498: Circles on a Lattice

On a square lattice, a circle can pass through 2, 3, or 4 points, as in the diagram below. The original question asks for a circle that passes through 5 points, but can you define a circle that passes through other numbers of points? and explain how the circle was created?

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On a square lattice, a circle can pass through exactly 2, 3, or 4 points. What about 5? #math #mathchat #MTBoS pic.twitter.com/CsMd6zzbLo

— Matt Enlow (@CmonMattTHINK) August 5, 2015