[ Weighing $\quad]$ [ Examples and Counterexamples. Constructions ]

What weights can three weights have so that with their help, it is possible to weigh any whole number of kilograms from 1 to 10 on balance scales (weights can be placed on both pans)? Provide an example.
Let's take, for example, weights of 3, 4, and 9 kg: $1=4-3, 2=9-3-4, 5=9-4, 6=9-3, 7=3+4, 8=3+9-4, 10=4+9-3$.

## Answer

For example, weights of 3, 4, and 9 kg.

![](https://cdn.mathpix.com/cropped/2024_05_06_7ebe57248fabc5f818b0g-33.jpg?height=109&width=1101&top_left_y=0&top_left_x=17)

In a corridor 100 m long, 20 rugs with a total length of 1 km have been laid. The width of each rug is equal to the width of the corridor.

What is the maximum possible total length of the uncovered sections of the corridor?

## Hint

What is the minimum length of the longest rug?

## Solution

Estimate. Out of 20 rugs with a total length of $1000=20 \cdot 50$ m, the length of at least one is not less than 50 m. Thus, even one rug covers at least 50 m.

Example. Take 20 rugs, each 50 m long, and place them exactly on top of each other.

## Answer

$50 \text{m}$.