Let T(n) be a triangular portion of the triangular grid with n points on a side. It is an unsolved problem what the maximum number of points is that can be selected from T(n) so that no three selected points are the vertices of an equilateral triangle. For small values of n the maximum appears to be 2n-2. However, for T(12), shown below, I found more than 2*12-2=22 points, no three of which are the vertices of an equilateral triangle! How many can you find?
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Examples:
Here's an example of 5 points selected from T(4), no three of which are the vertices of an equilateral triangle.
It turns out that this selection of 5 cannot be increased to 6 without three of the selected points being the vertices of an equilateral triangle. If we add the first point in the second row, we get
Notice that three of the 6 s's are the vertices of an equilateral triangle.
There is a better selection of 6 points in T(4) no three of which are the vertices of an equilateral triangle.
(In reply to
Needed Conjecture by owl)
Could we also settle the small cases? Tristan shot down my claim of 2n-2 for n<12. He got 17 for n=9; and it appears his trick yields 19 for n=10 and 21 for n=11.
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Posted by McWorter
on 2005-08-20 15:27:44 |
re: Needed Conjecture
