Proceed by induction on the number of pirates, starting with two.

2. The most senior will propose that he keep all the loot and since he will vote for his own proposal, the least senior pirate will get nothing.

3. If there are three pirates left, the most senior will need one vote. The least senior will vote for anything more than nothing, so the most senior will win by offering the least senior one gold coin. The second least senior will get nothing.

4. If there are four pirates, the most senior needs one vote, and he can get it by offering the second least senior one coin. The least senior and third least senior get nothing.

5. If there are five pirates, the most senior needs two votes. He can get them from the least senior and third least senior by offering them one coin each, keeping 98 coins for himself.

This pattern continues.

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