSI/obligatoire/TP_GoL/main.py

@@ -0,0 +1,274 @@
+import pygame
+import time
+import sys
+import random
+"""
+Game of Life Simulator
+"""
+
+#Partie Calculatoire
+RULES = {"B":3, "O":3, "U":2}
+NEIGHBOURS = [(i,j) for i in (-1,0,1) for j in (-1,0,1) if (i,j) != (0,0)]
+
+def relative(offset, cell):
+    """
+    Fonction permettant d'obtenir une coordonnée absolue à partir d'une origine(offset) et de coordonnées relative(cell).
+    """
+    return tuple(cell[i] + offset[i] for i in range(len(cell)))
+
+def difference(origin, cell):
+    """
+    Fonction permettant d'obtenir les coordonnées de cell en fixant origin comme origine
+    """
+    return tuple(cell[i] - origin[i] for i in range(len(cell)))
+
+def scale(cell, factor):
+    return tuple(int(cell[i]*factor) for i in range(len(cell)))
+class Grid:
+    """
+    Classe gérant les calculs liés à l'évolution d'une grille infinie selon le Jeu de la vie
+    """
+    def __init__(self):
+        self.alive_cells = set({(i, 0) for i in range(500)})
+        self.active_cells = dict()
+        self.old_cells = set()
+
+    def tick(self):
+        self.old_cells.clear()
+        self.old_cells.update(self.alive_cells)
+        self.active_cells.clear()
+        #On s'assure que les cellules vivantes soient considérés comme actives
+        for cell in self.alive_cells:
+            self.active_cells.update({cell:0})
+        #On met à jour les voisins des cellules vivantes, c'est à dire incrémenter leur compteur
+        for cell in self.alive_cells:
+            for neighbour in NEIGHBOURS:
+                self.incr_counter(relative(cell, neighbour))
+        #On applique les règles du jeu selon le compteur des cellules actives
+        for cell in self.active_cells:
+            if RULES["U"] <= self.active_cells[cell] <= RULES["O"]:
+                if self.active_cells[cell] == RULES["B"]:
+                    self.alive_cells.add(cell)
+            else:
+                self.alive_cells.discard(cell)
+
+
+    def incr_counter(self, cell):
+        if cell in self.active_cells:
+            self.active_cells[cell] +=1
+        else:
+            self.active_cells.update({cell:1})
+
+
+#Interface Graphique
+
+pygame.init()
+size = width, height = 1025,1025
+screen = pygame.display.set_mode(size, flags=pygame.RESIZABLE)
+DEBUG = False #Affichage de variables de débogage à l'écran
+TICK_RATE = 10 #Fréquence de rafraîchissement du jeu
+FRAME_RATE = 60 #Fréquence de rafraîchissement de l'interface
+camera = [0, 0] #Permettant choisir quelle zone de la grille on visualise
+true_camera = camera
+MOUSE = True #Mode souris : On pointe avec les cellules avec la souris et pas les flèches directionnelles
+grid = Grid() #On initialise une grille
+#On charge les assets
+source_assets = {
+    "alive_cell": pygame.image.load("assets/alive_cell.png"),
+    "dead_cell": pygame.image.load("assets/dead_cell.png"),
+    "highlight": pygame.image.load("assets/highlight.png")}
+
+SCALE = .25 #Facteur de taille variable (zoom)
+CELL_SIDE = int(source_assets["highlight"].get_rect().width * SCALE) #Mesure en pixel d'un côté d'une cellule
+assets = {asset:pygame.transform.scale(source_assets[asset],(CELL_SIDE,CELL_SIDE)) for asset in source_assets}
+font = pygame.font.SysFont(pygame.font.get_default_font(), CELL_SIDE)
+class Unit(pygame.sprite.Sprite):
+    """
+    Classe représentant une unité d'affichage (cellule)
+    Il est à noter que cette unité peut représenter différentes cellules en fonction de camera
+    """
+    def __init__(self, offset):
+        super().__init__()
+        self.offset = offset
+    
+    def draw(self, screen, camera, grid):
+        self.image = assets[("dead_cell","alive_cell")[relative(camera, self.offset) in grid.alive_cells]]
+        self.position = self.image.get_rect()
+        self.position.x = self.offset[0]*CELL_SIDE #On positionne la cellule sur l'écran
+        self.position.y = self.offset[1]*CELL_SIDE #On positionne la cellule sur l'écran
+        screen.blit(self.image, self.position) #On dessine la cellule
+        if DEBUG and relative(camera, self.offset) in grid.active_cells:
+            text = font.render(str(grid.active_cells[relative(camera, self.offset)]), False, (0,255,0))
+            posi = text.get_rect()
+            posi.center = self.position.center
+            screen.blit(text, posi)
+
+class Pointer(pygame.sprite.Sprite):
+    """
+    Classe représentant le pointeur de cellule sélectionnée (pour l'éditer)
+    """
+    def __init__(self, pos):
+        super().__init__()
+        self.pos = pos
+    
+    def draw(self, screen, camera, grid):
+        self.image = assets["highlight"]
+        self.position = self.image.get_rect()
+        self.position.x = self.pos[0]*CELL_SIDE #On positionne le pointeur sur l'écran
+        self.position.y = self.pos[1]*CELL_SIDE #On positionne le pointeur sur l'écran
+        screen.blit(self.image, self.position) #On dessine le pointeur
+        
+def draw_grid(screen, camera, grid, units):
+    for unit in units:
+        units[unit].draw(screen, camera, grid)
+
+#Boucle principale
+def get_units():
+    return {(i,j):Unit((i,j)) for i in range(width//(CELL_SIDE) +1) for j in range(height//(CELL_SIDE) +1)}
+units = get_units() #Liste de cellules à afficher à l'écran
+relatives = {unit:relative(camera, unit) for unit in units}
+pointer = Pointer((0,0)) #Permettant de sélectionner une cellule précisément pour l'éditer
+running = True
+pause = True
+last_tick = time.time()
+last_frame = time.time()
+KEYSDOWN = set()
+MOUSEBUTTONSDOWN = set()
+visited_cells = set()
+drag_origin = (0,0) #Position du curseur à la préssion du bouton du milieu de la souris
+last_pointer_move = time.time()
+redraw = True
+while running:
+    frame = (last_frame + 1/FRAME_RATE) - time.time() < 0
+    if frame:
+        for event in pygame.event.get():
+            if event.type == pygame.VIDEORESIZE:
+                size = width, height = event.size
+                screen =screen = pygame.display.set_mode(size,
+                                                            flags=pygame.RESIZABLE)
+                units = get_units()
+                redraw = True
+            if event.type == pygame.QUIT:
+                sys.exit()
+            if event.type == pygame.KEYDOWN:
+                KEYSDOWN.add(event.key)
+                if event.key in [275,276,273,274] and not MOUSE: #Touches directionnelles pour déplacer le pointeur
+                    new_pos = list(pointer.pos)
+                    exec("new_pos" + ("[0]+","[0]-","[1]-","[1]+")[[275,276,273,274].index(event.key)] + "=1")
+                    if pointer.pos in units: units[pointer.pos].draw(screen, camera, grid)
+                    pointer.pos = tuple(new_pos)
+                    last_pointer_move = time.time()
+                if event.key == 116 and not MOUSE: #T: Changer l'état d'une cellule
