If the potential difference between $$B$$ and $$D$$ is zero, the value of $$x$$ is $$\frac{1}{n}$$ $$\Omega$$. The value of $$n$$ is ______.

since potential difference between B and D is zero, the bridge is balanced

so we apply:

(left upper)/(left lower) = (right upper)/(right lower)

step 1: simplify each arm

left upper:
6 Ω ∥ 3 Ω = (6×3)/(6+3) = 18/9 = 2 Ω

left lower:
1 Ω + 2 Ω = 3 Ω

right upper:
x Ω ∥ 1 Ω = x/(x+1)

right lower:
x Ω

step 2: apply balance

2 / 3 = [x/(x+1)] / x

step 3: simplify RHS

[x/(x+1)] / x = 1/(x+1)

so:

2/3 = 1/(x+1)

step 4: solve

x + 1 = 3/2
x = 1/2