1. way

| 1. chief | $\ldots 10$ minutes |
| :--- | :--- |
| 2. chief | $\ldots 30$ minutes |
| 3. chief | ... 60 minutes |

They will pass the pipe every $x$ minutes, so the total duration of the council is $3 x$ minutes. Then we have:

$$
\begin{gathered}
\frac{x}{10}+\frac{x}{30}+\frac{x}{60}=1, \\
x \cdot\left(\frac{1}{10}+\frac{1}{30}+\frac{1}{60}\right)=1, \\
x \cdot \frac{6+2+1}{60}=1, \\
x=\frac{20}{3} .
\end{gathered}
$$

The total duration of the council is $3 x=3 \cdot \frac{20}{3}=20$ minutes.

2. way. Suppose the chiefs want to deliberate for 60 minutes. Then the first will smoke 6 pipes during this time, the second 2, and the third 1. In total, they need 9 pipes. Since they actually have only 3, their council must last only 20 minutes.

(Complete solution 6 points)
U ohně seděli náčelníci tří indiánských kmenů se třemi stejnými dýmkami. Měli válečnou poradu a kouřili. První z nich vykouří celou dýmku za deset minut, druhý za půl hodiny a třetí za hodinu. Jak si mají náčelníci mezi sebou měnit dýmky, aby se mohli radit co nejdéle.

(Bednářová)

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By the fire sat the chiefs of three Indian tribes with three identical pipes. They were having a war council and smoking. The first of them can smoke a whole pipe in ten minutes, the second in half an hour, and the third in an hour. How should the chiefs exchange pipes among themselves so that they can deliberate for as long as possible.

(Bednářová)

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