János, Péter, Tamás, Gyuri, István are brothers. Once, one of them broke a window. When their father asked who was the culprit, they said the following:

János: It was Péter or Tamás.

Péter: Neither Gyuri nor I did it.

Tamás: Both of you are lying.

István: No, one of you is telling the truth, but the other is not.

Gyuri: No, István, you are wrong.

Their mother added:

Three of my sons are indeed telling the truth, but I don't trust what the other two said.

Who broke the window?
I. solution. At least one of István and Gyuri is wrong, because their statements contradict each other. If Tamás were telling the truth, then at least three would not be telling the truth. This contradicts the mother's statement. Thus, Tamás is lying, and one of István and Gyuri is also lying, while János and Péter are telling the truth. Based on their statements, the culprit is Tamás. Therefore, Tamás broke the window, and Tamás and István are lying.

II. solution. The real culprit can be easily identified if we assume in turn that János, Péter, Tamás, István, or Gyuri was the culprit, and based on the assumption, we write down who is telling the truth and who is lying.

| The value of the given answers | The assumed culprit |  |  |  |  |
| :--- | :---: | :---: | :---: | :---: | :---: |
|  | János | Péter | Tamás | István | Gyuri |
| János | L | T | T | L | L |
| Péter | T | L | T | T | L |
| Tamás | L | L | L | L | T |
| István | T | T | L | T | L |
| Gyuri | L | L | T | L | T |

From the table, it can be read that the mother's statement is only true if the culprit is Tamás, since in all other cases, three would not be telling the truth.

Therefore, the culprit was Tamás.

Remarks. 1. The first solution solves the problem through "reasoning," while the second uses a "table." Due to the simple structure of the answers, the first type of solution is significantly faster and more easily adaptable to other examples than the second.

2. We draw attention to some inaccuracies found in the solutions. In the first solution, the fact that János and Péter are telling the truth does not follow from Tamás lying, since this only tells us that at least one is telling the truth.
3. Many prepared a solution similar to the second, based on "testing cases," for example, assuming that two boys are lying. Thus, this solution is incomplete but can be made complete by examining the cases where one or none of the boys are lying. (P. T.)