Not any less or any more

Geometry Level 3

It is easy to draw a circle which goes through two points of a regular grid, or 4, or even 8. But is there a circle which goes through exactly 3 of the grid points?

Yes No

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2 solutions

Marta Reece
Jul 1, 2017

There are many ways a circle can go through exactly three points of a grid.

Is it true, that for every n positive integer there exist a circle on the grid, which goes through exactly n grid points?

Áron Bán-Szabó - 3 years, 11 months ago

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That is the interesting question, isn't it?

Marta Reece - 3 years, 11 months ago

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Yes it is. I have a divination that the statement is true.

Áron Bán-Szabó - 3 years, 11 months ago

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@Áron Bán-Szabó If you have actual solution for some nice number, say 5, you could post that as a problem.

Marta Reece - 3 years, 11 months ago

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@Marta Reece Ok, I will.

Áron Bán-Szabó - 3 years, 11 months ago
Jon Haussmann
Jul 11, 2017

This is an instance of Schinzel's Theorem .

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