Question 31
Consider $2018$ points placed on the vertices of regular hexagons as shown in the picture:
A bee and a beetle play the following game: initially, the bee chooses one of the $2018$ points and paints it yellow; then, the beetle chooses one of the $2017$ points that have not been painted and paints it green. They continue playing in this way: the bee chooses one unpainted point and paints it yellow and, then, the beetle chooses one unpainted point and paints it green. When all points have been painted, if there is an equilateral triangle with its three vertices painted the same color, the bee wins. Otherwise, the beetle wins. Determine which of them has a winning strategy.
A bee and a beetle play the following game: initially, the bee chooses one of the $2018$ points and paints it yellow; then, the beetle chooses one of the $2017$ points that have not been painted and paints it green. They continue playing in this way: the bee chooses one unpainted point and paints it yellow and, then, the beetle chooses one unpainted point and paints it green. When all points have been painted, if there is an equilateral triangle with its three vertices painted the same color, the bee wins. Otherwise, the beetle wins. Determine which of them has a winning strategy.