Analyzing the Logic Puzzle with # and @ Symbols

This question asks us to find a hidden mathematical rule that connects the numbers and the symbols '#' and '@' based on the examples provided. Once we figure out the rule, we can apply it to the new set of numbers to find the result.

Examining the Given Examples

We are given two examples:

• 8 # 2 @ 5 = 21
• 4 # 6 @ 2 = 26

We need to determine what operations the symbols '#' and '@' represent. Let's assume the standard order of operations might apply, or perhaps the symbols represent operations performed in sequence from left to right.

Discovering the Pattern

Let's try different combinations of basic arithmetic operations for the symbols '#' and '@'. Common operations include addition (+), subtraction (-), multiplication ($\times$), and division ($\div$).

Let's consider the first example: 8 # 2 @ 5 = 21.

• If # is '+' and @ is '+': $8 + 2 + 5 = 15$ (Not 21)
• If # is '$\times$' and @ is '+': $(8 \times 2) + 5 = 16 + 5 = 21$ (This matches the first example)
• If # is '+' and @ is '$\times$': $8 + (2 \times 5) = 8 + 10 = 18$ (Not 21)
• If # is '$\times$' and @ is '$\times$': $8 \times 2 \times 5 = 80$ (Not 21)

The combination of multiplication for '#' and addition for '@' worked for the first example: $\left(\text{First Number } \# \text{ Second Number}\right) \ @ \ \text{Third Number} = \left(\text{First Number} \times \text{ Second Number}\right) + \text{Third Number}$.

Let's test this potential rule with the second example: 4 # 6 @ 2 = 26.

Using the rule: $\left(4 \times 6\right) + 2 = 24 + 2 = 26$.

This also matches the second example.

So, the pattern is confirmed: The operation '$#$' represents multiplication ($\times$), and the operation '$@$' represents addition (+).

Applying the Rule to 3 # 7 @ 9

Now we apply the discovered rule to the expression 3 # 7 @ 9.

According to our rule:

• Replace '#' with multiplication ($\times$).
• Replace '@' with addition (+).

The expression becomes $\left(3 \times 7\right) + 9$.

Calculating the Result

Let's perform the calculation following the standard order of operations (perform multiplication before addition):

First, calculate $3 \times 7$:

$\left(3 \times 7\right) + 9 = 21 + 9$

Next, perform the addition:

$21 + 9 = 30$

Final Answer

Based on the pattern found in the given examples, the value of 3 # 7 @ 9 is 30.

Revision Table: Logic Puzzle Operations

Additional Information: Logic Puzzle Solving Techniques

Logic puzzles often involve finding patterns or rules based on given examples. Here are some common techniques used to solve such problems:

• Arithmetic Operations: The symbols might represent basic operations like addition, subtraction, multiplication, or division.
• Sequence/Position: The operation or result might depend on the position of the numbers in the sequence.
• Digit Manipulation: Sometimes the rule involves operating on the individual digits of the numbers (e.g., sum of digits, product of digits).
• Comparison/Ordering: The symbols could represent comparison operators or ordering relationships.
• Combined Rules: More complex puzzles might combine multiple operations or rules.

A systematic approach, trying simple possibilities first and then moving to more complex ones, is often helpful in cracking these types of logic puzzles.