The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the first player, the second player, or neither player can force a win.
Problem
Source: Simon Marais 2017 A1
Tags: game, combinatorics
Eevee2002
29.04.2021 10:26
Player 2 can always win with the symmetry strategy
lorodane
26.07.2024 14:23
Eevee2002 wrote: Player 2 can always win with the symmetry strategy I don't think this works because the objective is to be the first creating a triangle. For player 2 to create a triangle with that strategy, player 1 must have done it first.
lorodane
26.07.2024 14:28
Player 1 wins by colouring thre edges from the same vertex red and then forming a triangle on turn 4. To be more specific, name vertices 1,2,3,4,5 and players A,B. A starts by colouring edge 12, B colours an edge which must be connected to 3,4, or 5. WLOG the edge is 5x (51, 52, 53 or 54). Now A colours 13. B is forced to block triangle 123 by colouring 23. Then A can color 14, B must block by colouring 24 or 34. Then A wins by colouring the other.