1. Martians love to dance dances where they have to hold hands. In the dance "Pyramid," no more than 7 Martians can participate, each of whom has no more than three hands. What is the maximum number of hands that the dancers can have if any hand of one Martian holds exactly one hand of another Martian?
Solution. In each handshake, two hands are involved, so the total number of hands shaken will be even. The maximum variant (7 Martians with three hands) gives 21 hands. Therefore, the greatest number of hands will be achieved if 6 Martians have three hands each and one has two hands. The dancers have 20 hands.

![](https://cdn.mathpix.com/cropped/2024_05_06_9f19007539cd17954bd3g-1.jpg?height=300&width=328&top_left_y=1415&top_left_x=836)

Comment. Justification of the even number of hands in the correct answer - 2 points. For the maximum score, a diagram illustrating the described situation should be constructed.