From 2b74511ddf2717cd6a91bc46a048f866e5f8627c Mon Sep 17 00:00:00 2001 From: William Driot Date: Fri, 29 Nov 2024 18:22:34 +0100 Subject: [PATCH] =?UTF-8?q?Ajout=20de=20Points=20color=C3=A9s=20sur=20un?= =?UTF-8?q?=20cercle?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- fiches/points-colories-fiche.tex | 212 ++++++++++++++++++++++++++++++- index.pdf | Bin 0 -> 278028 bytes index_avec_fiches.pdf | Bin 0 -> 261944 bytes src/points-colories.tex | 187 ++++++++++++++++++++++++++- 4 files changed, 393 insertions(+), 6 deletions(-) create mode 100644 index.pdf create mode 100644 index_avec_fiches.pdf diff --git a/fiches/points-colories-fiche.tex b/fiches/points-colories-fiche.tex index df50ad4..17b34ec 100644 --- a/fiches/points-colories-fiche.tex +++ b/fiches/points-colories-fiche.tex @@ -1,5 +1,213 @@ \section*{Eléments de réponse} -\q (Facile) Première réponse +\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. -\q (Moyen) Deuxieme réponse +\begin{center} + \begin{tikzpicture} + \begin{scope}[xshift=10cm] + \draw (0,0) circle (1.5cm); + + \coordinate (A1) at (0:1.5cm); + \coordinate (A2) at (140:1.5cm); + + \coordinate (B1) at (180:1.5cm); + \coordinate (B2) at (340:1.5cm); + + \node at (0:1.8cm) {1}; + \node at (180:1.8cm) {2}; + \node at (140:1.8cm) {3}; + \node at (340:1.8cm) {4}; + + \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, 140} { + \filldraw[fill=red] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm + } + + \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} + +\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 (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 $D$. + +\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); + + \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{center} + +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 $[0,\pi]$. + +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{tikzpicture} + \begin{scope}[xshift=10cm] + \draw (0,0) circle (1.5cm); + + \coordinate (A1) at (0:1.5cm); + \coordinate (A2) at (120:1.5cm); + + \coordinate (B1) at (140:1.5cm); + + \node at (0:2.5cm) {1er point}; + \node at (140:1.8cm) {$ \theta $}; + \node at (120:1.8cm) {$ \alpha \theta $}; + + % \draw[dotted] (0:5cm) -- (180:5cm); + + % \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, 120} { + \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]; + + \end{scope} + \end{tikzpicture} +\end{center} + +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. + +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. + +\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 (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. + +\begin{center} + \begin{tikzpicture} + \begin{scope}[xshift=10cm] + \draw (0,0) circle (1.5cm); + + % \draw[dotted] (0:5cm) -- (180:5cm); + + % \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, 40, 80, 120, 160, 200, 240, 280, 320} { + \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]; + + \end{scope} + \end{tikzpicture} +\end{center} + +Le fait qu'on considère les $\varepsilon$-approximations de ce polygône régulier fait qu'on considère un évènement de probabilité non-nulle. + +Dans ce cas, il est clair que pour $\varepsilon$ assez petit, Alice va être obligée de : + +\begin{itemize} + +\item Trouver l'arc primitif adverse le plus grand (qui est unique avec probabilité 1), + +\item Le couper, de sorte à l'annuler, et ce en placant deux points dans cet arc de sorte que l'arc formé soit alors plus grand que tous les autres arcs adverses (restants), + +\item couper un arc adverse sur deux, pour rendre tous les arcs adverses primitifs. + +\end{itemize} + +De cette manière, par construction, elle s'assure la victoire. On se convainc facilement qu'elle n'a pas d'autre solution : + +\begin{itemize} + +\item Tant qu'elle laisse l'arc primitif adverse maximal non coupé, elle ne peut pas gagner puisqu'elle ne peut pas former d'arc plus grand ailleurs (par maximalité du précédent) ; d'où la nécessité du premier point + +\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. + +\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. + +\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. + +\end{enumerate} + +\q (Moyen/Difficile) J'ai la preuve, mais il est tard, je l'écris plus tard + +\q (Difficile/Ouvert) Peut-être seulement difficile sans être infaisable. + +\q (Ouvert) + +\q (Ouvert) \ No newline at end of file diff --git a/index.pdf b/index.