1. In the marked points (see figure) there are 4 burrows. In them live hobbits: Frodo, Sam, Merry, and Pippin. Frodo's burrow is closer to Merry's burrow than to Pippin's. And Sam'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 $A, B, C, D$ be the holes marked in the order "from left to right and from top to bottom". Sam's hole is $C$ (it is not the farthest from the river, i.e., not $A$ and not $B$, but also not the closest to the forest strip, i.e., not $D$). From the conditions, it follows that Frodo does not live in $A$ (otherwise, Merry and Pippin would be at the same distance), Merry does not live in $D$, and Pippin does not live in $B$ (this follows from the location of Sam's hole). If Frodo lives in $D$, then Pippin is in $A$, Merry is in $B$, and the first condition is not satisfied. Therefore, Frodo lives in $B$, Merry in $A$, and Pippin in $D$.

Criteria. Full solution - 7 points.

Correct answer - 2 points.

Correct answer but with several omitted statements (e.g., "Sam is in the middle hole, as he is closer to the river, but without specifying the possibility of Sam being in the hole closest to the river") - 5 points.

Correct answer and a small number of logically correct statements, but with significant omissions of necessary transitions and cases - 3 points.

Justification of where Sam lives - 2 points.

Correct reasoning about who cannot be where, after explaining where Sam is - 2 points.

Incomplete case analysis - no more than 3 points.