Problem 11 · Find a grouped parity invariant · 3 points
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Nine lamps, numbered 1 through 9 clockwise around a circle, start off. A move flips any three consecutive lamps; which set could be exactly the lamps left on?
- {1, 2, 3, 4}
- {3, 4, 6, 7}
- {2, 4, 6, 7}
- {1, 2, 6, 8}
- {1, 3, 6, 9}
Show a hint
- A move touches three positions. Can you group the lamps so that every move treats the groups alike?
- Try the groups {1, 4, 7}, {2, 5, 8}, and {3, 6, 9}. What happens to the parity of the number of lit lamps in each?
Show the answer and solution
Answer: B — {3, 4, 6, 7}
Every move flips one lamp in each of the three groups, so their numbers of lit lamps must always have the same parity. For {3, 4, 6, 7}, those numbers are 2, 0, and 2; it is reachable by flipping {3, 4, 5} and then {5, 6, 7}. Every other choice has unequal group parities. In particular, {1, 2, 3, 4} has an even total of four lit lamps, but its group counts are 2, 1, and 1.