## 26. Gold Coins

The king is waiting for each of his 30 vassals, as in previous years, to present him with 30 gold coins. But the king knows that one of them has the unfortunate habit of presenting coins not weighing 10 g as required, but 9 g. How can the king, with a single weighing, identify the culprit to have his head cut off if he dares to cheat his suzerain again this year?
26. It is sufficient to weigh $(30 \cdot 31) / 2 = 465$ coins, where one coin is taken from the offering of the first vassal, two coins from the offering of the second, three coins from the third, ..., and finally, 30 coins are taken from the offering of the thirtieth vassal (i.e., all the coins of the 30th vassal are used). If all the coins weigh 10 g each, i.e., no one is guilty, then the total weight of all the coins is

$$
10(1+2+3+\ldots+30)=10 \frac{30(30+1)}{2}=4650 \text{g}
$$

If the weight of the coins turns out to be 1 g less than this amount, then the first vassal is guilty.

If the weight turns out to be 2 g less than 4650 g, then the second vassal is guilty, and so on.

If the weight of all the coins turns out to be only $4650-30=4620$ g, then the thirtieth vassal is guilty.