Let's evaluate how to determine the sum of all the numbers in a 3 by 3 magic square using a key value from the magic square. A magic square is a grid where the sum of numbers in each row, column, and diagonal are the same.
1. Identify the Magic Square Sum:
In a 3 by 3 magic square, it's given that the sum of numbers in any row, column, or diagonal is constant.
2. Analyze the Center Number's Role:
For any 3 by 3 magic square, the center number plays a crucial role. To illustrate, let’s look at the given examples:
- For the first magic square:
[tex]\[
\begin{array}{|l|l|l|}
\hline 8 & 1 & 6 \\
\hline 3 & 5 & 7 \\
\hline 4 & 9 & 2 \\
\hline
\end{array}
\][/tex]
The center number is 5.
- For the second magic square:
[tex]\[
\begin{array}{|l|l|l|}
\hline 3 & 2 & 7 \\
\hline 8 & 4 & 0 \\
\hline 1 & 6 & 5 \\
\hline
\end{array}
\][/tex]
The center number is 4.
- For the third magic square:
[tex]\[
\begin{array}{|c|c|c|}
\hline 10 & 9 & 14 \\
\hline 15 & 11 & 7 \\
\hline 8 & 13 & 12 \\
\hline
\end{array}
\][/tex]
The center number is 11.
3. Evaluate Total Sums:
For each 3 by 3 magic square, the total sum of all the numbers can be determined accurately by focusing on the center number in the square.
4. Establishing the Pattern:
When we observe and calculate, we see that the total sum of all the numbers in these 3 by 3 magic squares is effectively the center number multiplied by 9.
- For the first magic square:
The center number is 5.
The total sum of the numbers is [tex]\(5 \times 9 = 45\)[/tex].
- For the second magic square:
The center number is 4.
The total sum of the numbers is [tex]\(4 \times 9 = 36\)[/tex].
- For the third magic square:
The center number is 11.
The total sum of the numbers is [tex]\(11 \times 9 = 99\)[/tex].
5. Conclusion:
Therefore, the correct response is:
D. Multiply the center number by 9.
