On a secret island, there live two kinds of natives: one kind always speaks the truth (honest ones), and the other always lies (liars). Researchers there encountered several groups of natives and asked each member of the groups how many honest ones were in their group.

- From one four-member group, they received the same number from everyone.
- From another group, they received the numbers $0,1,3,3,3,4$.

How many honest ones could there be in one group and how many in the other? Determine all possibilities.

(M. Volfová)
If the first group consisted only of honest people, the researchers would receive the numbers 4, 4, 4, 4 as answers. If the group had three, two, or one honest person, the researchers could not receive four identical numbers as answers. If the group consisted only of liars, the researchers could receive any set of four numbers that do not include 0 as answers. Therefore, in the first group, there could either be four honest people or none.

Since all honest people in the group must (unlike liars) answer the same, the number $k$ can represent the answer of an honest person only if it repeats $k$ times. Therefore, the number of honest people in the second group could be either one or three. (If the group consisted only of liars, then the answer 0 would be true, which is not possible for liars.)

Evaluation. 2 points for determining the possibilities for each group; 2 points for completeness and quality of the commentary.

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