# 11.5. (7 points)

At a joint conference of the party of liars and the party of truth-tellers, 32 people were elected to the presidium and seated in four rows of eight. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars always lie, and truth-tellers always tell the truth. What is the minimum number of liars in the presidium for the described situation to be possible? (Two members of the presidium are considered neighbors if one of them sits to the left, right, in front, or behind the other).
Answer: with eight liars.

Solution: Divide all the seats in the presidium into eight groups as shown in the figure. If there are fewer than eight liars, then in one of these groups, only truth-tellers will be sitting, which is impossible. The contradiction obtained shows that there are no fewer than eight liars. The figure shows how eight liars can be seated in the presidium so that the condition of the problem is satisfied.
![](https://cdn.mathpix.com/cropped/2024_05_06_1fc28fbfec7452221237g-3.jpg?height=346&width=1438&top_left_y=1443&top_left_x=384)

Comments. Without an example of seating the liars - 5 points.