Maximum Number of Acute Angles in a 2001-gon
Date: 05/29/2002 at 10:19:25 From: Charlene Subject: Acute angles in a 2001-gon I am stumped with this question that came out recently in a mathematics competition: What is the largest possible number of acute angles a 2001-gon can have if no two sides cross each other? Thank you in advance.
Date: 05/29/2002 at 15:10:47 From: Doctor Rick Subject: Re: Acute angles in a 2001-gon Hi, Charlene. If sides were allowed to cross each other, then you could make a 2001-pointed star. Think of how we make 5-pointed stars (5 sides, each crossing two other sides). Do the same thing with 2001 points instead, and you've got a 2001-gon with 2001 acute angles! That's not allowed. But is a non-convex polygon (one with some interior angles greater than 180 degrees) allowed? Your question said nothing about a "convex 2001-gon". If it did, the answer would be a small number, in fact the same for polygons with any number of sides; you can discover the answer by considering the sum of the EXTERIOR angles of a polygon. If the 2001-gon doesn't have to be convex, then we can try the "outside" of a star. The outside of a 5-pointed star, for instance, has 10 vertices (5 acute and 5 "reflex", where an edge comes in and turns back outward). A 1000-pointed star is a 2000-gon, and it has 1000 acute angles. If you add one vertex by "breaking" one side and bending it slightly, you still have 1000 acute angles. It may be possible to do better than 1000 acute angles. I don't see an easy way to prove what is the greatest number, not yet. Why don't you keep thinking about it and tell me what you come up with? Here is a suggestion. Instead of a 2001-gon, think about an 11-gon. Maybe if we come up with a way to be sure of the answer in this case, we can apply the same reasoning to the 2001-gon. Have fun! I hope to hear back from you. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 05/30/2002 at 08:37:15 From: Doctor Rick Subject: Re: Acute angles in a 2001-gon Hi again, Charlene. I have some more thoughts on your problem. First, another Math Doctor (Dr. Douglas) came up with a 2001-gon that has more than 1000 acute angles. We still didn't have a proof that we couldn't do even better. But then I reconsidered the variant problem in which the polygon must be convex, and I realized that a second method for solving that problem could also be used to solve yours. I mentioned that, if the polygon must be complex, you could solve the problem by considering the sum of the exterior angles (which is 360 for *any* non-self-intersecting polygon). That method works because the exterior angles of a convex polygon are all positive. Therefore if the sum of any set of exterior angles exceeds 360 degrees, the sum of all the exterior angles must be even greater; but this can't be. In the case of a non-convex polygon, exterior angles get a little weird. The exterior angle is the supplement of the corresponding interior angle, that is, 180 degrees minus the interior angle. If the interior angle is reflex (greater than 180 degrees), then the exterior angle (by this definition) is negative! We don't usually talk about exterior angles in such a case. Now consider another way to solve the problem for convex polygons. We know that all the interior angles must be less than 180 degrees. (They cannot be reflex; that would make the polygon non-convex.) We also know that the acute interior angles must be less than 90 degrees, by definition. Finally, we know that the sum of the interior angles of an n-gon is 180(n-2) degrees, so the sum for a 2001-gon is 180(2001-2) = 180*1999. Now suppose there are k acute interior angles. Then there are (2001-k) non-acute angles, and the sum of the angles must be less than 90k + 180(2001-k) Thus, 90k + 180(2001-k) > 180*1999 180*2001 - 90k > 180*1999 180*2001 - 180*1999 > 90k 360/90 > k 4 > k Thus we have proved that the number of acute angles in a convex 2001-gon must be less than 4. (If you repeat this with n, the number of sides, in place of 2001, you'll see that n actually cancels out and the result is the same for any n, as I stated last time.) The same method can be used to show an upper limit on the number of acute interior angles of any 2001-gon, convex or not. The number you find this way is the same as the number of acute angles in the example that Dr. Douglas found, so we know that this number of acute angles is possible -- it is the greatest possible. Here is a hint to help you find this 2001-gon. Instead of making a star with alternating acute and reflex angles, you can make *two* acute angles (both nearly right angles) between reflex angles. That's not a complete description of the shape, but it will get you started thinking about it. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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