3
Problems(2)
16 competitors in a tournament
Source: New Zealand NZMOC Camp Selection Problems 2011 Juniors 3
9/18/2021
There are competitors in a tournament, all of whom have different playing strengths and in any match between two players the stronger player always wins. Show that it is possible to find the strongest and second strongest players in matches.
combinatorics
2-player game on a equilateral grid
Source: New Zealand NZMOC Camp Selection Problems 2011 Seniors 3
9/18/2021
Chris and Michael play a game on a board which is a rhombus of side length (a positive integer) consisting of two equilateral triangles, each of which has been divided into equilateral triangles of side length . Each has a single token, initially on the leftmost and rightmost squares of the board, called the “home” squares (the illustration shows the case ).
https://cdn.artofproblemsolving.com/attachments/e/b/8135203c22ce77c03c144850099ad1c575edb8.png
A move consists of moving your token to an adjacent triangle (two triangles are adjacent only if they share a side). To win the game, you must either capture your opponent’s token (by moving to the triangle it occupies), or move on to your opponent’s home square.
Supposing that Chris moves first, which, if any, player has a winning strategy?
combinatoricsEquilateralgridgame strategygamewinning strategy