Find the missing number in the following number matrix.
Show Hint
Look for row/column symmetries when dealing with 2x2 inner matrices.
Here, the sum of the bottom row ($5+3=8$) is exactly half of the outer bottom number ($16$).
By symmetry, the sum of the top row ($2+?$) should also be half of the outer top number ($12$), which is $6$.
Thus:
\[ 2 + ? = 6 \quad \implies \quad ? = 4 \]
This quick shortcut yields the correct answer in seconds.
Step 1: Understanding the Question:
This question requires identifying the mathematical rule that relates the numbers arranged inside and around a geometric grid.
By looking at the symmetries of the diagram, we can discover a pattern among the outer values, and then find how they relate to the values in the inner 2x2 grid. Step 2: Detailed Explanation:
• Analyze the Outer Quadrants:
Let us examine the relationships among the outer numbers in each quadrant of the diamond:
- Top-Left quadrant: Corners has $81$, and adjacent side numbers are $12$ and $21$.
Notice: $21 - 12 = 9 = \sqrt{81}$.
- Bottom-Left quadrant: Corner has $25$, and adjacent side numbers are $21$ and $16$.
Notice: $21 - 16 = 5 = \sqrt{25}$.
- Bottom-Right quadrant: Corner has $81$, and adjacent side numbers are $16$ and $7$.
Notice: $16 - 7 = 9 = \sqrt{81}$.
- Top-Right quadrant: Corner has $25$, and adjacent side numbers are $12$ and $7$.
Notice: $12 - 7 = 5 = \sqrt{25}$.
This confirms a consistent rule: the absolute difference of the adjacent side numbers equals the square root of the corner number.
• Relate the Inner Grid to the Outer Sides:
Now, let's look at the inner 2x2 grid containing the numbers:
Top-Left: $2$, Bottom-Left: $5$, Bottom-Right: $3$, Top-Right: $?$
We can relate the sums of the rows and columns in this 2x2 grid to the outer side numbers:
- Bottom Row Sum: The bottom row values are $5$ and $3$.
Sum $= 5 + 3 = 8$.
The outer bottom middle number is $16$, which is $2 \times 8$.
- Left Column Sum: The left column values are $2$ and $5$.
Sum $= 2 + 5 = 7$.
The outer left middle number is $21$, which is $3 \times 7$.
- Let us apply this rule to the Top Row and Right Column:
Let $x$ be the missing number in the top-right position.
- Top Row Sum: The values are $2$ and $x$.
Let us test the option $x = 4$:
Sum $= 2 + 4 = 6$.
The outer top middle number is $12$, which is $2 \times 6$ (factor $= 2$).
- Right Column Sum: The values are $4$ and $3$.
Sum $= 4 + 3 = 7$.
The outer right middle number is $7$, which is $1 \times 7$ (factor $= 1$).
This gives a highly consistent and symmetric set of factors:
The row factors are $(2, 2)$, which are equal.
The column factors are $(3, 1)$, which average to $2$.
This confirms $x = 4$ is the correct missing value. Step 3: Final Answer:
The missing number in the matrix is 4.
Therefore, the correct option is (B).
