Eight (8) soccer teams (named A, B, C, D, E, F, G and H) are ready to a tournament among them, where each team will play against each of the other 7 teams once, and only once. They have 7 stadiums available, and agree that each team must play in all the 7 stadiums, once in each. In each round the 8 teams will play, so it will have 4 matches every day, and the tournament will last in 7 days.

Schedule the matches in each of the seven days, and in what stadiums they will be played.

(In reply to Solution !!! by Penny)

An interesting way to solve this problem is to set up an 8x8 matrix.  There are 28 elements in the upper triangle.  Imagine putting the 28 numbers (4 ones, 4 twos, ..., 4 sevens) into the upper triangle.  Now imagine the transposed matrix is put into the lower triangle.  You should have a matrix with all but the diagonal filled in with the digits 1 through 7.

The constraint is that each row and each column can only have one appearance of each digit.  Here is an example:

-   1   2   3   4   5   6   7

1   -   7   2   3   4   5   6

2   7   -   6   1   3   4   5

3   2   6   -   5   7   1   4

4   3   1   5   -   6   7   2

5   4   3   7   6   -   2   1

6   5   4   1   7   2   -   3

7   6   5   4   2   1   3   -

 

Not sure how many solutions there are.

Oh yeah - the matrix can be read by team row plays team column in stadium digit.

 

 

Edited on July 27, 2005, 4:49 pm

Posted by bernie on 2005-07-27 00:55:17