The Clever King

The king can gain $\displaystyle\$63$ for the salary, in this manner:

• For $i=1,2,\ldots,63$ , on the $i$ -th day, he removes the salary of the $i$ -th citizen and increases the salaries of the $i+1$ -st and $i+2$ -nd citizens. Clearly this always passes; there are two "yes" and one "no". Note that after all $63$ days, we now have $63$ unpaid citizens ( $1,2,\ldots,63$ ) plus $2$ paid citizens ( $64,65$ ).
• On the $64$ th day, he removes the salaries of those two remaining people, and give $\$1$ salary to three other citizens. This also passes; there are three "yes" and two "no". This means the king spends $\$3$ in salary and keeps his own into $\$66 - \$3 = \boxed{\$63}$ .

Now we will show that the king must spend at least $\$3$ .

Note that on each day where the king gains some salary, there must be at least a single "no" (because the sum of the salaries of the citizens decrease), and so the king needs at least two "yes". Thus the citizens must have at least $\$2$ among them, the two dollars given to those who voted "yes". In addition, any day will have at least two people with salaries. Suppose otherwise, then let on some day, the number of people with salaries decrease from two or more into one. Thus the king only receives at most one "yes" vote from this person, but at least one "no" vote from everyone else, not able to pass it.

Now we will show spending only $\$2$ is not enough.

If the king only spends $\$2$ , then as above, these two dollars must be on different people; call them on citizens $1$ and $2$ . Before this day, one of these applies:

• The king spends less than $\$2$ , which is impossible due to the above.
• The king spends exactly $\$2$ . By above, these must be on different people, and thus we have the same case, only renaming the citizens.
• The king spends more than $\$2$ .

In the last case, by the above, we need two "yes". These can only come from those who has the salary, and thus before the last day, they are both empty-handed. There's at most one "no", so before this day, at most one person has a salary. But this contradicts our result.

Thus the king needs to spend at least $\$3$ .

Motivation: No matter how much we give or take, we'll only get either one "yes" or one "no" from each citizen. We can as well add as little as possible when we need a "yes", but take as much as possible when we need a "no". The first way that I thought of is below:

• On the first day, give $33$ of the paid citizens an increase to $\$2$ and remove the salaries of the other $32$ of those currently paid.
• On the second day, give $17$ of the paid citizens an increase to $\$3$ and remove the salaries of the other $16$ of those currently paid.
• On the third day, give $9$ of the paid citizens an increase to $\$4$ and remove the salaries of the other $8$ of those currently paid.
• On the fourth day, give $5$ of the paid citizens an increase to $\$5$ and remove the salaries of the other $4$ of those currently paid.
• On the fifth day, give $3$ of the paid citizens an increase to $\$6$ and remove the salaries of the other $2$ of those currently paid.
• On the sixth day, give $2$ of the paid citizens an increase to $\$7$ and remove the salary of the other $1$ of those currently paid.
• On the seventh day, give $3$ unpaid citizens a salary of $\$1$ and remove the salaries of the $2$ currently paid.

in the first suggestion he gives $2 to 33 people and 0 to rest including himself. next he gives $4 to 15 people and $3 to 2(notice money is getting concentrated with less and less people).
next, $7 to 9 and $3 to 1.
so on till two people have $33 each.
now he will give $1 to 3 bankrupt subjects and keep the rest for himself.
$66-$3= $66.