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 ________.
[The diagram shows five walkways forming a single closed loop shaped like a house outline (a pentagon), with a black dot at each of its five corners.]
Answer: B
The minimum number of guards is 3.
The walkways form one closed loop of five walkways joining five junctions in a ring. A guard at a junction covers exactly the two walkways meeting there, so two guards can cover at most four walkways. Five walkways therefore need at least three guards.
Three are enough. Number the junctions 1 to 5 round the loop and place guards at 1, 3 and 4: walkway 1-2 is watched from 1, walkways 2-3 and 3-4 from 3, walkway 4-5 from 4, and walkway 5-1 from 1. Any choice of three junctions that leaves no walkway with both ends unguarded works.
So 2 is too few, and 4 or 5 guards would work but are not the minimum.
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