Uncooperative Crickets

Trap-here

Trapper is visiting right now. So far, so good. He is currently still sleeping, since he only got 1.5 hours of sleep the night before he arrived. Once he wakes up, Skye, Trapper, and I will see the sights.

I have a bunch of fun math problems that I am working on. Here they are:

• This is one I gave to my geometry students for homework. Go to Taxicab Treasure Hunt and play the game. First, figure out a strategy that will minimize the number of moves you need to make. That is, you need to give me a list of instructions that I can follow so that I will always be guaranteed to find the treasure within, say, 4 moves. My students need to prove why their strategy guarantees that they find the treasure in (say) 4 moves or fewer, and then they need to prove that it cannot be done in fewer. "4" is not necessarily the correct answer, it is just an example.
  
  Once they have done that, they need to come up with a strategy to minimize the number of blocks that they travel on the way to the treasure. Same thing as above, but just minimize the number of blocks travelled rather than the number of moves. This is harder.
  
  I have proofs for both of these. I've known how to do the top one for a while, and I had a wishy-washy proof of the second one for a while, too. Last night, I figured out a solid proof for the second.
  
  
• Euclid proved that there are an infinite number of prime numbers. I want to prove that there are an infinite number of prime numbers p of the form p = 6n+1, where n is an integer.
  
  This is less accessible, and I don't have a proof yet. However, I think that it is similar to Euclid's proof, and uses that fact that ALL prime numbers p besides 2 and 3 either have the form 6n+1 or 6n-1 for some integer n.
  
  
• This is less accessible, too. Suppose that W is a finite dimensional vector space, and let V be an arbitrary vector space. Let L(W,V) denote the set of linear transformations that map W to V, and suppose that T is in L(W,V). Prove that T is injective if and only if there exists an S in L(V,W) such that ST is the identity on V.
  
  
• Skye gave me this one based on a problem I gave her. The problem I gave her is this: suppose that you have, say, n cookies in front of you. You are sitting across the table from an opponent. You and your opponent are going to take turns taking cookies. Each turn, you (or your opponent) must take exactly one or two cookies. The goal is to NOT take the last cookie.
  
  The question I gave Skye (and some of my former students) is this: is there a strategy that guarantees that you win (i.e. do NOT take the last cookie). Skye turned it around on me, and asked the total number of different ways the game could go. For instance, if there are three cookies, the first personcould take 1 cookie, the second takes 1 cookie on the second turn, and the first person would be forced to take the 1 remaining cookie on the second turn. Therefore, the first person loses, since s/he took the last cookie. This game could be described by (1,1,1), since each person took on cookie on the first turn. The first person could have taken 2 cookies on the first turn, leading to a (2,1) game. This would be smart, since it guarantees victory. Or the game could have gone (1,2). So the games could go (1,1,1), (2,1), or (1,2) for a total of three different games.
  
  If there are four cookies, then the games could go (1,1,1,1), (2,1,1), (1,2,1), (1,2,2), or (2,2). This gives a total of 5 different games (although some of the games involve people making really stupid moves).
  
  Skye noticed that two cookies leads to 3 possible games, three cookies leads to 5 possible games, four cookies leads to 8 possible games, five cookies leads to 13 possible games, and six cookies leads to 21 different games. The sequence 3,5,8,13,21 is made up of Finonacci numbers.
  
  The problem is to prove that for any number of cookies, the number of possible games will be the next Fibonacci number.
  

The first problem requires no formal math training. The last three probably do.

"Sometimes I think that I'm bigger than the sound."

posted by Dirk Awesome @ 12:14 PM