From b1141ea6ee9d46145a3a76a70628c107454d1d75 Mon Sep 17 00:00:00 2001 From: William Driot Date: Fri, 13 Dec 2024 13:19:18 +0100 Subject: [PATCH] =?UTF-8?q?Points=20colori=C3=A9s=20sur=20un=20cercle,=20r?= =?UTF-8?q?evisite?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- fiches/points-colories-fiche.tex | 158 +++++++++++++---------- index.pdf | Bin 379498 -> 0 bytes index_avec_fiches.pdf | Bin 373428 -> 0 bytes src/points-colories.tex | 215 ++++++++++++------------------- src/toboggan.tex | 2 - 5 files changed, 172 insertions(+), 203 deletions(-) delete mode 100644 index.pdf delete mode 100644 index_avec_fiches.pdf diff --git a/fiches/points-colories-fiche.tex b/fiches/points-colories-fiche.tex index 17b34ec..bdf5708 100644 --- a/fiches/points-colories-fiche.tex +++ b/fiches/points-colories-fiche.tex @@ -2,9 +2,10 @@ \q (Facile) Puisque son adversaire joue en premier, elle peut jouer pile en face de lui (sur le point diamétralement opposé au premier coup de son adversaire). Son adversaire n'a pas le droit de jouer un point déjà coloré, il joue, et il suffit alors à Lucie de faire un arc d'angle strictement plus grand que celui formé par son adversaire à son deuxième coup. Il est clair qu'elle le peut toujours. -\begin{center} - \begin{tikzpicture} - \begin{scope}[xshift=10cm] +\begin{figure}[h] + \centering + \begin{tikzpicture} + \begin{scope} \draw (0,0) circle (1.5cm); \coordinate (A1) at (0:1.5cm); @@ -28,70 +29,85 @@ \foreach \angle in {180, 340} { \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm } - \end{scope} - \end{tikzpicture} -\end{center} + \end{scope} + \end{tikzpicture} + \label{fig:PointsColoriesQuestion1} +\end{figure} \q \begin{enumerate} \item (Facile) Non. Lors de son deuxième coup, Lucie forme nécessairement un unique arc primitif. Il reste alors un coup à jouer à son adversaire, et, quelle qu'ait été la stratégie de Lucie pour placer ses deux points, elle ne pourra jamais s'assurer que son adversaire ne \og coupe \fg pas son unique arc, menant à un match nul. -m - \item (Moyen) Oui, la réponse à la question suivante en fournit une. + + \item (Facile/Moyen/Difficile/Ouvert) (Facile : trouver quelques petits résultats partiels, Moyen : Trouver des résultats partiels assez forts, Difficile : Aller très loin, Ouvert : Déterminer l'ensemble complet) - \item (Moyen+) On note $D$ la droite formée par le premier point placé par Alice et le centre du cercle. Après que l'adversaire ait joué son point $1$, Alice joue le symétrique de $1s$ par rapport à $D$. + Soit $S$ l'ensemble cherché. + + \textbf{Facile :} On peut trouver des éléments de $S$ et démontrer que d'autres n'y sont pas. -\begin{center} - \begin{tikzpicture} - \begin{scope}[xshift=10cm] - \draw (0,0) circle (1.5cm); - - \coordinate (A1) at (0:1.5cm); - \coordinate (A2) at (220:1.5cm); + Par exemple la question précédente montre que $ 1 \notin S $. En prenant des stratégies arbitraires, on trouve des éléments de $S$ en calculant leurs probabilités associées. Il est impossible de lister dans ce document toutes les stratégies possibles, mais il est clair qu'on peut toujours en trouver, calculer les probabilités associées et en déduire des des éléments de $S$. - \coordinate (B1) at (140:1.5cm); + \textbf{Moyen :} On a $ \frac 3 4 \in S $. - \node at (0:1.8cm) {1}; - \node at (140:1.8cm) {2}; - \node at (220:1.8cm) {3}; + Démonstration : - \draw[dotted] (0:2cm) -- (180:2cm); - - % \draw[very thick, red] (A1) arc[start angle=0, end angle=140, radius=1.5cm]; - % \draw[very thick, blue] (B1) arc[start angle=180, end angle=340, radius=1.5cm]; - - \foreach \angle in {0, 220} { - \filldraw[fill=red] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm - } + On note $D$ la droite formée par le premier point placé par Alice et le centre du cercle. Après que l'adversaire ait joué son point $2$, Alice joue le symétrique de $2$ par rapport à $D$. + + On définit $ \theta $ comme l'angle \textit{aigu} l'angle formé entre le premier point placé (par Lucie) et le second (par l'adversaire). Clairement, $\theta$ suit une loi uniforme sur $[0,\pi]$. - \foreach \angle in {140} { - \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm - } + \begin{figure}[h] + \centering + \begin{tikzpicture} + \begin{scope} + \draw (0,0) circle (1.5cm); + + \coordinate (A1) at (0:1.5cm); + \coordinate (A2) at (220:1.5cm); + + \coordinate (B1) at (140:1.5cm); + + \node at (0:1.8cm) {1}; + \node at (140:1.8cm) {2}; + \node at (220:1.8cm) {3}; + + \draw[dotted] (0:2cm) -- (180:2cm); + + % \draw[very thick, red] (A1) arc[start angle=0, end angle=140, radius=1.5cm]; + % \draw[very thick, blue] (B1) arc[start angle=180, end angle=340, radius=1.5cm]; + + \foreach \angle in {0, 220} { + \filldraw[fill=red] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm + } + + \foreach \angle in {140} { + \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm + } + + \draw[dotted] (0,0) -- (B1); + + \draw[thin] (0.4,0) arc[start angle=0, end angle=140, radius=0.4cm]; + + \node at (70:1cm) {$\theta$}; + + \end{scope} + \end{tikzpicture} + \end{figure} - \draw[dotted] (0,0) -- (B1); - - \draw[thin] (0.4,0) arc[start angle=0, end angle=140, radius=0.4cm]; + Alors, dans une telle configuration où l'angle vaut $ \theta $, on voit que Lucie gagne si et seulement si son adversaire ne joue pas de sorte à couper l'arc qu'elle a formé, et qu'il y a match nul sinon. + + Ceci arrive avec une probabilité $ \frac{2\pi - \theta}{2 \pi} $, parce que le numérateur de cette fraction est l'angle dans lequel l'adversaire doit jouer pour que Lucie gagne. + + Comme $ \theta $ est d'espérance $ \frac \pi 2 $, l'espérance vaut $ 1 - \frac{\pi/2}{2\pi} $, c'est à dire 3/4. - \node at (70:1cm) {$\theta$}; + \textbf{Moyen/Difficile :} On a $]\frac 1 4, \frac 1 2[ \subset S$ - \end{scope} - \end{tikzpicture} -\end{center} + Démonstration : -On définit alors $ \theta $ comme l'angle \textit{aigu} l'angle formé entre le premier point placé (par Lucie) et le second (par l'adversaire). Clairement, $\theta$ suit une loi uniforme sur $[0,\pi]$. + Soit $\alpha \in ]0,1[$. Si $\theta$ désigne toujours l'angle aigu formé entre le premier point placé (par Lucie) et le second (par l'adversaire), Lucie joue le point situé à l'angle $ \alpha \theta $ dans le même repère. -Alors, dans une telle configuration où l'angle vaut $ \theta $, on voit que Lucie gagne si et seulement si son adversaire ne joue pas de sorte à couper l'arc qu'elle a formé, et qu'il y a match nul sinon. - -Ceci arrive avec une probabilité $ \frac{2\pi - \theta}{2 \pi} $, parce que le numérateur de cette fraction est l'angle dans lequel l'ordinateur doit jouer pour que Lucie gagne. - -Comme $ \theta $ est d'espérance $ \frac \pi 2 $, l'espérance vaut $ 1 - \frac{\pi/2}{2\pi} $, c'est à dire 3/4. - - \item (Ouvert) Peut-être, mais pas sûr. Je ne sais pas. - - \item (Moyen/Difficile) Soit $\alpha \in ]0,1[$. Si $\theta$ désigne toujours l'angle aigu formé entre le premier point placé (par Lucie) et le second (par l'adversaire), Lucie joue le point situé à l'angle $ \alpha \theta $ dans le même repère. - -\begin{center} +\begin{figure}[h] + \centering \begin{tikzpicture} \begin{scope}[xshift=10cm] \draw (0,0) circle (1.5cm); @@ -124,25 +140,31 @@ Comme $ \theta $ est d'espérance $ \frac \pi 2 $, l'espérance vaut $ 1 - \frac \end{scope} \end{tikzpicture} -\end{center} +\end{figure} -En se placant toujours dans le \underline{même repère}, l'évènement \og ne pas perdre \fg revient clairement à \og l'adversaire joue un point dont l'angle (dans le même repère) est dans $[0,\theta + \alpha \theta]$\fg. +En se repérant toujours dans le même repère, l'évènement \og ne pas perdre \fg revient clairement à \og l'adversaire joue un point dont l'angle (dans le même repère) est dans $[0,\theta + \alpha \theta]$\fg. La probabilité associée est $ \frac{\theta + \alpha \theta}{2\pi} $, l'espérance vaut $(1+\alpha)/4$, qui décrit $]\frac 1 4, \frac 1 2[$ quand $\alpha$ décrit $]0,1[$. - \item et g) (Difficile/Ouvert) Il y a pas mal de choses à dire et à faire, en considérant par exemple des stratégies