studiot Posted May 6, 2023 Share Posted May 6, 2023 (edited) Thank you for clarifying that it must not be placeable. The nearest I have been able to come is to start with a 1 x 1 square, which obviously has area of 1 and is aligned to the grid blue points at its four corners. If we now reduce one side only by an infinitesimal amount δ then the area now becomes 1(1-δ), which is strictly less than 1, by an infinitesimal amount. This slants one side only away from its corner. Lifting the square off the base two corners and shifting it sideways by infinitesimal amounts leaves the square clear of all four blue grid points. Edited May 6, 2023 by studiot Link to comment Share on other sites More sharing options...
Genady Posted May 6, 2023 Author Share Posted May 6, 2023 9 minutes ago, studiot said: Thank you for clarifying that it must not be placeable. The nearest I have been able to come is to start with a 1 x 1 square, which obviously has area of 1 and is aligned to the grid blue points at its four corners. If we now reduce one side only by an infinitesimal amount δ then the area now becomes 1(1-δ), which is strictly less than 1, by an infinitesimal amount. This slants one side only away from its corner. Lifting the square off the base two corners and shifting it sideways by infinitesimal amounts leaves the square clear of all four blue grid points. I understand your description and there are no problems with it, but I am not clear about what it proves. I have tried a few posts above to reformulate the problem in a - hopefully - clearer way. Here I write it again: You have a plane with the grid points on it as described in the OP. Somebody gives you a transparency with an inkblot on it. You position the transparency on top of your plane in such a way that the inkblot does not cover or touch any of the grid points. Prove that such a positioning exists for inkblot of any shape as long as its area is less than unity. Link to comment Share on other sites More sharing options...
studiot Posted May 7, 2023 Share Posted May 7, 2023 Pick's Theorem Pick's theorem - Wikipedia Link to comment Share on other sites More sharing options...
Commander Posted May 8, 2023 Share Posted May 8, 2023 A shape like this Link to comment Share on other sites More sharing options...
Genady Posted May 8, 2023 Author Share Posted May 8, 2023 4 hours ago, Commander said: A shape like this If you shift the shape up, or the grid down as below, the shape does not intersect with the grid dots: Link to comment Share on other sites More sharing options...
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