Problem 12 · Invariants · 4 points
Made by AI and not reviewed by a real person.
Five lamps stand in a row, all off. Bea makes exactly three moves; each move switches two neighboring lamps, turning each on lamp off and each off lamp on. Which row could she make, from left to right?
- on-off-on-off-off
- off-on-off-on-off
- on-on-on-off-off
- on-on-off-on-on
- on-off-on-on-on
Show a hint
- Group positions 1, 3, 5 together and positions 2, 4 together.
- Each move switches one lamp in each group.
Show the answer and solution
Answer: E — on-off-on-on-on
Each move switches one lamp in each group. Since all lamps start off, after three moves each group must have an odd number of lamps on. Only on-off-on-on-on fits: positions 1, 3, 5 are on, and position 4 is on. Bea can make it by switching pairs (1, 2), then (2, 3), then (4, 5). The tempting on-off-on-off-off has two lamps on, but both are in the first group, so it cannot result from three moves.