MathDB

7

Part of 2023 Tuymaada Olympiad

Problems(2)

People exchange places

Source: Tuymaada 2023 Junior P7

7/7/2023
3n3n people forming nn families of a mother, a father and a child, stand in a circle. Every two neighbours can exchange places except the case when a parent exchanges places with his/her child (this is forbidden). For what nn is it possible to obtain every arrangement of those people by such exchanges? The arrangements differing by a circular shift are considered distinct.
combinatoricsTuymaada
Sliding Hexagonal Pieces

Source: 2023 Tuymaada Senior P7

7/12/2023
Hexagonal pieces numbered by positive integers are placed on the cells of a hexagonal board with side nn. Two adjacent cells are left empty, and thanks to it some pieces can be moved. Two pieces with common sides exchanged places (see an example in the attachment 2).
Prove that if n≥3n \ge 3 the second arrangement cannot be obtained from the first one by moving piece
Note. Moving a piece a requires two adjacent empty cells. For instance, if they are on the right of a (attachment 1, left figure), a can be moved right till it touches an angle (attachment 1, middle figure), and then it can be moved upward right or downward right (attachment 1, right figure)
combinatorics