To solve for the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex] in the given magic square, let's remember that the sum of the numbers in each row, column, and diagonal is the same. This sum is known as the magic constant.
Given magic square:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline 8 & a & 18 \\
\hline 26 & 16 & b \\
\hline c & d & 24 \\
\hline
\end{tabular}
\][/tex]
1. Determine the Magic Constant:
- We start by calculating the magic constant using the row or column with the maximum number of known values. Let's consider the second row:
[tex]\[
26 + 16 + b = S
\][/tex]
where [tex]\( S \)[/tex] is the magic constant.
2. Choose another row or column to equate to the magic constant with fewer unknowns:
- Consider the first row:
[tex]\[
8 + a + 18 = S
\][/tex]
- The third row:
[tex]\[
c + d + 24 = S
\][/tex]
3. Calculate [tex]\( a \)[/tex]:
- From the first row’s equation:
[tex]\[
8 + a + 18 = S \implies a + 26 = S \implies a = S - 26
\][/tex]
4. Calculate [tex]\( b \)[/tex]:
- From the second row’s equation:
[tex]\[
26 + 16 + b = S \implies b = S - 42
\][/tex]
5. Calculate [tex]\( c \)[/tex]:
- Use the first column’s values next:
[tex]\[
8 + 26 + c = S \implies c + 34 = S \implies c = S - 34
\][/tex]
6. Calculate [tex]\( d \)[/tex]:
- Use the diagonal sum property:
[tex]\[
8 + 16 + 24 = S \implies 48 = S
\][/tex]
With [tex]\( S = 48 \)[/tex], we substitute back to find [tex]\( a \)[/tex], [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex]:
- [tex]\( a = S - 26 = 48 - 26 = 22 \)[/tex]
- [tex]\( b = S - 42 = 48 - 42 = 6 \)[/tex]
- [tex]\( c = S - 34 = 48 - 34 = 14 \)[/tex]
- Confirm [tex]\( d \)[/tex] using the third row:
[tex]\[
c + d + 24 = 48 \implies 14 + d + 24 = 48 \implies 38 + d = 48 \implies d = 48 - 38 = 10
\][/tex]
Thus, the values are:
[tex]\[
a = 22, \quad b = 6, \quad c = 14, \quad d = 10
\][/tex]
Therefore, the completed magic square is:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline 8 & 22 & 18 \\
\hline 26 & 16 & 6 \\
\hline 14 & 10 & 24 \\
\hline
\end{tabular}
\][/tex]
