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Complete the following magic squares by replacing the asterisks with numbers so that the sums of the rows, columns, and diagonals are always equal.

\begin{tabular}{|l|c|l|l|}
\hline
9 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] & 8 & 14 \\
\hline
8 & 7 & [tex]$\ \textless \ /em\ \textgreater \ $[/tex] & 4 \\
\hline
7 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] & 12 & [tex]$\ \textless \ /em\ \textgreater \ $[/tex] \\
\hline
13 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] & [tex]$\ \textless \ /em\ \textgreater \ $[/tex] & 9 \\
\hline
\end{tabular}

\begin{tabular}{|l|l|l|c|}
\hline
9 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] & 14 & [tex]$\ \textless \ /em\ \textgreater \ $[/tex] \\
\hline
14 & 15 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] & [tex]$\ \textless \ /em\ \textgreater \ $[/tex] \\
\hline
7 & 21 & 10 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] \\
\hline
[tex]$\ \textless \ /em\ \textgreater \ $[/tex] & 15 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] & 8 \\
\hline
\end{tabular}

\begin{tabular}{|r|l|l|r|}
\hline
7 & 18 & 9 & 7 \\
\hline
9 & [tex]$\ \textless \ /em\ \textgreater \ $[/tex] & 23 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] \\
\hline
[tex]$\ \textless \ /em\ \textgreater \ $[/tex] & 7 & 9 & [tex]$\ \textless \ em\ \textgreater \ $[/tex] \\
\hline
[tex]$\ \textless \ /em\ \textgreater \ $[/tex] & 15 & 2 & [tex]$*$[/tex] \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
Para resolver el problema de los cuadros mágicos, debemos encontrar los valores que hacen que la suma de las filas, columnas y diagonales sea la misma para cada cuadro. Procederemos con cada cuadro uno por uno, estableciendo primero una constante [tex]\( M \)[/tex] (la suma mágica) para cada cuadro, y luego resolviendo las ecuaciones necesarias.

### Primer Cuadro Mágico
[tex]\[ \begin{array}{|c|c|c|c|} \hline 9 & * & 8 & 14 \\ \hline 8 & 7 & * & 4 \\ \hline 7 & & 12 & \\ \hline 13 & & & 9 \\ \hline \end{array} \][/tex]

Llama a los asteriscos [tex]\( x_1, x_2, x_3, x_4, x_5, x_6, x_7 \)[/tex] de la siguiente manera:
[tex]\[ \begin{array}{|c|c|c|c|} \hline 9 & x_1 & 8 & 14 \\ \hline 8 & 7 & x_2 & 4 \\ \hline 7 & x_3 & 12 & x_4 \\ \hline 13 & x_5 & x_6 & 9 \\ \hline \end{array} \][/tex]

Sabemos que:
1. [tex]\(9 + x_1 + 8 + 14 = M\)[/tex]
2. [tex]\(8 + 7 + x_2 + 4 = M\)[/tex]
3. [tex]\(7 + x_3 + 12 + x_4 = M\)[/tex]
4. [tex]\(13 + x_5 + x_6 + 9 = M\)[/tex]

Por las columnas:
5. [tex]\(9 + 8 + 7 + 13 = M\)[/tex]
6. [tex]\(x_1 + 7 + x_3 + x_5 = M\)[/tex]
7. [tex]\(8 + x_2 + 12 + x_6 = M\)[/tex]
8. [tex]\(14 + 4 + x_4 + 9 = M\)[/tex]

Por las diagonales:
9. [tex]\(9 + 7 + 12 + 9 = M\)[/tex]
10. [tex]\(14 + x_2 + x_3 + 13 = M\)[/tex]

Calcula [tex]\( M \)[/tex] de la suma de la columna 1:
[tex]\[ M = 9 + 8 + 7 + 13 = 37 \][/tex]

Usamos este valor [tex]\( M \)[/tex] para encontrar los valores restantes.
De [tex]\( 9 + x_1 + 8 + 14 = 37 \)[/tex]:
[tex]\[ 31 + x_1 = 37 \Rightarrow x_1 = 6 \][/tex]
De [tex]\( 8 + 7 + x_2 + 4 = 37 \)[/tex]:
[tex]\[ 19 + x_2 = 37 \Rightarrow x_2 = 18 \][/tex]
De [tex]\( 7 + x_3 + 12 + x_4 = 37 \)[/tex]:
[tex]\[ 19 + x_3 + x_4 = 37 \Rightarrow x_3 + x_4 = 18 \][/tex]
De [tex]\( 13 + x_5 + x_6 + 9 = 37 \)[/tex]:
[tex]\[ 22 + x_5 + x_6 = 37 \Rightarrow x_5 + x_6 = 15 \][/tex]
De [tex]\( x_1 + 7 + x_3 + x_5 = 37 \)[/tex]:
[tex]\[ 6 + 7 + x_3 + x_5 = 37 \Rightarrow x_3 + x_5 = 24 \][/tex]
De [tex]\( 8 + x_2 + 12 + x_6 = 37 \)[/tex]:
[tex]\[ 8 + 18 + 12 + x_6 = 37 \Rightarrow 38 + x_6 = 37 \Rightarrow x_6 = -1 \][/tex] (error porque no hay solución, revisar valores).

### Segundo Cuadro Mágico
[tex]\[ \begin{array}{|c|c|c|c|} \hline 9 & & 14 & \\ \hline 14 & 15 & & \\ \hline 7 & 21 & 10 & * \\ \hline & 15 & & 8 \\ \hline \end{array} \][/tex]

Del primer cuadro:
[tex]\[ M = 14+21+10+8 = 53 \][/tex]

[tex]\[ \begin{array}{|c|c|c|c|} \hline 9 & x_1 & 14 & x_2 \\ \hline 14 & 15 & x_3 & x_4 \\ \hline 7 & 21 & 10 & x_5 \\ \hline x_6 & 15 & x_7 & 8 \\ \hline \end{array} \][/tex]

Suma de filas, columnas y diagonales debe ser igual a M.

Si M= 53
Fila 1: 9 + x_1 + 14 + x_2 = 9 + x_1 + 14 + x_2 = 53
Fila 2: 14 + 15 + x_3 + x_4 = 14 + 15 + x_3 + x_4 = 53
Fila 3: 7 + 21 + 10 + x_5 = 7 + 21 + 10 + x_5 = 53
Fila 4: x_6 + 15 + x_7 + 8 = x_6 + 15 + x_7 + 8 = 53

Columna 1:9 + 14 + 7 + x_6 = 9 + 14 + 7 + x_6 = 53
Columna 2: x_1 + 15 + 21 + 15 = x_1 + 15 + 21 + 15 = 53
Columna 3: 14 + x_3 + 10 + x_7 = 14 + x_3 + 10 + x_7 = 53
Columna 4: x_2 + x_4 + x_5 + 8 = x_2 + x_4 + x_5 + 8 = 53

### Tercer Cuadro Mágico
[tex]\[ \begin{array}{|r|l|l|r|} \hline 7 & 18 & 9 & 7 \\ \hline 9 & & 23 & \\ \hline & 7 & 9 & \\ \hline & 15 & 2 & \\ \hline \end{array} \][/tex]

Suma de filas, columnas y diagonales debe ser igual a M. Similar con el Primero.

### Resolveremos los anteriores en Python.
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