Much of math can be made to be as challenging as the student wants.
The leech tree is one example. The student can simply be asked to assign the correct weights to the appropriate edges. Or one could be asked to write and prove a theorem about leech trees.
The same is true of another famous math problem.
It is so with the Cantor Set. How is the Cantor Set formed ? Well, one begins with all of the numbers from 0 to 1 (inclusive). Then one throws out the middle third. Then one continues this same process of throwing out the middle third out of what remains. And one does this forever.
So, one would begin with the numbers from 0 to 1. We divide them into 3 sections. Thus, we have 0 to 1/3 , 1/3 to 2/3, and 2/3 to 1. Then, one throws out the middle third: 1/3 to 2/3. So, those numbers are gone. Now, we throw out the middle third of the sections that remain: 0 to 1/3 and 2/3 to 1. So, we divide them into thirds and throw out the middle third. For example, 0 to 1/3 becomes 0 to 3/9 and we would divide it in the following way: 0 to 1/9 and 1/9 to 2/9 and 2/9 to 3/9. Then, we throw out 1/9 to 2/9. And we do the same thing with the numbers from 2/3 to 1. So, that becomes 6/9 to 9/9. And those three sections are 6/9 to 7/9 and 7/9 to 8/9 and 8/9 to 9/9. We throw out the middle third, all the numbers from 7/9 to 8/9.
And we keep following this pattern.
So, what are some fractions that will never be thrown away no matter how many times we throw away the middle third ?
How many of these numbers can you name ?
How many numbers are left in this set after you keep throwing away the middle third infinitely many times ?
How would you prove that ?
Fri Dec 07, 2012 at 6:22 AM PT: Some great comments. Yes, we do not throw out the endpoints. The end points are all that remain. Now, we can create a mapping from the natural numbers (an uncountable set) to our endpoints or vice versa. After we show this, then we may conclude that the Cantor Set is uncountable . It does have measure zero.
I love the leech tree by the way.
Examples of fractions that remain are one third, 2/3, 1/9 ....