Task 1. A rectangular metal plate has a mass of $10 \mathrm{~g}$. It needs to be divided (cut) into three parts, each having an integer mass. With the three resulting parts, it should be possible to measure any mass from 1 to 10 grams, which is a natural number. What should be the mass of each of the divided parts?
Solution. The weights at which the tile needs to be cut are 2 g, 3 g, 5 g $(2+3+5=10)$. The masses of $2 \mathrm{~g}, 3 \mathrm{~g}, 5 \mathrm{~g}, 7 \mathrm{~g}, 8 \mathrm{~g}, 10 \mathrm{~g}$ can be measured directly. The mass of $1 \mathrm{~g}$ will be the smallest of all, so when we place it on the scale with $2 \mathrm{~g}$, the side with 2 g will be heavier. The mass of 4 g, when placed on the scale with a mass of $3 \mathrm{~g}$, the side with $4 \mathrm{~g}$ will be heavier. When the mass of $4 \mathrm{~g}$ is placed on the scale with a mass of $5 \mathrm{~g}$, the side with $5 \mathrm{~g}$ will be heavier. The mass of $6 \mathrm{~g}$ will be lighter on the scale than the mass of $7 \mathrm{~g}$, but heavier than $5 \mathrm{~g}$. Similarly for 9 $\mathrm{g}$, since $8 \leq 9 \leq 10$.