In an abstract connection-counting model, 58 lamps must be connected to one…

Concept

When identical connector units are chained from one source, each unit contributes its total outlets but every connection between two units consumes one outlet.

For n four-outlet boards in one connected chain, there are n − 1 board-to-board connections, so the number of outlets left for lamps is 4n − (n − 1) = 3n + 1.

Application

1. Let n be the number of extension boards. To connect 58 lamps, the available lamp outlets must satisfy 3n + 1 ≥ 58.

2. Subtract 1 from both sides: 3n ≥ 57.

3. Divide by 3: n ≥ 19. Therefore, the least whole number of boards is 19.

Cross-check

With 19 boards there are 19 × 4 = 76 outlets on the boards. The 18 board-to-board connections consume 18 of them, leaving 76 − 18 = 58 outlets for the lamps. Thus, 19 extension boards are required.