Assignment 1

Just looking at assignment 1. It looks like the first question can be answered without induction using a standard Euclidean argument:

Draw lines from the points on the circle to the center.
That makes n triangles.
The total angles in each triangle is pi.
That gives n*pi angles
But the angles at the center add up to 2*pi since it's a circle
So total angles is (n-2)*pi

I guess I can shoehorn this into an inductive proof somehow.

Question 2 looks like it might best be handled recursively. Make a function that makes up combinations from some set. If the set is empty, return nothing. If set has one element, return that. If set has more than one element, build up subsets consisting of first element and the results of the function called recursively for all other elements. When you're all done, add the empty set as the final element.

Question 3 is harder than I thought at first. It's easy to show that, if n1/n2 = theta as defined in the problem, n1/n2 = n2/(n1-n2). Just some basic algebra. That leads us to n1^2 - n2^2 = n1n2. Something about that seems fishy, so I whipped up a little program to see if it is true. For the first 100,000 numbers, there is no pair that will satisfy the equation. So I think it's false. I just can't prove it. At. All.

Question 4? Looks like if n=0, NE trees = 1, n=1 NET = 1, n=2, NET = 3, n=3, NET=12, n=4,NET=72; n=5, NET=252. Can't see a pattern in that, recursive or otherwise.