Maximum the amount of coins the king can get Here is how it works: 
The king has $65$ citizens. Counting himself, each of them gets a coin every month. The king can give a suggestion of how the coins should be distributed, and evryone else cast a vote. It will pass if it has more than $50$% voted "yes". Every citizen is extremely selfish and does the following: if he/she gets more coin, then he/she will vote "yes". If he/she gets less coin, then he/she will vote "no". Otherwise, he/she won't vote at all. 
Question: What's the maximum amount of coin (out of 66) the king can get, and how is it achieved in minimal amount of suggestions?
Apprently the answer seem to be 63, and it's pretty obvious, but how can one formally prove it? Also, how can I find the answer to the second question?
Edit: here is an example that might help you to understand the question:
Suggestion $1$: $33$ citizens gets 2 coins each ($33$ "yes" $32$ "no")
Suggestion $2$: $17$ citizens gets 3/4 coins each ($17$ "yes" $16$ "no")
Suggestion $3$: $9$ citizens gets 6/7 coins each ($9$ "yes" $8$ "no")
Suggestion $4$: $5$ citizens gets 12/13 coins each ($5$ "yes" $4$ "no")
Suggestion $5$: $3$ citizens gets 22 coins each ($3$ "yes" $2$ "no")
Suggestion $6$: $2$ citizens gets 33 coins each ($2$ "yes" $1$ "no")
Suggestion $7$: $3$ citizens didn't get any coin the first round gets one coin each. The king gets the rest of 63 coins. ($3$ "yes" $2$ "no")
 A: Observe that, in the first step, the king is obliged to throw in his own coin in order to make any proposal pass (otherwise, a majority among the 65 citizens can never be reached). Therefore, all 66 coins are divided among the 65 citizens, and at least one citizen will be duped when coins are transferred to the king (meaning that at least two citizens will have to benefit from the voting round, and thus at least get one coin each). Also observe that at any point in time, at least two citizens will have coins (otherwise, there can be no majority among the citizens). Since no citizen will ever possess all coins, a scenario in which two coins are given to two citizens (one each) to sway the vote, is impossible. Therefore, the king can never obtain 64 coins or more. Finally, observe that if no citizen has one coin (a state which can be achieved after one voting round), money can continuously be transferred from one citizen to two others, until only two citizens with coins remain: one with 65 coins, one with one coin. Giving 63 coins to the king, two to the citizen with one coin and one to a citizen with no coins then results in the optimal solution.
As for the shortest solution: at least seven voting rounds are needed. This can be deducted as follows: if there are $n$ citizens with at least one coin, we need to benefit at least $\lfloor \frac{n}{2}\rfloor + 1$ citizens in the next round. Given that there are $65$ citizens, we have:
$$65 \to 33 \to 17 \to 9 \to 5 \to 3 \to 2$$
As soon as we have two citizens holding all coins (each one having at least two coins), the optimal solution is achieved by giving 63 coins to the king, and one coin to three other citizens. Your suggested solution is thus the shortest one.
