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| -rw-r--r-- | controle-2020qcm.tex | 62 |
1 files changed, 62 insertions, 0 deletions
diff --git a/controle-2020qcm.tex b/controle-2020qcm.tex index 0ea03cb..dd09605 100644 --- a/controle-2020qcm.tex +++ b/controle-2020qcm.tex @@ -859,6 +859,68 @@ $4$ \begin{question} +Quel joueur a une stratégie gagnante dans le jeu combinatoire +(impartial, à information parfaite) associé au graphe orienté +acyclique représenté ci-dessous, la position de départ étant +notée $s$ ? + +\begin{center} +\begin{tikzpicture}[>=stealth,thick,text width=5bp,text height=5bp,text depth=0bp] +\node (l0) at (-40bp,0bp) [draw,circle] {}; +\node (c0) at (0bp,0bp) [draw,circle] {}; +\node (r0) at (40bp,0bp) [draw,circle] {}; +\node (l1) at (-40bp,-40bp) [draw,circle] {}; +\node (c1) at (0bp,-40bp) [draw,circle] {}; +\node (r1) at (40bp,-40bp) [draw,circle] {}; +\node (l2) at (-40bp,-80bp) [draw,circle] {}; +\node (c2) at (0bp,-80bp) [draw,circle] {}; +\node (r2) at (40bp,-80bp) [draw,circle] {}; +\node (l3) at (-40bp,-120bp) [draw,circle] {}; +\node (c3) at (0bp,-120bp) [draw,circle] {}; +\node (r3) at (40bp,-120bp) [draw,circle] {}; +\node (l4) at (-40bp,-160bp) [draw,circle] {}; +\node (c4) at (0bp,-160bp) [draw,circle] {}; +\node (r4) at (40bp,-160bp) [draw,circle] {}; +\node (c5) at (0bp,-200bp) [draw,circle] {$s$}; +\draw[->] (l0) -- (c0); \draw[->] (r0) -- (c0); +\draw[->] (l1) -- (l0); \draw[->] (l1) -- (c0); +\draw[->] (c1) -- (l0); \draw[->] (c1) -- (r0); +\draw[->] (r1) -- (r0); \draw[->] (r1) -- (c0); +\draw[->] (l1) -- (c1); \draw[->] (r1) -- (c1); +\draw[->] (l2) -- (l1); \draw[->] (l2) -- (c1); +\draw[->] (c2) -- (l1); \draw[->] (c2) -- (r1); +\draw[->] (r2) -- (r1); \draw[->] (r2) -- (c1); +\draw[->] (l2) -- (c2); \draw[->] (r2) -- (c2); +\draw[->] (l3) -- (l2); \draw[->] (l3) -- (c2); +\draw[->] (c3) -- (l2); \draw[->] (c3) -- (r2); +\draw[->] (r3) -- (r2); \draw[->] (r3) -- (c2); +\draw[->] (l3) -- (c3); \draw[->] (r3) -- (c3); +\draw[->] (l4) -- (l3); \draw[->] (l4) -- (c3); +\draw[->] (c4) -- (l3); \draw[->] (c4) -- (r3); +\draw[->] (r4) -- (r3); \draw[->] (r4) -- (c3); +\draw[->] (l4) -- (c4); \draw[->] (r4) -- (c4); +\draw[->] (c5) -- (l4); \draw[->] (c5) -- (r4); +\end{tikzpicture} +\end{center} + +\rightanswer +le second joueur + +\answer +le premier joueur + +\answer +aucun des deux + +\end{question} + + +% +% +% + +\begin{question} + Alice et Bob jouent au jeu suivant sur un échiquier $8\times 8$ sur lequel est positionné un unique pion (commun aux deux joueurs) : \begin{itemize} |