dépendant d'un paramètre qu'on fait varier, ce qui permet de trouver certains intervalles inclus dans les ensembles recherchés. - - Pour les inclusions réciproques, notamment pour le premier, trouver sa borne supérieure (ce qui était l'objet de d)) a l'air assez difficile. + \textbf{Difficile :} Il y a pas mal de choses à dire et à faire, en considérant par exemple des variantes de la stratégie précédente, dépendant d'un paramètre qu'on fait varier, ce qui permet de trouver certains intervalles inclus dans l'ensemble cherché. + + \textbf{Ouvert :} La détermination de l'ensemble complet. Notamment, trouver la borne supérieure de $S$ et établit des inclusions réciproques à celles établies ci-dessus a l'air assez difficile. + + \item On peut faire des raisonnement assez similaires à ceux de la question précédentes pour obtenir des résultats partiels de la même forme. Les raisonnements sont différents, la combinatoire change, mais on peut obtenir des résultats aussi partiels que ceux de la question précédente, avec les mêmes ordres de difficulté. Je pense que les élèves vont imaginer tellement de stratégies différentes, et obtenir tellement de résultats (partiels) différents, qu'il n'est pas forcément nécessaire d'en montrer plus que ça et de développer des pages et des pages de calculs dans cette fiche solution. Simplement, voilà la tête des solutions qu'on peut donner à ces questions. \end{enumerate} -\q (Difficile/Ouvert) Pareil, on peut trouver des intervalles inclus dans les ensembles recherchés en considérant certaines stratégies dépendant d'un paramètre qu'on fait varier, les inclusions réciproques étant sans nul doute plus difficiles. +\q (Difficile/Ouvert) On peut établir des résultats partiels, mais c'est plus difficile parce qu'on a beaucoup plus de mal à voir les choses vu que $n \geqslant 3$ tours sont joués. On peut trouver des intervalles inclus dans les ensembles recherchés en considérant certaines stratégies dépendant d'un paramètre qu'on fait varier, les inclusions réciproques étant plus difficiles, et la détermination complète de l'ensemble ouverte. -\q (Moyen) Ceci découle de l'observation suivante : L'adversaire peut toujours jouer quelque chose de très proche, à $ \varepsilon $ près, du (redouté) polygôme régulier à $p$ côtés. +\q (Moyen/Ouvert) (Moyen : Trouver des conditions nécessaires, Ouvert : Trouver la CNS) -\begin{center} + \textbf{Facile (à voir)/Moyen-Difficile (à bien justifier):} Il est assez naturel de voir que $ n \geqslant 1 + \left \lfloor \frac{p+1}{2} \right \rfloor $. + + Démonstration : Ceci découle de l'observation suivante : L'adversaire peut toujours jouer quelque chose de très proche, à $ \varepsilon $ près, du (redouté) polygôme régulier à $p$ côtés. + +\begin{figure}[h] \begin{tikzpicture} - \begin{scope}[xshift=10cm] + \begin{scope} \draw (0,0) circle (1.5cm); % \draw[dotted] (0:5cm) -- (180:5cm); @@ -160,9 +182,9 @@ La probabilité associée est $ \frac{\theta + \alpha \theta}{2\pi} $, l'espéra \end{scope} \end{tikzpicture} -\end{center} +\end{figure} -Le fait qu'on considère les $\varepsilon$-approximations de ce polygône régulier fait qu'on considère un évènement de probabilité non-nulle. +Le fait qu'on considère les $\varepsilon$-approximations de ce polygône régulier fait qu'on considère un évènement de probabilité non-nulle. (L'évènement "Il existe un polygone régulier $P$ à $n$ cotés inscrit dans le cercle tel que le polygone joué par l'adversaire a tous ses sommets à distance $\leqslant \varepsilon$ de $P$" est de probabilité non-nulle). Dans ce cas, il est clair que pour $\varepsilon$ assez petit, Alice va être obligée de : @@ -184,27 +206,31 @@ De cette manière, par construction, elle s'assure la victoire. On se convainc f \item Tant qu'elle ne forme pas d'arc dépassant la taille de tous les autres arcs primitifs, elle ne peut pas gagner. Or, toujours par maximalité, elle ne peut pas faire de tel arc ailleurs qu'au sein de l'arc primitif adverse maximal ; d'où la nécessité du second point -\item Dès que $varepsilon$ est petit (i.e. très que la figure adverse est $\varepsilon$-proche du polygône régulier pour $\varepsilon$ petit), il va être nécessaire de couper un arc sur deux, sinon deux arcs consécutifs (parce que $\varepsilon$ est petit) formeront un arc non primitif dépassant l'arc du point précédent ; d'où la nécessité du troisième point. +\item Dès que $\varepsilon$ est petit (i.e. dès que la figure adverse est $\varepsilon$-proche du polygône régulier pour $\varepsilon$ petit), il va être nécessaire de couper un arc sur deux, sinon deux arcs consécutifs (parce que $\varepsilon$ est petit) formeront un arc non primitif dépassant l'arc du point précédent ; d'où la nécessité du troisième point. \end{itemize} -Pour faire tout cela, il faut un point pour un arc sur deux (arrondi au supérieur si $p$ est impair) : $ \left \lfloor \frac{p+1}{2} \right \rfloor $, plus 1 pour construire l'arc du second point précédent, d'où le résultat. +Pour faire tout cela, il faut des points pour couper un arc sur deux (arrondi au supérieur si $p$ est impair) : $ \left \lfloor \frac{p+1}{2} \right \rfloor $, plus 1 pour construire l'arc du second point précédent. + +L'évènement "Elle est obligée de faire tout ça" est de probabilité strictement positive, car il contient l'évènement "les sommets joués par l'adversaire sont à distance $\leqslant \varepsilon$ d'un polygône régulier à $n$" (Il existe un polygone régulier $P$ à $n$ cotés inscrit dans le cercle tel que le polygone joué par l'adversaire a tous ses sommets à distance $\leqslant \varepsilon$ de $P$). Il reste à justifier que cet évènement est bien de probabilité $>0$, pas la peine de développer les calculs ici je pense ; on a condition nécessaire annoncée au début. + +Je ne sais pas si c'est suffisant, je pense que non. Je n'ai pas trouvé d'autres conditions. \q \begin{enumerate} - \item (Facile) Elle a évidemment envie de jouer le polygône régulier. - \item (Moyen) On cherche l'évènement contraire \og L'adversaire a bien coupé au moins une fois chaque arc \fg. La probabilité de gagner est un dénombrement au cours duquel il est très facile de s'exciter trop vite et de se tromper ! Cette question saura distinguer les équipes sachant raisonner droit de celles trébuchant au moindre obstacle. La réponse est 1 si $ p < n $ et ... sinon. + \item (Difficile/Ouvert) Je pense que c'est faisable, mais c'est vraiment dur, surtout en Terminale. + \end{enumerate} -\q (Moyen/Difficile) J'ai la preuve, mais il est tard, je l'écris plus tard +\q (Moyen/Difficile) \q (Difficile/Ouvert) Peut-être seulement difficile sans être infaisable. diff --git a/index.pdf b/index.pdf deleted file mode 100644 index 68dc0ed09899296cad018dc7d1ba1c53d4b76565..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 379498 zcma&NQ>-vbw5_>p+qP}nwr$(CZQHhO+qUiXuikyrZ|5X^XHrQ$R%O=AsC*fNR6#_H zmXVGXigaORbOVZwm4Jc3-pC4yhX;yY#?;Q-#e#s5gN2>o{}dFxn5B)2sS^Rcn2n)} zsfekuy@@FlA0L#ni<7CLEtJRBDaLftPTON$eMZhmB@$teM8XtlTF-Tolyzr`Y?YE4 z+y)^_QIv4qM?;N|Z&%DUvj%^I$jE!COZ2jczYj*Z+zui6n9X;C+10_>zN2lS|3HD%Sn(06)BI7p1 z;1o@AL<4S)e>cyR9h~BMzX3s)F~^=w zA%Z%5qLNmqJozNcHQcw8&UI1VcXKsxj}6#s!%t;v!%KjRfC8gn2814!(haFrO?imT z-fCc*jK$>-201bq_fqd}iv{OWN&(92oasGCxAC#8mZ8sM+n^Y{3@ 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Les règles sont les suivantes. -\medskip - -Le jeu se déroule sur un cercle. Au départ, les points du cercle sont \textit{non colorés}. L'un des deux adversaires est désigné pour jouer en premier. Chacun leur tour, Lucie et son adversaire jouent (c'est à dire, choisissent) un point, qui sera alors coloré en leur couleur respective : Rouge pour Lucie, Bleu pour son adversaire. Lorsqu'ils jouent, il leur est interdit de choisir un point qui a déjà été coloré par l'un d'eux. Lucie convient à l'avance du nombre de coups que la partie durera. Tous deux jouent le même nombre de coups, de sorte que le \textit{nombre de coups} est un entier pair, noté $2n$. Par exemple, si le \textit{nombre de coups} vaut $2n = 6$ coups, ils joueront $n=3$ coups chacun. La partie s'arrête donc lorsque les $2n$ coups sont