The difference between maximum and minimum resistance that can be obtained by…

When several resistors are connected together, the arrangement itself changes the total resistance. Connecting resistors in series always gives the largest possible combined resistance, since each resistor's value adds directly to the total. Connecting the same resistors in parallel always gives the smallest possible combined resistance, since parallel branches let current take multiple paths, and the reciprocal of the total resistance equals the sum of the reciprocals of each branch.

Applying this to three resistors of 1/3 ohm each:

1. Maximum resistance (series): adding the three resistor values directly gives 1/3 + 1/3 + 1/3 = 1 ohm.

2. Minimum resistance (parallel): for three equal resistors of value R in parallel, the combined resistance is R divided by 3, giving (1/3) ÷ 3 = 1/9 ohm.

3. Required difference: subtracting the minimum from the maximum gives 1 − 1/9 = 9/9 − 1/9 = 8/9 ohm.

Using the general result for n identical resistors of value R, the difference between the series and parallel totals is R(n − 1/n). For n = 3 and R = 1/3 ohm, this gives (1/3) × (3 − 1/3) = (1/3) × (8/3) = 8/9 ohm, confirming the same value. Since a series combination always exceeds a parallel combination of the same resistors, this difference of 8/9 ohm is necessarily positive.