The plane is tiled with equilateral triangles of side length 1, forming a triangular grid like the one shown in the picture. The figure shows that it is possible to form a hexagon and an equilateral triangle whose vertices lie on points of the triangular grid.
Is it possible to form a square whose vertices also lie on points of the grid? If not, is it possible to draw a square with side length at least 1 such that each of its vertices is at most a distance away from some grid point? What if we require the distance to be at most ?
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PROMYS Europe (2026)
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