To obtain a combined resistance of 4 Ω by placing a wire in parallel with a 12 Ω resistance, we need to find the resistance of the wire that needs to be added.

When resistances are connected in parallel, the formula for calculating the combined resistance is:

1/Rcombined = 1/R1 + 1/R2 + ...

Let`s substitute the given values into the formula and solve for the resistance of the wire:

1/Rcombined = 1/12 + 1/Rwire

We need to rearrange the equation to solve for Rwire:

1/Rwire = 1/Rcombined - 1/12

Combining the fractions on the right side:

1/Rwire = (12 - Rcombined) / (12 * Rcombined)

To simplify the equation, we can multiply both sides by Rwire * 12:

12 = (12 - Rcombined) * Rwire

Expanding the equation:

12 = 12Rwire - Rcombined * Rwire

Since we know that the combined resistance is 4 Ω, we can substitute it into the equation:

12 = 12Rwire - 4Rwire

Simplifying the equation:

12 = 8Rwire

Now we can solve for Rwire