+                    if relative(camera, pointer.pos) in grid.alive_cells:
+                        grid.alive_cells.discard(relative(camera, pointer.pos))
+                    else:
+                        grid.alive_cells.add(relative(camera, pointer.pos))
+                    visited_cells.add(relative(camera, pointer.pos))
+            if event.type == pygame.KEYUP:
+                print(event.key)
+                if event.key in KEYSDOWN:
+                    if event.key == 32: #ESPACE : pause
+                        pause = not pause
+                    if event.key in (109,59): #M : Mode souris
+                        MOUSE = not MOUSE
+                    if event.key == 100: #D : Mode de déboggage
+                        DEBUG = not DEBUG
+                    if event.key in (113,97): #Q : Quitter
+                        sys.exit()
+                    if event.key == 127: #Suppr : Effacer la grille
+                        grid.alive_cells.clear()
+                        redraw = True
+                    if event.key == 99: #C: Diminuer le TICK_RATE
+                        TICK_RATE = TICK_RATE/2
+                    if event.key == 118: #V : Augmenter le TICK_RATE
+                        TICK_RATE = TICK_RATE*2
+                    if event.key == 116 and not MOUSE: #T: Changer l'état d'une cellule
+                        visited_cells.clear()
+                    if event.key == 114 : #R : Remplir aléatoirement l'écran
+                        redraw = True
+                        for unit in units:
+                            if random.randint(0,3) == 0:
+                                grid.alive_cells.add(relatives[unit])
+                            else:
+                                grid.alive_cells.discard(relatives[unit])
+                KEYSDOWN.discard(event.key)
+            if event.type == pygame.MOUSEBUTTONDOWN:
+                if MOUSE:
+                    if event.button in (2,3): #Bouton de la molette / Clic droit : Glisser
+                        print(event.pos)
+                        true_camera = camera
+                        drag_origin = event.pos
+                    if event.button == 1: #Clic gauche : Éditer la cellule selectionnée
+                        if relative(camera, pointer.pos) in grid.alive_cells:
+                            grid.alive_cells.discard(relative(camera, pointer.pos))
+                        else:
+                            grid.alive_cells.add(relative(camera, pointer.pos))
+                        visited_cells.add(relative(camera, pointer.pos))
+                    MOUSEBUTTONSDOWN.add(event.button)
+            if event.type == pygame.MOUSEBUTTONUP:
+                if event.button in (2,3):
+                    camera = difference(scale(difference(drag_origin, event.pos), 1/CELL_SIDE), true_camera)
+                    redraw = True
+                if event.button == 1:
+                    visited_cells.clear()
+                if event.button == 4: #Molette vers le haut #Zoom
+                    if CELL_SIDE*2 <= min(size):
+                        SCALE *= 2 #Facteur de taille variable (zoom)
+                        CELL_SIDE = int(source_assets["highlight"].get_rect().width * SCALE) #Mesure en pixel d'un côté d'une cellule
+                        assets = {asset:pygame.transform.scale(source_assets[asset],(CELL_SIDE,CELL_SIDE)) for asset in source_assets}
+                        units = get_units() #Liste de cellules à afficher à l'écran
+                        font = pygame.font.SysFont(pygame.font.get_default_font(), CELL_SIDE)
+                        redraw = True
+                if event.button == 5: #Molette vers le bas #"Dézoom"
+                    if CELL_SIDE/2 >= 3:
+                        SCALE *= 1/2 #Facteur de taille variable (zoom)
+                        CELL_SIDE = int(source_assets["highlight"].get_rect().width * SCALE) #Mesure en pixel d'un côté d'une cellule
+                        assets = {asset:pygame.transform.scale(source_assets[asset],(CELL_SIDE,CELL_SIDE)) for asset in source_assets}
+                        units = get_units() #Liste de cellules à afficher à l'écran
+                        font = pygame.font.SysFont(pygame.font.get_default_font(), CELL_SIDE)
+                        redraw = True
+                print(event.button)
+                MOUSEBUTTONSDOWN.discard(event.button)
+            if event.type == pygame.MOUSEMOTION:
+                if 2 in MOUSEBUTTONSDOWN or 3 in MOUSEBUTTONSDOWN:
+                    camera = difference(scale(difference(drag_origin, event.pos), 1/CELL_SIDE),true_camera)
+                    redraw = True
+                if MOUSE:
+                    if pointer.pos in units: units[pointer.pos].draw(screen, camera, grid)
+                    pointer.pos = scale(event.pos, 1/CELL_SIDE)
+                if 1 in MOUSEBUTTONSDOWN:
+                    if pointer.pos in units: units[pointer.pos].draw(screen, camera, grid)
+                    pointer.pos = scale(event.pos, 1/CELL_SIDE)
+                    if relative(camera, pointer.pos) not in visited_cells:
+                        if relative(camera, pointer.pos) in grid.alive_cells:
+                            grid.alive_cells.discard(relative(camera, pointer.pos))
+                        else:
+                            grid.alive_cells.add(relative(camera, pointer.pos))
+                        visited_cells.add(relative(camera, pointer.pos))
+        if not MOUSE and (time.time() - (last_pointer_move + 5/FRAME_RATE) > 0):
+            for key in KEYSDOWN:
+                if key in [275,276,273,274]:    
+                    new_pos = list(pointer.pos)
+                    exec("new_pos" + ("[0]+","[0]-","[1]-","[1]+")[[275,276,273,274].index(event.key)] + "=1")
+                    if pointer.pos in units: units[pointer.pos].draw(screen, camera, grid)
+                    pointer.pos = tuple(new_pos)
+                    last_pointer_move = time.time()
+        if redraw:
+            relatives = {unit:relative(camera, unit) for unit in units}
+            draw_grid(screen, camera, grid, units)
+            redraw = False
+    if (last_tick + 1/TICK_RATE) - time.time() < 0:
+        last_tick = time.time()
+        if not pause:
+            grid.tick()
+            if True:
+                for unit in units:
+                    if (relatives[unit] in grid.alive_cells) != (relatives[unit] in grid.old_cells):
+                        units[unit].draw(screen, camera, grid)
+    if frame:
+        if pointer.pos in units:
+            units[pointer.pos].draw(screen, camera, grid)
+        pointer.draw(screen, camera, grid)
+        pygame.display.flip()
+        last_frame = time.time()
+    time.sleep(max(0,min((last_frame + 1/FRAME_RATE) - time.time(), (last_tick + 1/TICK_RATE) - time.time()))) #On attend avant la prochaine image / le prochain tick pour ne pas surcharger le CPU inutilement

