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/