joués. +Le jeu se déroule sur un cercle. Au départ, les points du cercle sont non colorés. L'un des deux adversaires est désigné pour jouer en premier. Chacun leur tour, Lucie et son adversaire jouent (c'est à dire, choisissent) un point, qui sera alors coloré en leur couleur respective : Orange pour Lucie, Bleu pour son adversaire. Lorsqu'ils jouent, il leur est interdit de choisir un point qui a déjà été coloré par l'un d'eux. Lucie convient à l'avance du \textit{nombre de coups} que la partie durera. Tous deux jouent le même nombre de coups, de sorte que le nombre de coups est un entier pair, noté $2n$. Par exemple, si le nombre de coups vaut $2n = 6$ coups, ils joueront $n=3$ coups chacun. La partie s'arrête donc lorsque les $2n$ coups sont joués. \medskip -À la fin de la partie, le cercle est découpé en arcs de cercle dont les extrémités sont soit rouge, soit bleu. Dans une telle configuration, un $\textit{arc primitif}$ est un arc dont les deux extrémités sont colorées (en rouge ou en bleu) et dont tous les autres points ne sont pas colorés (par exemple, le cercle tout entier, vu comme un arc de cercle, n'est jamais primitif). Les arcs primitifs dont les deux extrémités sont rouges sont alors colorés en rouge, et ceux dont les extrémités sont toutes deux bleues sont colorés en bleu. Le gagnant est alors celui étant parvenu à former l'arc de cercle \textbf{non nécessairement primitif} le plus long entièrement coloré de sa propre couleur. S'il y a égalité de tels arcs, ou s'il n'en existe aucun, la partie est déclarée nulle. +À la fin de la partie, le cercle est découpé en arcs de cercle dont les extrémités sont soit orange, soit bleu. Dans une telle configuration, un \textit{arc primitif} est un arc dont les deux extrémités sont colorées (en orange ou en bleu) et dont tous les autres points ne sont pas colorés (par exemple, le cercle tout entier, vu comme un arc de cercle, n'est jamais primitif). Les arcs primitifs dont les deux extrémités sont orange sont alors colorés en orange, et ceux dont les extrémités sont toutes deux bleues sont colorés en bleu. Le gagnant est alors celui étant parvenu à former l'arc de cercle non nécessairement primitif le plus long entièrement coloré de sa propre couleur. S'il y a égalité de tels arcs, ou s'il n'en existe aucun, la partie est déclarée nulle. -\begin{center} -\begin{tikzpicture} - \begin{scope} +\begin{figure}[h] + \centering + \begin{tikzpicture} + \begin{scope} - \draw (0,0) circle (1.5cm); + \draw (0,0) circle (1.5cm); - \foreach \angle in {45, 120, 200} { - \filldraw[fill=red] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm - } + \foreach \angle in {45, 120, 200} { + \filldraw[fill=orange] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm + } - \foreach \angle in {80, 160, 270} { - \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm - } - - \end{scope} + \foreach \angle in {80, 160, 270} { + \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm + } + + \end{scope} - \begin{scope}[xshift=5cm] - \draw (0,0) circle (1.5cm); + \begin{scope}[xshift=5cm] + \draw (0,0) circle (1.5cm); - \coordinate (A1) at (30:1.5cm); - \coordinate (A2) at (300:1.5cm); - \coordinate (A3) at (150:1.5cm); - \coordinate (B1) at (219:1.5cm); - \coordinate (B2) at (319:1.5cm); - \coordinate (B3) at (15:1.5cm); - - \draw[very thick, red] (A1) arc[start angle=30, end angle=150, radius=1.5cm]; - \draw[very thick, blue] (B2) arc[start angle=319, end angle=375, radius=1.5cm]; - - - \foreach \angle in {30, 300, 150} { - \filldraw[fill=red] (\angle:1.5cm) circle (2pt); - } + \coordinate (A1) at (30:1.5cm); + \coordinate (A2) at (300:1.5cm); + \coordinate (A3) at (150:1.5cm); + \coordinate (B1) at (219:1.5cm); + \coordinate (B2) at (319:1.5cm); + \coordinate (B3) at (15:1.5cm); + + \draw[very thick, orange] (A1) arc[start angle=30, end angle=150, radius=1.5cm]; + \draw[very thick, blue] (B2) arc[start angle=319, end angle=375, radius=1.5cm]; + + \foreach \angle in {30, 300, 150} { + \filldraw[fill=orange] (\angle:1.5cm) circle (2pt); + } - \foreach \angle in {219, 319, 15} { - \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); - } - - \end{scope} + \foreach \angle in {219, 319, 15} { + \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); + } + + \end{scope} - \begin{scope}[xshift=10cm] - \draw (0,0) circle (1.5cm); - - \coordinate (A1) at (10:1.5cm); - \coordinate (A2) at (60:1.5cm); - \coordinate (A3) at (180:1.5cm); - \coordinate (A4) at (290:1.5cm); - \coordinate (A5) at (330:1.5cm); + \begin{scope}[xshift=10cm] + \draw (0,0) circle (1.5cm); + + \coordinate (A1) at (10:1.5cm); + \coordinate (A2) at (60:1.5cm); + \coordinate (A3) at (180:1.5cm); + \coordinate (A4) at (290:1.5cm); + \coordinate (A5) at (330:1.5cm); - \coordinate (B1) at (70:1.5cm); - \coordinate (B2) at (95:1.5cm); - \coordinate (B3) at (155:1.5cm); - \coordinate (B4) at (200:1.5cm); - \coordinate (B5) at (260:1.5cm); - - \draw[very thick, blue] (A4) arc[start angle=290, end angle=420, radius=1.5cm]; - \draw[very thick, red] (B1) arc[start angle=70, end angle=155, radius=1.5cm]; - \draw[very thick, red] (B4) arc[start angle=200, end angle=260, radius=1.5cm]; - - \foreach \angle in {10, 60, 180, 290, 330} { - \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm - } + \coordinate (B1) at (70:1.5cm); + \coordinate (B2) at (95:1.5cm); + \coordinate (B3) at (170:1.5cm); + \coordinate (B4) at (190:1.5cm); + \coordinate (B5) at (280:1.5cm); + + \draw[very thick, blue] (A4) arc[start angle=290, end angle=420, radius=1.5cm]; + \draw[very thick, orange] (B1) arc[start angle=70, end angle=170, radius=1.5cm]; + \draw[very thick, orange] (B4) arc[start angle=190, end angle=280, radius=1.5cm]; + + \foreach \angle in {10, 60, 180, 290, 330} { + \filldraw[fill=blue] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm + } - \foreach \angle in {70, 95, 155, 200, 260} { - \filldraw[fill=red] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm - } - \end{scope} + \foreach \angle in {70, 95, 170, 190, 280} { + \filldraw[fill=orange] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm + } + \end{scope} + \end{tikzpicture} + \caption{Deux fins de partie pour $2n = 6$ et une pour $2n = 10$. À gauche, il n'y a pas d'arcs colorés (pas d'arc primitif) ; la partie est donc nulle. Au centre, Lucie (en orange) gagne : elle a réussi à construire un arc de taille maximale. À droite, l'adversaire (en bleu) gagne car il a formé un arc (non primitif) bleu de taille maximale.} + \label{fig:Exemple} +\end{figure} - \node[below] at (0,-2) { - \begin{tabular}{ll} - Exemple 1 - \end{tabular} - }; +Dans tout le problème, on appelle \textit{stratégie} une manière déterministe de décrire quoi jouer en fonction des coups qui ont été joués précédemment. Autrement dit, une stratégie est un algorithme qui indique quel coup jouer en fonction de la situation courante, de sorte que, dans deux situations identiques, il indiquera toujours le même coup à jouer. - - \node[below] at (5,-2) { - \begin{tabular}{ll} - Exemple 2 - \end{tabular} - }; - - \node[below] at (10,-2) { - \begin{tabular}{ll} - Exemple 3 - \end{tabular} - }; - -\end{tikzpicture} - -\end{center} - -Les configurations ci-dessus représentent deux fins de partie pour $2n = 6$ et une pour $2n = 10$. Dans le premier exemple, il n'y a pas d'arcs colorés, parce qu'il n'y a pas d'arc primitif. La partie est alors nulle. Dans le second exemple, Lucie (en rouge) gagne car elle a réussi à construire un arc de taille maximale. Dans le troisième exemple, l'adversaire (en bleu) gagne parce qu'il a formé un arc (non primitif) bleu de taille maximale. - -\medskip - -Dans tout le problème, on appelle \textit{stratégie} une manière déterministe de décrire de jouer en fonction des coups qui ont été joués précédemment. Autrement dit, une stratégie est un algorithme qui indique quel coup jouer en fonction de la situation courante, de sorte que, dans deux situations identiques, il indiquera toujours le même coup à jouer. - -\medskip - -Comme Lucie n'aime pas perdre, elle commence par se choisir pour adversaire l'Idiot du Village. Ce dernier