Aurumban the value of worked gold items is proportional to the square of their mass. Thieves steal a gold item worth 100 petaks, and from this they make identical medals whose total value is 10 petaks. A jeweler buys the medals and from them (not necessarily of the same mass) makes necklaces in such a way that an integer number of medals are used for each necklace. The total value of the necklaces is 46 petaks. How much are the individual necklaces worth?
Solution. The mass of the stolen gold object is 10 units because its value is proportional to the square of its mass. Let the mass of each of the medallions made from it be $x$ units, then the number of medallions is $\frac{10}{x}$. Writing the total value: $10=\frac{10}{x} \cdot x^{2}$, from which $x=1$. Therefore, the bracelets also weigh whole units, their total mass is 10 units, and their value (the sum of the squares of their masses) is 46 pataks. The heaviest bracelet cannot be more than 6 units in mass, because its value cannot be more than 46 pataks.

| Heaviest  mass | Heaviest  value | Total mass of others | Total value of others | Solution |
| :---: | :---: | :---: | :---: | :---: |
| 6 | 36 | 4 | 10 | $6,1,3$ |
| 5 | 25 | 5 | 21 | no |
| 4 | 16 | 6 | 30 | no |

If the heaviest bracelet weighs only 3 or 2 units, then the remaining value cannot be represented as the sum of squares of smaller numbers, considering that the sum of the numbers is 10.

The value of the bracelets is 1, 9, and 36 pataks.