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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1441010909619--
%%--Carrés magiques--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
$\vspace*{-0.35cm}$

\begin{enumerate}
\item En calculant la somme des nombres inscrits sur chaque ligne, chaque
colonne, chaque diagonale du carr\'{e} suivant, vous saurez ce qu'on appelle
un carr\'{e} magique d'ordre $5$ et de somme $65$.%

\[
\left(
\begin{array}
[c]{ccccc}%
3 & 16 & 9 & 22 & 15\\
20 & 8 & 21 & 14 & 2\\
7 & 25 & 13 & 1 & 19\\
24 & 12 & 5 & 18 & 6\\
11 & 4 & 17 & 10 & 23
\end{array}
\right)
\]


\item Compl\'{e}ter les carr\'{e}s suivants afin qu'ils soient magiques.%

%TCIMACRO{\TeXButton{BeginMulticols}{\begin{multicols}{4}}}%
%BeginExpansion
\begin{multicols}{4}%
%EndExpansion


$\left(
\begin{array}
[c]{ccc}%
1 & -1 & 0\\
-1 & . & 1\\
0 & 1 & -1
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
0 & 1 & -1\\
-1 & . & .\\
1 & -1 & 0
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
1 & . & -1\\
-2 & . & 2\\
1 & . & -1
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
3 & 0 & .\\
-2 & 1 & .\\
2 & . & .
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
\frac{3}{2} & . & \frac{4}{3}\\
. & 1 & \frac{7}{6}\\
. & . & .
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
\sqrt{32} & . & .\\
. & -\sqrt{18} & .\\
. & . & \sqrt{50}%
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
. & 0 & .\\
0 & 1 & 2\\
. & 2 & .
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
0 & . & .\\
1 & . & .\\
3 & . & .
\end{array}
\right)  $%

%TCIMACRO{\TeXButton{EndMulticols}{\end{multicols}}}%
%BeginExpansion
\end{multicols}%
%EndExpansion


\item Compl\'{e}ter les carr\'{e}s suivants afin qu'ils soient magiques de
sommes respectives $12$, $24$, $36$ et $(-24)$.
%TCIMACRO{\TeXButton{BeginMulticols}{\begin{multicols}{4}}}%
%BeginExpansion
\begin{multicols}{4}%
%EndExpansion


$\left(
\begin{array}
[c]{ccc}%
. & 1 & .\\
2 & . & .\\
. & . & .
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
. & 1 & .\\
2 & . & .\\
. & . & .
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
. & 3 & .\\
6 & . & .\\
. & . & .
\end{array}
\right)  $

$\left(
\begin{array}
[c]{ccc}%
. & -1 & .\\
-2 & . & .\\
. & . & .
\end{array}
\right)  $%

%TCIMACRO{\TeXButton{EndMulticols}{\end{multicols}}}%
%BeginExpansion
\end{multicols}%
%EndExpansion


\item Trouver un carr\'{e} magique d'ordre $3$ avec les $9$ premiers entiers
naturels non nuls. Pr\'{e}ciser sa somme.

\item Trouver un carr\'{e} magique d'ordre $3$ diff\'{e}rent de $\left(
\begin{array}
[c]{ccc}%
0 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{array}
\right)  $ comportant le maximum de $0$.

\item Trouver, s'il en existe, des carr\'{e}s magiques d'ordre $3$ \`{a}
coefficients entiers naturels de somme $25$, de somme~$27$.

\item On consid\`{e}re le carr\'{e} magique $\left(
\begin{array}
[c]{ccc}%
14 & 4 & 6\\
0 & 8 & 16\\
10 & 12 & 2
\end{array}
\right)  $. Trouver, \`{a} partir de celui-ci, un carr\'{e} magique d'ordre
$3$ \`{a} coefficients entiers naturels tous diff\'{e}rents de somme $2424$,
de somme $3333$.
\end{enumerate}
\end{exercise}


\end{document}