1. In the marked points (see figure), there are 6 burrows. In four of the burrows live hobbits: Frodo, Sam, Merry, and Pippin. The other two burrows are empty, and both are closer to Sam's burrow than Frodo's. Frodo's burrow is closer to the river than Merry's, but farther from the tree line than Pippin's. Who lives where? Justify your answer.
Solution. Let $A, B, C, D, E, F$ be the holes marked in the order "from left to right and from top to bottom". Frodo's hole can be $D, E, F$ from the condition that it is closer to the river than Merry's hole, and $B, C, E$ from the condition that it is further from the forest than Pippin's hole. Therefore, Frodo lives in hole $E$. Sam can then live in $-A, B$ or $D, C$ or $F$. But $B$ or $D, C$ or $F$ is ruled out because it would violate the conditions with the empty holes. Therefore, Sam lives in $A$, Pippin in $F$, and Merry in $C$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b8b9e2e35c54c762f4edg-1.jpg?height=304&width=293&top_left_y=409&top_left_x=1710)

Criteria. Full solution - 3 points.

1 point - Frodo's hole is correctly identified.

2 points - Correct answer.