Pirate Gold Division: Extending the Pattern

Consider the classic pirate gold division problem. There are $n$ pirates ranked by seniority (Pirate $n$ is the most senior). They must divide 100 gold coins. The rules are: - The most senior surviving pirate proposes a division of the coins. - All pirates (including the proposer) vote. If at least half vote in favor, the proposal passes. - If the proposal fails, the proposer is thrown overboard, and the next most senior pirate proposes. - Pirates are perfectly rational, greedy (prefer more gold), and above all else prefer survival. Work through the optimal proposals for $n = 6$ and $n = 7$ pirates using backward induction from the known cases ($n = 1$ through $5$). Then identify the general pattern: how does the proposer's share and the set of bribed pirates change as $n$ increases? Finally, what is the maximum number of pirates that can play this game with 100 gold coins such that the most senior pirate still survives?