portant bien son nom, il joue ses coups \textbf{aléatoirement}, sans réfléchir. Chaque coup joué suit alors une \textit{loi uniforme} sur le cercle. Lucie cherche alors des stratégies qui \textit{maximisent} sa probabilité de gagner contre cet adversaire. - -% \underline{Rappel :} Si un nombre est choisi dans un intervalle aléatoirement selon une loi uniforme, tout évènement de la forme \og Le nombre choisi vaut x \fg est de probabilité nulle. Plus généralement, on considèrera le fait que tout évènement de la forme \og Le point suivant, choisi aléatoirement (selon une loi uniforme) sur le cercle, tombe pile ici \fg, est de probabilité nulle, et ne se produira donc pas. Par exemple, sous les mêmes hypothèses, la probabilité qu'il existe deux paires de points formant les mêmes distances, est nulle. La probabilité que deux points \textit{Bref, ne vous embêtez pas trop avec ces petits détails, et concentrez vous sur le gros du travail !} - -\medskip +Comme Lucie n'aime pas perdre, elle commence par se choisir pour adversaire l'Idiot du Village. Ce dernier portant bien son nom, il joue ses coups aléatoirement, sans réfléchir. Chaque coup joué suit alors une loi uniforme sur le cercle. Lucie cherche alors des stratégies qui maximisent sa probabilité de gagner contre cet adversaire. Lucie et son adversaire conviennent de commencer par fixer $2n = 4$. -\q Montrer que si Lucie laisse son adversaire jouer en premier, elle peut gagner à tous les coups. +\q Montrer que si Lucie laisse son adversaire jouer en premier, elle dispose d'une stratégie lui permettant de gagner avec probabilité 1. \q Après qu'elle ait gagné une partie, son adversaire la laisse jouer en premier. \begin{enumerate} - \item Lucie dispose-t-elle d'une stratégie lui garantissant de gagner ? + \item Lucie dispose-t-elle dispose d'une stratégie lui permettant de gagner quoi qu'il advienne ? - \item Lucie dispose-t-elle d'une stratégie lui garantissant de ne jamais perdre ? + \item Étudier l'ensemble des $ p \in [0,1] $ tels qu'il existe une stratégie permettant à Lucie de gagner avec une probabilité exactement $p$. - \item Démontrer que Lucie dispose d'une stratégie lui garantissant une probabilité de gagner de 3/4. - - \item Peut-elle faire mieux ? - - \item Démontrer que, pour tout $p \in ] \frac 1 4, \frac 1 2 [ $, il existe une stratégie garantissant à Lucie de gagner avec une probabilité de $p$. - - \item Déterminer l'ensemble des $ p \in [0,1] $ tels qu'il existe une stratégie permettant à Lucie de gagner avec une probabilité de $p$. Cet ensemble est-il égal à $[0,1]$ ? - - \item Même question pour pour les probabilités de ne pas perdre. + \item Même question pour des probabilités de ne pas perdre. \end{enumerate} @@ -143,46 +104,30 @@ Lucie et son adversaire conviennent de commencer par fixer $2n = 4$. \begin{enumerate} - \item Étudier l'ensemble des $p \in [0,1]$ tels qu'il existe une stratégie permettant à Lucie de gagner avec une probabilité de $p$. + \item Reprendre la question précédente en étudiant dans ce nouveau contexte l'ensemble des $ p \in [0,1] $ tels qu'il existe une stratégie permettant à Lucie de gagner avec une probabilité exactement $p$. On pourra commencer par le cas $2n = 6$. - \item Et pour les probabilités de ne pas perdre ? + \item Même question pour des probabilités de ne pas perdre. \end{enumerate} -\medskip +Lucie propose un pacte à son adversaire. Ils conviennent d'un entier $p \geqslant n $, et les règles sont changées de sorte que l'adversaire de Lucie place $p$ points plutôt que $n$. Comme la règle du tour-par-tour est alors difficile à appliquer, ils conviennent que l'adversaire de Lucie placera tous ses points en premier. Ce dernier joue toujours aléatoirement sur le cercle, mais avant Lucie, de sorte que cette dernière a alors toute la liberté de choisir où placer ses points. Lucie a donc plus d'information que son adversaire, mais en contrepartie, ce dernier peut placer plus de points qu'elle. -Lucie propose un pacte à son adversaire. Ils conviennent d'un entier $p \geqslant n $, et les règles sont changées de sorte que l'adversaire de Lucie place $p$ points plutôt que $n$. Comme la règle du tour-par-tour est alors difficile à appliquer, ils conviennent que l'adversaire de Lucie placera tous ses points en premier. Ce dernier joue toujours aléatoirement sur le cercle, mais \textit{avant} Lucie, de sorte que cette dernière a alors toute la liberté de choisir où placer ses points. Lucie a donc plus d'information que son adversaire, mais en contrepartie, ce dernier peut placer plus de points qu'elle. +\q Déterminer des conditions nécessaires et suffisantes sur $n,p$ pour que Lucie dispose d'une stratégie lui permettant de gagner avec probabilité 1. -\q +\q Pour essayer, Lucie et son adversaire reprennent exactement la même configuration que la précédente, mais en échangeant les rôles. Lucie place $p$ points, son adversaire en place $n$. Ce dernier joue toujours aléatoirement, et Lucie place tous ses points en premier. De plus, cette fois-ci, on n'impose plus $ p \geqslant n $. \begin{enumerate} - \item Montrer que s'il existe une stratégie permettant à Lucie de gagner à coup sûr, alors $ n \geqslant 1 + \left \lfloor \frac{p+1}{2} \right \rfloor $. + \item Lucie est fortement tentée de jouer le polygône régulier à $p$ côtés inscrit dans le cercle. Quelle est alors sa probabilité de gagner ? - \item Lucie dispose-t-elle d'une stratégie optimale ? Discuter selon la valeur de $n$ et $p$. + \item Cette stratégie est-elle optimale ? \end{enumerate} -Pour essayer, Lucie et son adversaire reprennent exactement la même configuration que la précédente, mais en échangeant les rôles. Lucie place $p$ points, son adversaire en place $n$. Ce dernier joue toujours aléatoirement, et Lucie place tous ses points en premier. De plus, cette fois-ci, on n'impose plus $ p \geqslant n $. +Fatiguée de jouer avec l'Idiot du Village, Lucie se trouve un adversaire à sa taille : Lucien. L'un des deux joueurs est désigné pour jouer en premier, et $ 2n \in \mathbb N^* $ est fixé. La règle du tour-par-tour est alors appliquée. Lucien commence à jouer. -\q +\q L'un des deux joueurs dispose-t-il d'une stratégie gagnante ? -\begin{enumerate} - - \item Quelle configuration naturelle Lucie est-elle alors fortement tentée de jouer ? Quelle est alors sa probabilité de gagner ? - - \item Cette stratégie est-elle optimale ? Si oui, le démontrer. - -\end{enumerate} - -Fatiguée de jouer avec l'Idiot du Village, Lucie se trouve un adversaire à sa taille : Lucien. L'un des deux joueurs est désigné pour jouer en premier, et $ 2n \in \mathbb N^* $ est fixé. - -\q Démontrer qu'aucun des deux joueurs ne dispose alors de stratégie gagnante. - -Lucie et Lucien parviennent à convaincre $ k \geqslant 1 $ joueurs supplémentaires de jouer avec eux. Ils ont chacun une couleur et jouent chacun leur tour, toujours suivant les mêmes règles ; le nombre total $(k+2)n \in \mathbb N^* $ de points joués au cours de la partie étant fixé. - -\q L'un des joueurs a-t-il une stratégie gagnante ? - -\q Lucie se demande : que dire des questions précédentes si elle avait convenu dès le départ que le gagnant était non pas celui ayant l'arc le plus long, mais celui étant parveni à maximiser la somme des longueurs des arcs de sa couleur ? +\q Lucie se demande : que dire des questions précédentes si elle avait convenu dès le départ que le gagnant était non pas celui ayant l'arc le plus long, mais celui étant parvenu à maximiser la somme des longueurs des arcs primitifs de sa couleur ? \q Proposer et étudier d'autres pistes de recherche. \ No newline at end of file diff --git a/src/toboggan.tex b/src/toboggan.tex index ce4d0fe..d1d7d22 100644 --- a/src/toboggan.tex +++ b/src/toboggan.tex @@ -78,8 +78,6 @@ On trouvera un exemple sur la figure~\ref{fig:ToboggansSansY}. \medskip - - \q Emmy souhaite que de l'eau sorte par toutes les sorties en même temps. Pour quelle(s) valeur(s) de $N$ et $H$ cela est-il possible si \begin{enumerate} \item elle peut choisir où rentre l’eau ;

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