Discussion
Consider an art gallery whose walkways are shown as lines in the diagram. A black dot represents a junction of two walkways. A guard may be placed at a junction to watch over the walkways that join at that junction. The minimum number of guards needed to watch all the walkways is ___.
Answer: B
The minimum number of guards is 3.
The walkways form a single closed loop of 5 junctions and 5 walkways. Each guard covers at most the 2 walkways that meet at his junction, so 2 guards can watch at most 4 walkways, which is not enough for 5.
Three guards suffice: number the junctions 1 to 5 round the loop and place guards at 1, 3 and 4. Guard 1 covers walkways 5-1 and 1-2, guard 3 covers 2-3 and 3-4, and guard 4 covers 4-5.
4 and 5 guards also work, but they are not the minimum.
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