Question
A treasure of $100$ gold coins must be divided among five pirates, ranked by seniority from 1 (least senior) to 5 (most senior). They follow these rules to split the coins: $$ $$ 1. The most senior pirate proposes how many coins to distribute to each pirate. $$ $$ 2. All pirates vote on the proposal, including the pirate that made the proposal. If $50\%$ or more of the pirates accept the proposal, the coins are distributed as proposed. $$ $$ 3. Otherwise, if more than $50\%$ of the pirates reject the proposal, the most senior pirate is tossed overboard. The process restarts with the next most senior pirate making a proposal and the remaining pirates casting votes. $$ $$ 4. This process repeats until a proposal is accepted or only one pirate remains, in which case the last pirate gets all $100$ gold coins. $$ $$ $$ $$ Pirates prioritize survival over wealth. The pirates will act rationally to maximize their gain while ensuring they survive. How many coins does the most senior pirate (pirate 5) get if he makes an optimal proposal? If the most senior pirate is always thrown overboard, answer $0$.