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For n = 1, starting with a dot, the game will end after 2 moves. Starting with a cross, it will end after 2 moves if the first player adds a dot, after 3 moves if they add a cross: hence the first player has a winning strategy for both the normal and the misère version. For n > 1, the analysis is not completed.
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When counting dots near the boundary of the shape as 1/2, there are 69 interior dots and 20 boundary dots for an estimated area of 79, close to the actual area of 25 π ≈ 78.54. A dot planimeter is a device used in planimetrics for estimating the area of a shape, consisting of a transparent sheet containing a square grid of dots. To estimate ...
The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied ...
The problem of counting integer points in convex polyhedra arises in several areas of mathematics and computer science. [25] In application areas, the dot planimeter is a transparency-based device for estimating the area of a shape by counting the grid points that it contains. [26]