1. You have a bag of granulated sugar, a balance scale, a 1 kg weight, and paper bags in which you can package the sugar. You need to measure out 50 kg of sugar, using no more than 6 weighings. How can you do this?
Solution. Let $\Gamma$ denote a 1 kg weight. The right and left sides of the equalities correspond to the right and left pans of the balance.

1 step: 1 kg (sugar $)=\Gamma$;

2 step: 1 kg $+\Gamma=2$ kg;

3 step: 1 kg $+2 \kappa 2+\Gamma=4$ kg;

4 step: 1 kg $+2 \kappa 2+4 \kappa z+\Gamma=8$ kg;

5 step: 1 kg $+2 \kappa 2+4 \kappa z+8 \kappa z+\Gamma=16$ kg;

6 step: 1 kg $+2 \kappa 2+4 \kappa z+8 \kappa z+16 \kappa z+\Gamma=32$ kg.

Bags of 32 kg, 16 kg, and 2 kg sum up to the required 50 kg of sugar.

Evaluation criteria. Any sequence of steps that meets the conditions of the problem is scored 7 points.