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4b772bb675fe19a4e5c4aee56e018c9e807b7b0a GIT binary patch literal 278028 zcmeEuby!tR*Eb-cSfB!eAO}QR;BYvFDBUO>64EUpt)PS=0@AGl0*V4kC{juogp^2! zga|4n0*WB@%|09S7|(s*&+|Ul_kMqTT-O<9&z@NGn>B0JT6<;=s>#S5LJFX%54N;d zcTZvr!Z7@7hHXw~Ivnso~>Ll}=FL z1_gvx2{GI6FC3wm>-ScX!hhZD&%H&W|I{j>hXPQ&hC(9XCnTf=1%x&UtrMX5(NiSm zZ`u_8si*iqwJ8j>@Ncv!Ec7>R{)DbD>Q8N={@phJhLr?Fi2Eu0M^+N~cia3MHWEM+ zw~6{A7oq>nHc@}aLzti1{38!x{@phJj)$;6wfRRLBFxo)gNH=^j)!o>ZHoMnhj733 zlOQYzbCxh+2=kLLYh0nM8#BrrdLZN?hz&yQCkgXYZo&gW$iR8lrzIhU(7;dCT0(Ub zG7?((MJ|c_9T($&>h&MF82?+Z|5`$b{0lBN5P=3#10g79>t;kGg)u*9q!v`MbhLJ} zK_D?$6oT+a4YHqdvvdK)PMNw{%2=8^Sy)m78Sd)lVrl9??bSQFw;{S6H`}bcaEXs> zYx$|3Nz0(%kJ~TXCi)G8CHft?G|24b zl+C3r7E-y!PJ7+<1cf0Z?VQYJozYU6=PEWYR8Ddx?Yd(Bbj6+7vFt1FdvENNuu;72 zgCRSOw98?_iZ8O$KMj*JsJsuhQlbr#c+9g)CEF{?f~&jhk<#sF!t>Jmo*S{76s~yR z3UlBZ*_0iyDci70u0Sclzc84o!cCcVp0oV^-1H6q?!MA{23=oiUo`b?KKV46TdbTs zL&Az}YhH;3vsBcJkhp`gL*fih0~KB?b_qPUG@i5zG~iY9PChP|h@=#E-Y;NWvHJB| z{?aJ>-4Lqlx_O@W$&c&49Bc_2(U2EUxLuKh_)IgV!f->0@j_sH_exW1)U$^tgG%b( zm`I#%=H(&lc==(McHL(WI(zLqZcJyjYRM{bT-EcH@s?xf63 zjh}gG#G}dX@Zl^|nWbaU(dcQX@<-NX1Lo&xUsasR%cZBx&+b-}d^5bvRyk=-WN~esrcaXi7}K z{Y!&DSC+#gl2e!GK5*wEE7V5~Z;Y`UoKT1>yd9~Rm$mCPSBzr+a0W-DVe`z6{3y+m zqI%k^5J}|J)`_IxSP&Ec#BG*sfo<$wmJAbUNJ85XhuPB4h*qx?N@Xo_W z;x29->AV?pI+uEfd1al3&!+Qn?o2@P4ShuHo)70y87Mc&6x8B{p?UJo* zzsz`_$JJ*kF4QJRZ`a3MHzQq*v8I_|tegk@suqH9=hoQ&o{+q8emQPsrk5E_0#YS{1tAkLh8}Zk=7ZM|2$bNVWPrHD|a% zjeoPPqq{jgmh{%`d-)U*TH|-$9vER1xXSckI}2GA_oum54VGOq1>8~Os9jGomhx5Z zknX1^)A+hMaV5W&WMJVax%H=4ZuPqYzZBVLxYE2wBd;39N(G|=9(p`bH4}^ve0gTI zz;(&IVpD7i*Py_s?n9d0XXMFFzGdBSW78rWxioHyye-J}@}=?d^H;UJ>=0qK^H#wT zY=R`FZ0))CSSk>o>veOsY~jRjdU=@3U)=hoZ^OVU`GdfgEGwUhIBZCqhU_l-i-w;i z`8z!JPmbiIR~>k4OOrE@V#j(ZnB0J@VQzTxbl9`2{4J@b+qY2eV^uO4+wQx|1Ys7? 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Les règles sont les suivantes. -\q Première question +\medskip -\q Deuxième question +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. + +\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. + +\begin{center} +\begin{tikzpicture} + \begin{scope} + + \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 {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); + + \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); + } + + \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); + + \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 + } + + \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} + + \node[below] at (0,-2) { + \begin{tabular}{ll} + Exemple 1 + \end{tabular} + }; + + + \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 + +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 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 d'une stratégie lui garantissant de ne jamais perdre ? + + \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. + +\end{enumerate} + +\q Lucie et son adversaire choisissent désormais $2n > 4$. + +\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 Et pour les 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 \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 + +\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 dispose-t-elle d'une stratégie optimale ? Discuter selon la valeur de $n$ et $p$. + +\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 $. + +\q + +\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 Proposer et étudier d'autres pistes de recherche. \ No newline at end of file

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