Let's examine each of the 3 by 3 magic squares given and calculate the sum of their four corner entries.
The three magic squares given are:
1.
[tex]\[
\begin{array}{|c|c|c|}
\hline
8 & 1 & 6 \\
\hline
3 & 5 & 7 \\
\hline
4 & 9 & 2 \\
\hline
\end{array}
\][/tex]
2.
[tex]\[
\begin{array}{|c|c|c|}
\hline
3 & 2 & 7 \\
\hline
8 & 4 & 0 \\
\hline
1 & 6 & 5 \\
\hline
\end{array}
\][/tex]
3.
[tex]\[
\begin{array}{|c|c|c|}
\hline
10 & 9 & 14 \\
\hline
15 & 11 & 7 \\
\hline
8 & 13 & 12 \\
\hline
\end{array}
\][/tex]
We will calculate the sum of the corner entries for each magic square.
For the first magic square:
- Top left corner: 8
- Top right corner: 6
- Bottom left corner: 4
- Bottom right corner: 2
[tex]\[
\text{Sum of corners} = 8 + 6 + 4 + 2 = 20
\][/tex]
For the second magic square:
- Top left corner: 3
- Top right corner: 7
- Bottom left corner: 1
- Bottom right corner: 5
[tex]\[
\text{Sum of corners} = 3 + 7 + 1 + 5 = 16
\][/tex]
For the third magic square:
- Top left corner: 10
- Top right corner: 14
- Bottom left corner: 8
- Bottom right corner: 12
[tex]\[
\text{Sum of corners} = 10 + 14 + 8 + 12 = 44
\][/tex]
Putting these sums together, we get:
[tex]\[
\text{Sum of corners for all three} = [20, 16, 44]
\][/tex]
Next, we'll identify which key number can be used to determine the sum of the four corners for these magic squares. Each option in the given responses is a potential relationship:
A. The sum of the four corners is 4 times the number in the center.
B. The sum of the four corners is 4 more than three times the number in the top left corner.
C. The sum of the four corners is 5 times the number in the bottom left corner.
D. The sum of the four corners is twice the largest number.
We simplify the key numbers in each option for all three squares:
- For option A, the center numbers are 5, 4, and 11 respectively:
- 4 5 = 20
- 4 4 = 16
- 4 * 11 = 44
Given that the resulting sums exactly match with our calculated sums of corners, the correct condition is:
[tex]\[
\mathbf{A. \; The \; sum \; of \; the \; four \; corners \; is \; 4 \; times \; the \; number \; in \; the \; center.}
\][/tex]
