Answered

Examine the 3 by 3 magic squares. For a 3 by 3 magic square, how can you determine the sum of all the numbers in the square by using a key value in the magic square?

\begin{tabular}{|l|l|l|}
\hline 8 & 1 & 6 \\
\hline 3 & 5 & 7 \\
\hline 4 & 9 & 2 \\
\hline
\end{tabular}
\begin{tabular}{|l|l|l|}
\hline 3 & 2 & 7 \\
\hline 8 & 4 & 0 \\
\hline 1 & 6 & 5 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|}
\hline 10 & 9 & 14 \\
\hline 15 & 11 & 7 \\
\hline 8 & 13 & 12 \\
\hline
\end{tabular}

Choose the correct response below.
A. Multiply the largest number by 5.
B. Multiply the bottom middle number by 6.
C. Multiply the top left number by 10.
D. Multiply the center number by 9.



Answer :

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.