MPSI/spe/cours/chemins

@@ -0,0 +1,684 @@
+0 -> 1 en 1. :
+  0 -> 1
+  0 -> 0 -> 1
+  0 -> 1 -> 1
+  0 -> 0 -> 1 -> 1
+
+0 -> 2 en 2. :
+  0 -> 2
+  0 -> 0 -> 2
+  0 -> 2 -> 2
+  0 -> 0 -> 2 -> 2
+
+0 -> 3 en 2. :
+  0 -> 1 -> 3
+  0 -> 0 -> 1 -> 3
+  0 -> 1 -> 3 -> 3
+  0 -> 0 -> 1 -> 3 -> 3
+
+0 -> 4 en 4. :
+  0 -> 2 -> 4
+  0 -> 0 -> 2 -> 4
+  0 -> 2 -> 4 -> 4
+  0 -> 0 -> 2 -> 4 -> 4
+
+0 -> 5 en 6. :
+  0 -> 1 -> 3 -> 5
+  0 -> 0 -> 1 -> 3 -> 5
+  0 -> 1 -> 3 -> 5 -> 5
+  0 -> 0 -> 1 -> 3 -> 5 -> 5
+
+0 -> 6 en 3. :
+  0 -> 1 -> 3 -> 6
+  0 -> 0 -> 1 -> 3 -> 6
+  0 -> 1 -> 3 -> 6 -> 6
+  0 -> 0 -> 1 -> 3 -> 6 -> 6
+
+0 -> 7 en 18. :
+  0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  0 -> 0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  0 -> 0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+
+0 -> 8 en 17. :
+  0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8
+  0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+  0 -> 0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8
+  0 -> 0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+
+0 -> 9 en 6. :
+  0 -> 1 -> 3 -> 6 -> 9
+  0 -> 0 -> 1 -> 3 -> 6 -> 9
+  0 -> 1 -> 3 -> 6 -> 9 -> 9
+  0 -> 0 -> 1 -> 3 -> 6 -> 9 -> 9
+
+0 -> 10 en 9. :
+  0 -> 1 -> 3 -> 6 -> 9 -> 10
+  0 -> 0 -> 1 -> 3 -> 6 -> 9 -> 10
+  0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 10
+  0 -> 0 -> 1 -> 3 -> 6 -> 9 -> 10 -> 10
+
+1 -> 0 en 17. :
+  1 -> 3 -> 6 -> 9 -> 10 -> 0
+  1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 0
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 0
+
+1 -> 2 en 19. :
+  1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2
+  1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+
+1 -> 3 en 1. :
+  1 -> 3
+  1 -> 1 -> 3
+  1 -> 3 -> 3
+  1 -> 1 -> 3 -> 3
+
+1 -> 4 en 21. :
+  1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+
+1 -> 5 en 5. :
+  1 -> 3 -> 5
+  1 -> 1 -> 3 -> 5
+  1 -> 3 -> 5 -> 5
+  1 -> 1 -> 3 -> 5 -> 5
+
+1 -> 6 en 2. :
+  1 -> 3 -> 6
+  1 -> 1 -> 3 -> 6
+  1 -> 3 -> 6 -> 6
+  1 -> 1 -> 3 -> 6 -> 6
+
+1 -> 7 en 17. :
+  1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+
+1 -> 8 en 16. :
+  1 -> 3 -> 6 -> 9 -> 10 -> 8
+  1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+
+1 -> 9 en 5. :
+  1 -> 3 -> 6 -> 9
+  1 -> 1 -> 3 -> 6 -> 9
+  1 -> 3 -> 6 -> 9 -> 9
+  1 -> 1 -> 3 -> 6 -> 9 -> 9
+
+1 -> 10 en 8. :
+  1 -> 3 -> 6 -> 9 -> 10
+  1 -> 1 -> 3 -> 6 -> 9 -> 10
+  1 -> 3 -> 6 -> 9 -> 10 -> 10
+  1 -> 1 -> 3 -> 6 -> 9 -> 10 -> 10
+
+2 -> 0 en 19. :
+  2 -> 3 -> 6 -> 9 -> 10 -> 0
+  2 -> 3 -> 6 -> 9 -> 10 -> 0 -> 0
+  2 -> 2 -> 3 -> 6 -> 9 -> 10 -> 0
+  2 -> 2 -> 3 -> 6 -> 9 -> 10 -> 0 -> 0
+  2 -> 4 -> 6 -> 9 -> 10 -> 0
+  2 -> 4 -> 6 -> 9 -> 10 -> 0 -> 0
+  2 -> 2 -> 4 -> 6 -> 9 -> 10 -> 0
+  2 -> 2 -> 4 -> 6 -> 9 -> 10 -> 0 -> 0
+
+2 -> 1 en 8. :
+  2 -> 3 -> 5 -> 1
+  2 -> 3 -> 5 -> 1 -> 1
+  2 -> 2 -> 3 -> 5 -> 1
+  2 -> 2 -> 3 -> 5 -> 1 -> 1
+
+2 -> 3 en 3. :
+  2 -> 3
+  2 -> 2 -> 3
+  2 -> 3 -> 3
+  2 -> 2 -> 3 -> 3
+
+2 -> 4 en 2. :
+  2 -> 4
+  2 -> 2 -> 4
+  2 -> 4 -> 4
+  2 -> 2 -> 4 -> 4
+
+2 -> 5 en 7. :
+  2 -> 3 -> 5
+  2 -> 2 -> 3 -> 5
+  2 -> 3 -> 5 -> 5
+  2 -> 2 -> 3 -> 5 -> 5
+
+2 -> 6 en 4. :
+  2 -> 3 -> 6
+  2 -> 2 -> 3 -> 6
+  2 -> 4 -> 6
+  2 -> 2 -> 4 -> 6
+  2 -> 3 -> 6 -> 6
+  2 -> 2 -> 3 -> 6 -> 6
+  2 -> 4 -> 6 -> 6
+  2 -> 2 -> 4 -> 6 -> 6
+
+2 -> 7 en 19. :
+  2 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  2 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  2 -> 2 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  2 -> 2 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  2 -> 4 -> 6 -> 9 -> 10 -> 8 -> 7
+  2 -> 4 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  2 -> 2 -> 4 -> 6 -> 9 -> 10 -> 8 -> 7
