5. During the investigation of the stolen pie case, Bolvanshchik stated that the pie was stolen by the March Hare. The March Hare, in turn, testified that Sonya did not do it. Sonya stated that the pie was stolen by Bolvanshchik, and the March Hare could not have done it. It was later found out that the testimony of exactly one of them was false and at least one of them stole the pie. Can the guilty party be identified or an innocent party be excluded based on this information? (The statement "A and (a) B" is false if and only if at least one of the statements A or B is false)
# Solution

If Bolvanshchik lied, then the conditions "March Hare did not steal," "Sonya did not steal," and "Bolvanshchik stole and March Hare did not steal" must all be true. These conditions are consistent with each other. Therefore, Bolvanshchik remains a suspect.

If March Hare lied, then the conditions "March Hare stole," "Sonya stole," and "Bolvanshchik stole and March Hare did not steal" must all be true. These conditions contradict each other, so March Hare did not lie. Since his statements are true, Sonya did not steal the pie.

If Sonya lied, then according to her statements, there are three possible cases: "Bolvanshchik did not steal and March Hare stole," "Bolvanshchik did not steal and March Hare did not steal," and "Bolvanshchik stole and March Hare stole." At the same time, the conditions in the statements of the others must be met: "March Hare stole," "Sonya did not steal." From this, we conclude that March Hare and Bolvanshchik remain suspects, while Sonya is innocent.

Thus, it is impossible to determine the guilty party based on this data, but it is possible to exclude the innocent.

## Evaluation Criteria

Only the correct answer - 0 points.

Correct and complete solution to the problem - 7 points.

Only the correct and complete explanation that "Sonya is innocent" - 2 points.

Correct answer, but the full enumeration is not done - 2 points.

Correct answer and the full enumeration is done, but a logical error is made (for example, one of the cases is not mentioned when assuming Sonya lied) - 4 points.

Correct answer and the full enumeration is done, but there is no justification for a particular conclusion (for example, in one of the cases, the statements of all suspects are not compared) - 4 points.