In number puzzle questions, look for standard mathematical patterns first (squares, cubes, factorials, arithmetic series). If a single rule doesn't apply to all pairs directly, look for a "shifted" or "mapped" pattern where the result of an operation on one number corresponds to a different number's pair.
Step 1: Understanding the Question:
The circle shows pairs of numbers in opposite sectors. We need to find the logical pattern that connects these pairs to determine the missing number.
The pairs are: (3, 210), (4, 60), (5, 24), and (6, ?). Step 2: Key Formula or Approach:
Let's analyze the relationship within the pairs. It is common for such puzzles to involve arithmetic operations like powers, cubes, factorials, etc. Let's test the function \( f(n) = n^3 - n \). Step 3: Detailed Explanation:
Let's apply the function \( f(n) = n^3 - n \) to the numbers in the outer ring (3, 4, 5, 6):
For n = 3: \( f(3) = 3^3 - 3 = 27 - 3 = 24 \)
For n = 4: \( f(4) = 4^3 - 4 = 64 - 4 = 60 \)
For n = 5: \( f(5) = 5^3 - 5 = 125 - 5 = 120 \)
For n = 6: \( f(6) = 6^3 - 6 = 216 - 6 = 210 \)
Now let's see how these calculated values correspond to the numbers in the circle.
The value opposite to 4 is 60, which is \( f(4) \).
The value opposite to 3 is 210, which is \( f(6) \).
The value opposite to 5 is 24, which is \( f(3) \).
The pattern is that the value opposite a number `n` is the result of the function `f(x)` where `x` is another number from the sequence. Following this rotational mapping:
The result of f(6) is opposite 3.
The result of f(4) is opposite 4.
The result of f(3) is opposite 5.
Therefore, the result of f(5) should be opposite 6.
Step 4: Final Answer:
The number that replaces the question mark is opposite 6. Based on our discovered pattern, this should be the value of \( f(5) \).
\[ f(5) = 5^3 - 5 = 120 \]
So, the missing number is 120.