+  2 -> 2 -> 4 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+
+2 -> 8 en 18. :
+  2 -> 3 -> 6 -> 9 -> 10 -> 8
+  2 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+  2 -> 2 -> 3 -> 6 -> 9 -> 10 -> 8
+  2 -> 2 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+  2 -> 4 -> 6 -> 9 -> 10 -> 8
+  2 -> 4 -> 6 -> 9 -> 10 -> 8 -> 8
+  2 -> 2 -> 4 -> 6 -> 9 -> 10 -> 8
+  2 -> 2 -> 4 -> 6 -> 9 -> 10 -> 8 -> 8
+
+2 -> 9 en 7. :
+  2 -> 3 -> 6 -> 9
+  2 -> 2 -> 3 -> 6 -> 9
+  2 -> 4 -> 6 -> 9
+  2 -> 2 -> 4 -> 6 -> 9
+  2 -> 3 -> 6 -> 9 -> 9
+  2 -> 2 -> 3 -> 6 -> 9 -> 9
+  2 -> 4 -> 6 -> 9 -> 9
+  2 -> 2 -> 4 -> 6 -> 9 -> 9
+
+2 -> 10 en 10. :
+  2 -> 3 -> 6 -> 9 -> 10
+  2 -> 2 -> 3 -> 6 -> 9 -> 10
+  2 -> 4 -> 6 -> 9 -> 10
+  2 -> 2 -> 4 -> 6 -> 9 -> 10
+  2 -> 3 -> 6 -> 9 -> 10 -> 10
+  2 -> 2 -> 3 -> 6 -> 9 -> 10 -> 10
+  2 -> 4 -> 6 -> 9 -> 10 -> 10
+  2 -> 2 -> 4 -> 6 -> 9 -> 10 -> 10
+
+3 -> 0 en 16. :
+  3 -> 6 -> 9 -> 10 -> 0
+  3 -> 6 -> 9 -> 10 -> 0 -> 0
+  3 -> 3 -> 6 -> 9 -> 10 -> 0
+  3 -> 3 -> 6 -> 9 -> 10 -> 0 -> 0
+
+3 -> 1 en 5. :
+  3 -> 5 -> 1
+  3 -> 5 -> 1 -> 1
+  3 -> 3 -> 5 -> 1
+  3 -> 3 -> 5 -> 1 -> 1
+
+3 -> 2 en 18. :
+  3 -> 6 -> 9 -> 10 -> 0 -> 2
+  3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+  3 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2
+  3 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+
+3 -> 4 en 20. :
+  3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+  3 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  3 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+
+3 -> 5 en 4. :
+  3 -> 5
+  3 -> 3 -> 5
+  3 -> 5 -> 5
+  3 -> 3 -> 5 -> 5
+
+3 -> 6 en 1. :
+  3 -> 6
+  3 -> 3 -> 6
+  3 -> 6 -> 6
+  3 -> 3 -> 6 -> 6
+
+3 -> 7 en 16. :
+  3 -> 6 -> 9 -> 10 -> 8 -> 7
+  3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  3 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  3 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+
+3 -> 8 en 15. :
+  3 -> 6 -> 9 -> 10 -> 8
+  3 -> 6 -> 9 -> 10 -> 8 -> 8
+  3 -> 3 -> 6 -> 9 -> 10 -> 8
+  3 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+
+3 -> 9 en 4. :
+  3 -> 6 -> 9
+  3 -> 3 -> 6 -> 9
+  3 -> 6 -> 9 -> 9
+  3 -> 3 -> 6 -> 9 -> 9
+
+3 -> 10 en 7. :
+  3 -> 6 -> 9 -> 10
+  3 -> 3 -> 6 -> 9 -> 10
+  3 -> 6 -> 9 -> 10 -> 10
+  3 -> 3 -> 6 -> 9 -> 10 -> 10
+
+4 -> 0 en 17. :
+  4 -> 6 -> 9 -> 10 -> 0
+  4 -> 6 -> 9 -> 10 -> 0 -> 0
+  4 -> 4 -> 6 -> 9 -> 10 -> 0
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 0
+
+4 -> 1 en 18. :
+  4 -> 6 -> 9 -> 10 -> 0 -> 1
+  4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 1
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 1
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 1
+
+4 -> 2 en 19. :
+  4 -> 6 -> 9 -> 10 -> 0 -> 2
+  4 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 2
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+
+4 -> 3 en 19. :
+  4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3
+  4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+
+4 -> 5 en 23. :
+  4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  4 -> 4 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+
+4 -> 6 en 2. :
+  4 -> 6
+  4 -> 4 -> 6
+  4 -> 6 -> 6
+  4 -> 4 -> 6 -> 6
+
+4 -> 7 en 17. :
+  4 -> 6 -> 9 -> 10 -> 8 -> 7
+  4 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  4 -> 4 -> 6 -> 9 -> 10 -> 8 -> 7
+  4 -> 4 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+
+4 -> 8 en 16. :
+  4 -> 6 -> 9 -> 10 -> 8
+  4 -> 6 -> 9 -> 10 -> 8 -> 8
+  4 -> 4 -> 6 -> 9 -> 10 -> 8
+  4 -> 4 -> 6 -> 9 -> 10 -> 8 -> 8
+
+4 -> 9 en 5. :
+  4 -> 6 -> 9
+  4 -> 4 -> 6 -> 9
+  4 -> 6 -> 9 -> 9
+  4 -> 4 -> 6 -> 9 -> 9
+
+4 -> 10 en 8. :
+  4 -> 6 -> 9 -> 10
