Bad Deal

The expected loss when you pay $p$ coins, $E(p)$ , can be calculated as follows:

When $p$ is smaller than the value of the stone, $v$ : expected loss of $p$ with probability $1-\frac{p(p+1)}{2}$ .

When $p$ is greater than or equal to $v$ : expected loss of $p-v$ with probability $\frac{v}{55}$ for all $1≤v≤p$ .

This means that $E(p)=p(1-\frac{p(p+1)}{2})+\sum_{i=1}^p \frac{i}{55} (p-i)$ .

This can be simplified to $E(p)=p-\sum_{i=1}^p \frac{i^{2}}{55}=p-\frac{2p^{3}+3p^{2}+p}{330}$ .

When the derivative of this function is zero, we find a maximum point. The derivative is $1-\frac{6p^{2}+6p+1}{330}$ and is zero when $p=6.922$ .

As $E(6)=4.345$ and $E(7)=4.455$ , you will have the greatest expected loss when you pay $\boxed{7}$ coins.