Understanding the Symbol Substitution Puzzle

This problem requires us to decode a set of symbols that represent standard mathematical operations. We are given a specific mapping for each symbol. Our task is to substitute these symbols in the provided options and find the equation that holds true according to the standard order of mathematical operations.

Symbol-to-Operation Mapping

The question provides the following key for symbol substitution:

Evaluating Each Option

To solve this, we will substitute the symbols with their corresponding operations and evaluate each expression using the standard order of operations (PEMDAS/BODMAS): Multiplication and Division are performed before Addition and Subtraction, working from left to right.

Option 1: $7 \leftarrow 43 \uparrow 6 \downarrow 1 \text{ } 7 \text{ } 4$

The given expression is: $7 \leftarrow 43 \uparrow 6 \downarrow 1 \text{ } 7 \text{ } 4$.

Substituting the symbols based on the provided mapping:

$7 - 43 / 6 * 1 = 4$

Now, we evaluate this equation using the order of operations.

Step-by-step evaluation:

1. Division: $43 \div 6 = \frac{43}{6}$
2. Multiplication: $\frac{43}{6} \times 1 = \frac{43}{6}$
3. Subtraction: $7 - \frac{43}{6} = \frac{42}{6} - \frac{43}{6} = -\frac{1}{6}$

The equation becomes $-\frac{1}{6} = 4$. This is false.

Option 2: $3 \downarrow 6 \uparrow 2 \rightarrow 3 \leftarrow 6 \text{ } 7 \text{ } 5$

The given expression is: $3 \downarrow 6 \uparrow 2 \rightarrow 3 \leftarrow 6 \text{ } 7 \text{ } 5$.

Substituting the symbols:

$3 * 6 / 2 + 3 - 6 = 5$

Step-by-step evaluation:

1. Multiplication: $3 \times 6 = 18$
2. Division: $18 \div 2 = 9$
3. Addition: $9 + 3 = 12$
4. Subtraction: $12 - 6 = 6$

The resulting equation is $6 = 5$, which is false.

Option 3: $5 \rightarrow 7 \leftarrow 3 \uparrow 2 \text{ } 7 \text{ } 4$

The given expression is: $5 \rightarrow 7 \leftarrow 3 \uparrow 2 \text{ } 7 \text{ } 4$.

Substituting the symbols:

$5 + 7 - 3 / 2 = 4$

Step-by-step evaluation:

1. Division: $3 \div 2 = \frac{3}{2} = 1.5$
2. Addition: $5 + 7 = 12$
3. Subtraction: $12 - 1.5 = 10.5$

The resulting equation is $10.5 = 4$, which is false.

Option 4: $2 \downarrow 5 \leftarrow 6 \rightarrow 2 \text{ } 7 \text{ } 6$

The given expression is: $2 \downarrow 5 \leftarrow 6 \rightarrow 2 \text{ } 7 \text{ } 6$.

Substituting the symbols:

$2 * 5 - 6 + 2 = 6$

Now, we evaluate this equation using the order of operations.

Step-by-step evaluation:

1. Multiplication: $2 \times 5 = 10$

The expression simplifies to:

$10 - 6 + 2 = 6$

Continuing with Addition and Subtraction from left to right:

1. Subtraction: $10 - 6 = 4$
2. Addition: $4 + 2 = 6$

The resulting equation is $6 = 6$, which is true.

Final Determination

By carefully substituting the given symbols with their corresponding mathematical operations and evaluating each option according to the order of operations, we find that Option 4 is the only expression that correctly satisfies the equality.