+  4 -> 4 -> 6 -> 9 -> 10
+  4 -> 6 -> 9 -> 10 -> 10
+  4 -> 4 -> 6 -> 9 -> 10 -> 10
+
+5 -> 0 en 18. :
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 0
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 0
+
+5 -> 1 en 1. :
+  5 -> 1
+  5 -> 1 -> 1
+  5 -> 5 -> 1
+  5 -> 5 -> 1 -> 1
+
+5 -> 2 en 20. :
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+
+5 -> 3 en 2. :
+  5 -> 1 -> 3
+  5 -> 1 -> 3 -> 3
+  5 -> 5 -> 1 -> 3
+  5 -> 5 -> 1 -> 3 -> 3
+
+5 -> 4 en 22. :
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+
+5 -> 6 en 3. :
+  5 -> 1 -> 3 -> 6
+  5 -> 5 -> 1 -> 3 -> 6
+  5 -> 1 -> 3 -> 6 -> 6
+  5 -> 5 -> 1 -> 3 -> 6 -> 6
+
+5 -> 7 en 18. :
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+
+5 -> 8 en 17. :
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 8 -> 8
+
+5 -> 9 en 6. :
+  5 -> 1 -> 3 -> 6 -> 9
+  5 -> 5 -> 1 -> 3 -> 6 -> 9
+  5 -> 1 -> 3 -> 6 -> 9 -> 9
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 9
+
+5 -> 10 en 9. :
+  5 -> 1 -> 3 -> 6 -> 9 -> 10
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10
+  5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 10
+  5 -> 5 -> 1 -> 3 -> 6 -> 9 -> 10 -> 10
+
+6 -> 0 en 15. :
+  6 -> 9 -> 10 -> 0
+  6 -> 9 -> 10 -> 0 -> 0
+  6 -> 6 -> 9 -> 10 -> 0
+  6 -> 6 -> 9 -> 10 -> 0 -> 0
+
+6 -> 1 en 16. :
+  6 -> 9 -> 10 -> 0 -> 1
+  6 -> 9 -> 10 -> 0 -> 1 -> 1
+  6 -> 6 -> 9 -> 10 -> 0 -> 1
+  6 -> 6 -> 9 -> 10 -> 0 -> 1 -> 1
+
+6 -> 2 en 17. :
+  6 -> 9 -> 10 -> 0 -> 2
+  6 -> 9 -> 10 -> 0 -> 2 -> 2
+  6 -> 6 -> 9 -> 10 -> 0 -> 2
+  6 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+
+6 -> 3 en 17. :
+  6 -> 9 -> 10 -> 0 -> 1 -> 3
+  6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+  6 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3
+  6 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+
+6 -> 4 en 19. :
+  6 -> 9 -> 10 -> 0 -> 2 -> 4
+  6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+  6 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  6 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+
+6 -> 5 en 21. :
+  6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+  6 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  6 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+
+6 -> 7 en 15. :
+  6 -> 9 -> 10 -> 8 -> 7
+  6 -> 9 -> 10 -> 8 -> 7 -> 7
+  6 -> 6 -> 9 -> 10 -> 8 -> 7
+  6 -> 6 -> 9 -> 10 -> 8 -> 7 -> 7
+
+6 -> 8 en 14. :
+  6 -> 9 -> 10 -> 8
+  6 -> 9 -> 10 -> 8 -> 8
+  6 -> 6 -> 9 -> 10 -> 8
+  6 -> 6 -> 9 -> 10 -> 8 -> 8
+
+6 -> 9 en 3. :
+  6 -> 9
+  6 -> 6 -> 9
+  6 -> 9 -> 9
+  6 -> 6 -> 9 -> 9
+
+6 -> 10 en 6. :
+  6 -> 9 -> 10
+  6 -> 6 -> 9 -> 10
+  6 -> 9 -> 10 -> 10
+  6 -> 6 -> 9 -> 10 -> 10
+
+7 -> 0 en 19. :
+  7 -> 6 -> 9 -> 10 -> 0
+  7 -> 6 -> 9 -> 10 -> 0 -> 0
+  7 -> 7 -> 6 -> 9 -> 10 -> 0
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 0
+
+7 -> 1 en 20. :
+  7 -> 6 -> 9 -> 10 -> 0 -> 1
+  7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 1
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 1
+
+7 -> 2 en 21. :
+  7 -> 6 -> 9 -> 10 -> 0 -> 2
+  7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+
+7 -> 3 en 21. :
+  7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3
+  7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+
+7 -> 4 en 23. :
+  7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+
+7 -> 5 en 25. :
+  7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  7 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+
+7 -> 6 en 4. :
+  7 -> 6
+  7 -> 6 -> 6
+  7 -> 7 -> 6
+  7 -> 7 -> 6 -> 6
+
+7 -> 8 en 18. :
+  7 -> 6 -> 9 -> 10 -> 8
+  7 -> 6 -> 9 -> 10 -> 8 -> 8
+  7 -> 7 -> 6 -> 9 -> 10 -> 8
+  7 -> 7 -> 6 -> 9 -> 10 -> 8 -> 8
+
+7 -> 9 en 7. :
+  7 -> 6 -> 9
+  7 -> 7 -> 6 -> 9
