Problem 830

It is advisable to introduce designations for the ages of the participants:

Let $k, s, m, m_{1}, m_{2}$, and $m_{j}$ be the ages of the captain, the helmsman, the engineer, as well as the 1st, 2nd, and the youngest sailor. The statements then yield the following system of equations:

| Helmsman: | $s=2 m_{j}=m+6$, |
| :--- | :--- |
| 1st Sailor: | $m_{1}=m_{2}+4$, |
| 1st Sailor: | $m_{1}-m_{j}=m-m_{1}$, |
| 2nd Sailor: | $m_{2}=20$, |
| Average: | $k+s+m+m_{1}+m_{2}+m_{j}=6 \cdot 28=168$. |

From (4) and (2) it follows that $m_{1}=24$, and from this and (3) it follows that $m_{j}=2 m_{1}-m=48-m$. Substituting the latter into (1) yields $m=30$ and $s=36$. Substituting these results into (5) leads to

$$k+36+30+24+20+18=168 \quad \Longrightarrow \quad k=40$$

Thus, the captain is 40 years old.