To solve this problem, let's denote the three sides of the triangle as $x$, $x+2$, and $x+4$, where $x$ represents the shortest side (as all three sides are consecutive odd integers).

Given that the perimeter of the triangle is 63 inches, we can set up the following equation:

$x + (x + 2) + (x + 4) = 63$

Simplify and solve for $x$:

$3x + 6 = 63$

Subtract 6 from both sides:

$3x = 57$

Divide by 3:

$x = 19$

Therefore, the value of the shortest side is 19 inches.

Would you like details or have any questions? Here are 5 related questions:

1. What would the side lengths be if the perimeter was 75 inches?
2. How do you determine if a set of side lengths can form a valid triangle?
3. What is the formula for finding the perimeter of a triangle?
4. What are the properties of odd integers?
5. How would the problem change if the sides were consecutive even integers?

Tip: Remember, the perimeter of a triangle is simply the sum of all its sides, which is useful for solving problems involving unknown side lengths.