Dots-and-Polygons

TL;DR

This paper explores two new variations of the Dots-and-Boxes game, applying geometric and combinatorial strategies, including Pick's theorem, to analyze gameplay and potential winning tactics.

Contribution

It introduces Dots-and-Triangles and Dots-and-Polygons, extending game analysis methods to new lattice-based variations with geometric calculations.

Findings

Strategies similar to Dots-and-Boxes are effective in the new variations

Pick's theorem is useful for area calculation in these games

The paper provides insights into optimal play tactics

Abstract

Dots-and-Boxes is a popular children's game whose winning strategies have been studied by Berlekamp, Conway, Guy, and others. In this article we consider two variations, Dots-and-Triangles and Dots-and-Polygons, both of which utilize the same lattice game board structure as Dots-and-Boxes. The nature of these variations along with this lattice structure lends itself to applying Pick's theorem to calculate claimed area. Several strategies similar to those studied in Dots-and-Boxes are used to analyze these new variations.