Thursday, October 17, 2013

Player Selections

A player of Football Squares decides how many squares to obtain and where to position the squares assuming there are plenty of squares in the 10 x 10 grid still available. If the player plays only 1 square, the player should feel free to place it anywhere. Since the rows and column numbers are chosen after the squares, it doesn't matter where one would place a single square.

If the player plays two squares, there are three ways the player can decide to play. Each square on an independent row and column, or both on the same row, or both on the same column.

If the player plays three squares, there are six ways to arrange the squares knowing that interchanging rows and columns would not matter as each have its numbers chosen at random. The six ways include placing a single square randomly and then placing the other two squares as above (3 ways). The other other three ways have all the squares connected by row or column either all vertical, all horizontal, or such that there is a corner square and then one above and one next. The figure below shows the 1, 3, 6 possible ways to choose 1, 2, or 3 squares. Note: squares sitting side-by-side represent any choice where those two squares sit on the same row and squares above-and-below similarly represent any choice where two squares are on the same column. A square that is diagonally offset represents any choice where it has a unique row and column relative to the other squares. The letter above a configuration is just a unique name to be used to refer to that specific configuration.


If the player plays four squares, it turns out there are 16 possible ways to uniquely arrange four squares such that interchanging rows and columns do not matter. Those patterns are shown below.
 

If the player selects five squares, there are 34 possible ways. Those include 16 ways where an additional square with a unique row and column is chosen along with one of the patterns above for four squares. Then there are these 18 additional  patterns.
 

It is an interesting mathematical problem to work out how many different ways could squares be taken where interchanging rows and columns doesn't matter. The series has been worked out. For six squares, there are 90, seven 211, and eight squares can be arranged 558 unique ways. The numbers grow quite fast beyond that (see http://oeis.org/A049311).

No comments:

Post a Comment