Consider the player ? > ?. At most, ? players fall to their death. This happens only if, as above, all of the first ? players choose to hop onto a weak glass panel in their first hop onto an unvisited column. This outcome would reveal the path for safe traversal over the bridge for the remaining ? − ? players behind them. Therefore, player ? is guaranteed to survive and ?(??) = ? for ? > ?.

Sophie Maclean is a recent maths graduate from the University of Cambridge and very much misses her degree. She has no free time—she is a Chalkdust editor.
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