Understanding the Number Pattern Puzzle

This question presents a series of equations that don't follow standard multiplication rules. The goal is to identify the underlying pattern or logic connecting the numbers on the left side (operands) to the result on the right side.

We are given:

• \(2 × 3 = 13\)
• \(4 × 5 = 41\)
• \(6 × 7 = ?\)

We need to find the rule that applies to the first two equations and then apply it to the third one.

Identifying the Mathematical Pattern

Let's analyze the first equation: \(2 × 3 = 13\).

Standard multiplication gives \(2 × 3 = 6\). This is not 13.

Let's try squaring the numbers and adding them:

• Square of the first number: \( 2^2 = 4 \)
• Square of the second number: \( 3^2 = 9 \)
• Sum of the squares: \( 4 + 9 = 13 \)

This matches the result given!

Verifying the Pattern with the Second Equation

Let's check if this pattern holds for the second equation: \(4 × 5 = 41\).

Using the same rule (sum of squares):

• Square of the first number: \( 4^2 = 16 \)
• Square of the second number: \( 5^2 = 25 \)
• Sum of the squares: \( 16 + 25 = 41 \)

The pattern \( a \times b = a^2 + b^2 \) works for the second equation as well.

Applying the Pattern to Find the Missing Value

Now, we apply the confirmed pattern \( a \times b = a^2 + b^2 \) to the final equation: \(6 × 7 = ?\).

• Here, \( a = 6 \) and \( b = 7 \).
• Square of the first number: \( 6^2 = 36 \)
• Square of the second number: \( 7^2 = 49 \)
• Sum of the squares: \( 36 + 49 = 85 \)

Therefore, based on the identified pattern, \(6 × 7 = 85\).

Conclusion

The sequence follows the rule where the result is the sum of the squares of the two numbers. Applying this rule to \(6 × 7\) gives 85.