Four football players were chatting on the train, and four young students were listening to them. When three football players got off, the boys asked the fourth one which team he played for. The answer: "One of us plays for Honvéd, one for Vasas, one for Dorog, the rest you can figure out from the conversation you overheard." - The boys then opined as follows:

I. From the conversation, I gathered that if the bearded one is from Dorog, then the tall one is from Vasas, because he said that $\ldots$

II. I gathered that if the tall one plays for Vasas, then the brown-haired one is from Dorog, because . . .

III. If the brown-haired one is not a member of Honvéd, then the bearded one is from Vasas, because ...

IV. I think that if the tall one is from Honvéd, then the brown-haired one is from Vasas.

The fourth football player said: "It's enough that all four of your statements are correct. By working together, you can already get the answer, and in fact, one of your statements is not even necessary to solve the question."

Which team does the tall, the brown-haired, and the bearded player belong to, and which of the boys' statements can be omitted?
We can easily list all the possibilities for distributing the 3 players into 3 teams, as there are only 6 such possibilities. Indeed, starting with the mustachioed player, we open 3 columns to indicate the team, and we always divide these into two, because the long-haired player can belong to either of the remaining two teams, and the brown-haired player is then taken as a member of the team that has not yet appeared.

|  | 1. | 2. | 3. | 4. | 5. | 6. |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Mustachioed | Honvéd |  | Vasas |  | Dorog |  |
| Long-haired | Vasas | Dorog | Honvéd | Dorog | Honvéd | Vasas |
| Brown-haired | $\overline{\text { Dorog }}$ | Vasas | Dorog | Honvéd | Vasas | Honvéd |
| to be discarded  why | III. | III. | IV. |  | I., III. | II. |

The assumption of the first boy is only true for the 5th and 6th distributions, and his conclusion is only true for the 6th, so the 5th distribution cannot be correct. Similarly, according to the second boy, the 1st and 6th distributions are possible, but the 6th cannot be correct. The assumption of the third boy is true for all distributions except the 4th and 6th, and his conclusion is only true for the 3rd, so the 1st, 2nd, and 5th distributions must be discarded. Finally, according to the fourth, the distribution cannot be the 3rd. Therefore, only the 4th distribution is possible, and only this one fits the reality. The long-haired player is in Honvéd, the brown-haired player is in Dorog, and the mustachioed player is in Vasas.

Only the first boy's statement can be omitted in the answer, because the 5th distribution he excluded is also excluded by the third, and moreover, omitting any other would leave more distributions possible, making the question undecidable.

Kornél Medgyesi (Pannonhalma, Benedek-rendi g. II. o. t.)