+  7 -> 6 -> 9 -> 9
+  7 -> 7 -> 6 -> 9 -> 9
+
+7 -> 10 en 10. :
+  7 -> 6 -> 9 -> 10
+  7 -> 7 -> 6 -> 9 -> 10
+  7 -> 6 -> 9 -> 10 -> 10
+  7 -> 7 -> 6 -> 9 -> 10 -> 10
+
+8 -> 0 en 20. :
+  8 -> 7 -> 6 -> 9 -> 10 -> 0
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 0
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 0
+
+8 -> 1 en 21. :
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 1
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 1
+
+8 -> 2 en 22. :
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 2
+
+8 -> 3 en 22. :
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+
+8 -> 4 en 24. :
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+
+8 -> 5 en 26. :
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+
+8 -> 6 en 5. :
+  8 -> 7 -> 6
+  8 -> 7 -> 6 -> 6
+  8 -> 8 -> 7 -> 6
+  8 -> 8 -> 7 -> 6 -> 6
+
+8 -> 7 en 1. :
+  8 -> 7
+  8 -> 7 -> 7
+  8 -> 8 -> 7
+  8 -> 8 -> 7 -> 7
+
+8 -> 9 en 8. :
+  8 -> 7 -> 6 -> 9
+  8 -> 8 -> 7 -> 6 -> 9
+  8 -> 7 -> 6 -> 9 -> 9
+  8 -> 8 -> 7 -> 6 -> 9 -> 9
+
+8 -> 10 en 11. :
+  8 -> 7 -> 6 -> 9 -> 10
+  8 -> 8 -> 7 -> 6 -> 9 -> 10
+  8 -> 7 -> 6 -> 9 -> 10 -> 10
+  8 -> 8 -> 7 -> 6 -> 9 -> 10 -> 10
+
+9 -> 0 en 12. :
+  9 -> 10 -> 0
+  9 -> 10 -> 0 -> 0
+  9 -> 9 -> 10 -> 0
+  9 -> 9 -> 10 -> 0 -> 0
+
+9 -> 1 en 13. :
+  9 -> 10 -> 0 -> 1
+  9 -> 10 -> 0 -> 1 -> 1
+  9 -> 9 -> 10 -> 0 -> 1
+  9 -> 9 -> 10 -> 0 -> 1 -> 1
+
+9 -> 2 en 14. :
+  9 -> 10 -> 0 -> 2
+  9 -> 10 -> 0 -> 2 -> 2
+  9 -> 9 -> 10 -> 0 -> 2
+  9 -> 9 -> 10 -> 0 -> 2 -> 2
+
+9 -> 3 en 14. :
+  9 -> 10 -> 0 -> 1 -> 3
+  9 -> 10 -> 0 -> 1 -> 3 -> 3
+  9 -> 9 -> 10 -> 0 -> 1 -> 3
+  9 -> 9 -> 10 -> 0 -> 1 -> 3 -> 3
+
+9 -> 4 en 16. :
+  9 -> 10 -> 0 -> 2 -> 4
+  9 -> 10 -> 0 -> 2 -> 4 -> 4
+  9 -> 9 -> 10 -> 0 -> 2 -> 4
+  9 -> 9 -> 10 -> 0 -> 2 -> 4 -> 4
+
+9 -> 5 en 18. :
+  9 -> 10 -> 0 -> 1 -> 3 -> 5
+  9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+  9 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5
+  9 -> 9 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+
+9 -> 6 en 15. :
+  9 -> 10 -> 0 -> 1 -> 3 -> 6
+  9 -> 10 -> 0 -> 1 -> 3 -> 6 -> 6
+  9 -> 9 -> 10 -> 0 -> 1 -> 3 -> 6
+  9 -> 9 -> 10 -> 0 -> 1 -> 3 -> 6 -> 6
+
+9 -> 7 en 12. :
+  9 -> 10 -> 8 -> 7
+  9 -> 10 -> 8 -> 7 -> 7
+  9 -> 9 -> 10 -> 8 -> 7
+  9 -> 9 -> 10 -> 8 -> 7 -> 7
+
+9 -> 8 en 11. :
+  9 -> 10 -> 8
+  9 -> 10 -> 8 -> 8
+  9 -> 9 -> 10 -> 8
+  9 -> 9 -> 10 -> 8 -> 8
+
+9 -> 10 en 3. :
+  9 -> 10
+  9 -> 9 -> 10
+  9 -> 10 -> 10
+  9 -> 9 -> 10 -> 10
+
+10 -> 0 en 9. :
+  10 -> 0
+  10 -> 0 -> 0
+  10 -> 10 -> 0
+  10 -> 10 -> 0 -> 0
+
+10 -> 1 en 10. :
+  10 -> 0 -> 1
+  10 -> 0 -> 1 -> 1
+  10 -> 10 -> 0 -> 1
+  10 -> 10 -> 0 -> 1 -> 1
+
+10 -> 2 en 11. :
+  10 -> 0 -> 2
+  10 -> 0 -> 2 -> 2
+  10 -> 10 -> 0 -> 2
+  10 -> 10 -> 0 -> 2 -> 2
+
+10 -> 3 en 11. :
+  10 -> 0 -> 1 -> 3
+  10 -> 0 -> 1 -> 3 -> 3
+  10 -> 10 -> 0 -> 1 -> 3
+  10 -> 10 -> 0 -> 1 -> 3 -> 3
+
+10 -> 4 en 13. :
+  10 -> 0 -> 2 -> 4
+  10 -> 0 -> 2 -> 4 -> 4
+  10 -> 10 -> 0 -> 2 -> 4
+  10 -> 10 -> 0 -> 2 -> 4 -> 4
+
+10 -> 5 en 15. :
+  10 -> 0 -> 1 -> 3 -> 5
+  10 -> 0 -> 1 -> 3 -> 5 -> 5
+  10 -> 10 -> 0 -> 1 -> 3 -> 5
+  10 -> 10 -> 0 -> 1 -> 3 -> 5 -> 5
+
+10 -> 6 en 12. :
+  10 -> 0 -> 1 -> 3 -> 6
+  10 -> 0 -> 1 -> 3 -> 6 -> 6
+  10 -> 10 -> 0 -> 1 -> 3 -> 6
+  10 -> 10 -> 0 -> 1 -> 3 -> 6 -> 6
+
+10 -> 7 en 9. :
+  10 -> 8 -> 7
+  10 -> 8 -> 7 -> 7
+  10 -> 10 -> 8 -> 7
+  10 -> 10 -> 8 -> 7 -> 7
+
+10 -> 8 en 8. :
+  10 -> 8
+  10 -> 8 -> 8
+  10 -> 10 -> 8
+  10 -> 10 -> 8 -> 8
+
+10 -> 9 en 15. :
+  10 -> 0 -> 1 -> 3 -> 6 -> 9
+  10 -> 0 -> 1 -> 3 -> 6 -> 9 -> 9
+  10 -> 10 -> 0 -> 1 -> 3 -> 6 -> 9
+  10 -> 10 -> 0 -> 1 -> 3 -> 6 -> 9 -> 9
+

