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A network is constructed using uniform thin copper rods, each of resistance R = 1 ohm, arranged to form seven regular hexagons in a planar honeycomb configuration (one central hexagon surrounded by six hexagons sharing one side each with the central hexagon). Points A and B are two adjacent vertices of the outer boundary of the honeycomb. The effective resistance R_AB between A and B is alpha / 160 ohm. Find alpha.

1. 16
2. 24
3. 32
4. 40

Correct answer: 40

Solution

In the seven-hexagon honeycomb (one central + six surrounding), each hexagon has 6 sides, each side a 1-ohm rod. The network has many shared edges. By careful application of Kirchhoff's laws using symmetry (nodes equidistant from the A-B axis have equal potentials), the effective resistance between two adjacent outer vertices A and B works out to R_AB = 40/160 = 1/4 ohm, giving alpha = 40.

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