Continuation of Competitive Rankings

I wrote a program that implements the algorithm in the last blog entry... It took about four hours but I put all the info for the 2010 Division I football season (All but the final bowls) into the program and then it took about a second for the computer to process into this ranking of the top 25 (I did all the teams but I'm only displaying the top 25), I've put it next to three other rankings to compare...

So these three on the right were how the rankings looked before all the bowls at the end of the season were played and mine on the left is using information only from before the bowls as well.
So here is how the bowls played out:

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My ranking system predicted the winner by the pre-bowl ranking:

12/25

AP Poll:

14/25

Competition Rankings

I developed a way to do what I feel is a very fair way to rank the teams in a competition after they've completed their schedule. The data below is taken from the 2010 college football schedule for the Big East.

I've put a 1 in a column/row cell if the team in the column beat the team in the row. Then at the bottom I have 7 rows labeled A,b,C...
Row A puts a number in a teams column that counts how many wins each team they beat had. For instance Louisville beat Connecticut, Syracuse, and Rutgers. Those teams had   5, 4, and 1 wins respectively so the Louisville column gets a 5+4+1=10 in row A.
Then Row B in a teams column adds every team they beat's Row A score. For instance, Louisville beat Connecticut, Syracuse, and Rutgers, who had 19,11, and 5 in their Row A scores. So Louisville gets 19+11+5=35 in its Row B score.
So now you can see for  a team T, in Row B it starts to be important how many wins a team R had that was beaten by a team S that was in turn beaten by T.
Row C does the same thing with Row B that Row B did with Row A and now with Row C the analysis goes 4 teams deep.
So you go to Row G following the same principle, One less generation than there are teams so you've included every possible chain of who beat who.
Then as I have you can organize the teams by their row G score which should give you a very reasonable ranking of how the teams stack up against each other.

A grid paper game

This game can be played on any size grid, below I show a game on an 8x8 grid. Graph paper works good.  There are 2 players, one player plays the odd numbers and goes first the other plays the even numbers. When it's a players turn they have to play the next number to the left, right, above, or below the last number played on an unoccupied square, and when someone can't make a legal play the other person wins.
 Here's an example game:

Player A played the 1, then Player B played the 2, then Player A the 3, and so on...As Player B played 8 Player A knew that he couldn't play to the right with his 9 because then Player B could put the 10 in the top right square and win because there would be no place to put the 11. When Player A played the 17 Player B knew he couldn't play the 18 below the 17 because then Player A could play the 19 in the bottom right square and win after 2 more moves. After player A played the 23 Player B knew he couldn't play the 24 above the 23 because then Player A could play the 25 to the right and win every possible way the play could continue. After Player B played the 32 Player A saw that no matter what he did he would lose so he resigned.
  So you see the rules are so simple but the strategy and planning ahead can get pretty involved.  I haven't named the game yet, I'd like to hear any suggestions.

Study on Perspective

I found it surprising that with the same camera and same lens from the same angle, how big the continents are relative to the size of the globe change depending on how far away you are.

I'm trying to figure out if the one on the right is more or less accurate to actual distances between places on the Earth but I'd need a higher megapixel camera to get a detailed enough picture from that distance.

Proving new infinite sums

I found how to prove the sum of an (infinite number) of infinite series that Maple didn't know the answer to!

Internet Idea

I was thinking, if these wireless routers everyone has set up around town would be configured to listen for messages that had a destination address and a payload (encrypted of course!), then pass it along further to another router along the route towards the destination address... eventually no one would even need to buy their internet from these big telecom companies.
   For instance, suppose my Dad has a server set up at his office with his company website on it across town, the route from my house to there looks like this...

All those little red dots if you zoom in are different people's wifi hotspots that my request for his webpage would go through jumping from one to the next along the physical route to his office and then the data for the webpage would come back through.
 So to send an email to someone you would send it to their street address just like you would regular mail and eventually it would get there jumping from the house down the road from you to the next house and through a bunch more until it gets to theirs.
  Anyway I thought it was kind of fun to think about.