6. Jack Sparrow needed to distribute 150 piastres into 10 purses. After placing a certain number of piastres in the first purse, he put more in each subsequent purse than in the previous one. As a result, it turned out that the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many piastres are in the 6th purse?
# Answer: 16

Solution: Let there be $x$ piastres in the first purse. Then in the second there are no less than $\mathrm{x}+1$, in the third - no less than $\mathrm{x}+2$... in the 10th - no less than $\mathrm{x}+9$. Thus, on one side $x+$ $x+1+\cdots+x+9=10 x+45 \leq 150$, From which $x \leq 10$. On the other side $x \geq(x+$ 9 )/2, from which $x \geq 9$. Therefore, in the first there are 9 or 10 piastres. But 9 cannot be, because then in the last one there will be no more than 18 and the sum will not be 150.

So in the 1st there are 10, in the second - no less than 11, in the 3rd - no less than 12, in the 4th - no less than 13, in the 5th no less than 14 and in the $6-$th - no less than 15. But with 15 the sum is less than 150. So it must be 16. And it cannot be more because then in the last purse there will be 21.

For graders: Partial credit can be given for a correct example of distribution (trial and error) without proof of uniqueness.