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Dots and Triangles

This article is a sequel to Ernest R. Ranucci’s “Dots and Squares” which appeared in the January 1969 issue of JRM (pp. 57–60). The original article dealt with dots and squares played on a square array of points. This game is played the same way but with dots and equilateral triangles on a triangular array. Two players take turns connecting a pair of points until a triangle is enclosed. The person completing the triangle writes his initial in it. He then connects two more points. The game ends when all triangles are claimed. Of strategic importance is the decision of accepting or rejecting an opportunity for the completion of a triangle. The refusal of one or two triangles could lead to opportunities for completing a greater number of them. Two questions are therefore suggested:

What is the greatest number of moves that may be made before one of the players is presented with an opportunity for completion of at least one triangle?

What is the greatest number of moves that may be made before one of the players is forced to complete one triangle?