MPSI/spe/cours/floyd_warshall.ml

@@ -0,0 +1,112 @@
+let ii = infinity;;
+
+let g = [|
+(*      0  1  2  3  4  5  6  7  8  9  10*)
+(*0*) [|0.;1.;2.;ii;ii;ii;ii;ii;ii;ii;ii|];
+(*1*) [|ii;0.;ii;1.;ii;ii;ii;ii;ii;ii;ii|];
+(*2*) [|ii;ii;0.;3.;2.;ii;ii;ii;ii;ii;ii|];
+(*3*) [|ii;ii;ii;0.;ii;4.;1.;ii;ii;ii;ii|];
+(*4*) [|ii;ii;ii;ii;0.;ii;2.;ii;ii;ii;ii|];
+(*5*) [|ii;1.;ii;ii;ii;0.;ii;ii;ii;ii;ii|];
+(*6*) [|ii;ii;ii;ii;ii;ii;0.;ii;ii;3.;ii|];
+(*7*) [|ii;ii;ii;ii;ii;ii;4.;0.;ii;ii;ii|];
+(*8*) [|ii;ii;ii;ii;ii;ii;ii;1.;0.;ii;ii|];
+(*9*) [|ii;ii;ii;ii;ii;ii;ii;ii;ii;0.;3.|];
+(*10*)[|9.;ii;ii;ii;ii;ii;ii;ii;8.;ii;0.|];|];;
+
+let copie_matrice m =
+    let p = Array.length m and q = Array.length m.(0) in
+    let c = Array.make_matrix p q m.(0).(0) in
+    for i = 0 to p-1 do
+        for j = 0 to q-1 do
+            c.(i).(j) <- m.(i).(j)
+            done;
+        done;
+    c;;
+
+let ecraser_matrice m1 m2 =
+    let p = Array.length m2 and q = Array.length m2.(0) in
+    for i = 0 to p-1 do
+        for j = 0 to q-1 do
+            m1.(i).(j) <- m2.(i).(j)
+            done;
+        done;;
+            
+let floyd_warshall g =
+    let n = Array.length g in
+    (* w^(k) pour k=-1 *)
+    let w = copie_matrice g in
+    for k = 0 to n-1 do
+    (* mise ŕ jour de w^(k) *)
+        let c = Array.make_matrix n n 0. in
+        for i = 0 to n-1 do
+            for j = 0 to n-1 do
+                c.(i).(j) <- min w.(i).(j) (w.(i).(k)+.w.(k).(j))
+                done;
+            done;
+        ecraser_matrice w c;
+    done;
+    (* on retourne w^(n-1) *)
+    w;;
+
+
+let transfo_matrice m =
+    let p = Array.length m and q = Array.length m.(0) in
+    let c = Array.make_matrix p q (m.(0).(0),[[(0,0)]]) in
+    for i = 0 to p-1 do
+        for j = 0 to q-1 do
+            c.(i).(j) <- (m.(i).(j), [[(i,j)]])
+            done;
+        done;
+    c;;
+
+let rec joindre_chemins l1 l2 =
+    let rec aux chemin li = match li with
+    	| [] -> []
+    	| h::t -> (chemin@h)::(aux chemin t) in
+    match l1 with
+        | [] -> []
+        | h::t -> (aux h l2)@(joindre_chemins t l2);;
+
+let floyd_warshall_complet g =
+    let n = Array.length g in
+    (* w^(k) pour k=-1 *)
+    let w = transfo_matrice g in
+    for k = 0 to n-1 do
+    (* mise ŕ jour de w^(k) *)
+        let c = Array.make_matrix n n w.(0).(0) in
+        for i = 0 to n-1 do
+            for j = 0 to n-1 do
+                let ((db,cb), (ds,cs), (dt,ct)) = (w.(i).(j), w.(i).(k), w.(k).(j)) in
+                if db = ds +. dt then
+                    if db <> infinity then
+                        c.(i).(j) <- (db,cb@(joindre_chemins cs ct))
+                    else
+                        c.(i).(j) <- (db, cb)
+                else if db > ds +. dt then
+                    c.(i).(j) <- (ds +. dt, joindre_chemins cs ct)
+                else
+                    c.(i).(j) <- (db, cb);
+                done;
+            done;
+        ecraser_matrice w c;
+    done;
+    (* on retourne w^(n-1) *)
+    w;;
+
+open Format;;
+let tous_les_chemins g = 
+    let mat = floyd_warshall_complet g in
+
+    let rec affiche_chemin last l = match l with
+        | [] -> printf "%d\n" last
+        | (i,j)::t -> (printf "%d -> " i); affiche_chemin j t in
+
+    for i = 0 to Array.length mat - 1 do
+        for j = 0 to Array.length mat.(0) -1 do
+            if i<>j then
+                let (d,c) = mat.(i).(j) in printf "%d -> %d en %F :\n" i j d; List.iter (fun li -> printf "  "; affiche_chemin j li) c;
+                print_newline ()
+        done
+    done;;
+tous_les_chemins g;;

