Let's solve this step by step.

Let the three consecutive odd integers be:

• $x$ (the first integer),
• $x + 2$ (the middle integer),
• $x + 4$ (the third integer).

According to the problem, three times the middle integer is six more than the sum of the first and third integers. This can be written as:

$3(x + 2) = (x + (x + 4)) + 6$

Step 1: Simplify both sides of the equation.

On the left side: $3(x + 2) = 3x + 6$

On the right side: $(x + (x + 4)) + 6 = 2x + 4 + 6 = 2x + 10$

Step 2: Set the two expressions equal to each other.

$3x + 6 = 2x + 10$

Step 3: Solve for $x$.

Subtract $2x$ from both sides: $x + 6 = 10$

Now, subtract 6 from both sides: $x = 4$

Step 4: Find the three consecutive odd integers.

If $x = 4$, the three consecutive odd integers would be:

• $x = 4$ (but this is not odd!),
• $x + 2 = 6$,
• $x + 4 = 8$.

This seems incorrect, because the integers are supposed to be odd. Let's revisit the assumption that $x$ represents the first odd integer.

We should instead define $x = 2n + 1$ to ensure the numbers are odd integers. Would you like to continue exploring that possibility?

Alternatively, here are five questions that relate:

1. What would happen if we redefined $x$ to ensure the integers are odd?
2. Could a different approach help confirm whether the