[ Planar graphs. Euler's formula ]

In the country of Lake, there are seven lakes connected by ten non-intersecting channels, such that one can sail from any lake to any other. How many islands are there in this country?

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Consider a planar graph, where the vertices correspond to lakes, and the edges - to channels. Let it divide the plane into $F$ pieces. By Euler's formula (problem $\underline{30759}$) $7-10+F=2$, so $F=5$. One of these pieces is the mainland, and the rest are islands.

## Answer

4 islands.