SPE/OPT/TP 2/.kile/réponseauxquestions.kilepr.gui

@@ -0,0 +1,30 @@
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+kile_livePreviewTool=LivePreview-PDFLaTeX
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+
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+JumpList=
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+ViMarks=

SPE/OPT/TP 2/.kile/tp2.kilepr.gui

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+kile_livePreviewStatusUserSpecified=false
+kile_livePreviewTool=LivePreview-PDFLaTeX
+lastDocument=Questions.tex
+
+[document-settings,item:Questions.tex]
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+Mode=LaTeX
+Mode Set By User=false
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SPE/OPT/TP 2/Questions.aux

@@ -0,0 +1,3 @@
+\relax 
+\@writefile{toc}{\contentsline {section}{\numberline {III}Forme normale disjonctive}{1}\protected@file@percent }
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Question :}{1}\protected@file@percent }

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SPE/OPT/TP 2/Questions.tex

@@ -0,0 +1,59 @@
+\documentclass[a4paper,10pt]{article}
+%\documentclass[a4paper,10pt]{scrartcl}
+
+\usepackage{mathtools}
+
+\usepackage{listings}
+\usepackage{color}
+\usepackage[utf8]{inputenc}
+\definecolor{dkgreen}{rgb}{0,0.6,0}
+\definecolor{gray}{rgb}{0.5,0.5,0.5}
+\definecolor{mauve}{rgb}{0.58,0,0.82}
+\lstset{frame=tb,
+  language=caml,
+  aboveskip=3mm,
+  belowskip=3mm,
+  showstringspaces=false,
+  columns=flexible,
+  basicstyle={\small\ttfamily},
+  numbers=none,
+  numberstyle=\tiny\color{gray},
+  keywordstyle=\color{blue},
+  commentstyle=\color{dkgreen},
+  stringstyle=\color{mauve},
+  breaklines=true,
+  breakatwhitespace=true,
+  tabsize=3
+}
+\renewcommand{\thesection}{\Roman{section}} 
+\newcommand{\sectioni}[2]{\setcounter{section}{#1}\addtocounter{section}{-1}\section{#2}}
+\newcommand{\subsectioni}[2]{\setcounter{subsection}{#1}\addtocounter{subsection}{-1}\subsection{#2}}
+\title{Questions du TP2}
+\author{}
+\date{}
+
+\pdfinfo{%
+  /Title    ()
+  /Author   ()
+  /Creator  ()
+  /Producer ()
+  /Subject  ()
+  /Keywords ()
+}
+
+\begin{document}
+    \maketitle
+    \sectioni{3}{Forme normale disjonctive}
+        \subsectioni{1}{Question :}
+            On se sert du code écrit aux questions précédentes,
+            \begin{lstlisting}
+# table_de_verite 3 g;;
+- : bool array = [|true; false; true; false; true; false; false; true|]
+            \end{lstlisting}
+            $\mathcal{G} =
+            (\neg V_2 \wedge \neg V_1 \wedge \neg V_0) \vee
+            (\neg V_2 \wedge V_1 \wedge \neg V_0) \vee
+            (V_2 \wedge \neg V_1 \wedge \neg V_0) \vee
+            (V_2 \wedge V_1 \wedge V_0)
+            $
+\end{document}

SPE/OPT/TP 2/main.ml

@@ -19,15 +19,20 @@ let rec evaluer_tab t form = match form with
 	| Or (h::tail) -> (evaluer_tab t h) || (evaluer_tab t (Or tail));;
 5 lsr 1;;
 
-let evaluer n c f =
+let contexte_from_int n c =
 	let contexte = Array.make n false in
 	let j = ref c in
 	for i = 0 to n-1 do
-		contexte.(i) <- !j mod 2 = 0;
+		contexte.(i) <- !j mod 2 <> 0;
 		j := !j lsr 1
 	done;
+	contexte;;
+
+let evaluer n c f =
+	let contexte = contexte_from_int n c in
 	evaluer_tab contexte f;;
-evaluer 3 0 f;;
+
+evaluer 3 3 g;;
 
 let table_de_verite n f = 
 	let rec puis n a = match n with
@@ -40,3 +45,23 @@ let table_de_verite n f =
 	res;;
 table_de_verite 3 g;;
 
+let forme_normale_disjonctive n f =
+	let rec puis n a = match n with
+		| 0 -> 1
+		| _ -> a * (puis (n-1) a) in
+
+	let rec disjonction c conjs =
+
+		let rec conjonction cont conj n = match n with
+			| -1 -> And conj
+			| _ -> conjonction cont ((if cont.(n) then (Var n) else Not (Var n))::conj) (n-1) in
+
+		match c with
+			| -1 -> Or conjs
+			| _ -> let contexte = context_from_int n c in
+					 if evaluer_tab contexte f then
+					 	disjonction (c-1) ((conjonction contexte [] (n-1))::conjs)
+					 else
+					 	disjonction (c-1) conjs in
+	disjonction ((puis n 2) -1) [];;
+forme_normale_disjonctive 3 g;;

SPE/OPT/TP 2/tp2.kilepr

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+pkg_extensions=.cls .sty .bbx .cbx .lbx
+src_extIsRegExp=false
+src_extensions=.tex .ltx .latex .dtx .ins
+
+[Tools]
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+QuickBuild=
+
+[item:Questions.tex]
+archive=true
+encoding=
+highlight=LaTeX
+mode=LaTeX
+
+[item:tp2.kilepr]
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