The goal is not to minimize the number of byes.  The goal is to
maximize fairness and competetiveness.

In the last round, when it's down to two teams, the greater the
discrepency in the number of games those two teams have played in the
tournment the less fair it is and the less competetive it is.
Therefore those final two teams should have either played the same
number of games or differ by at most one game played.

If byes can occur after the first round, then that stipulation will not
necessarily be true.

Therefore, byes can only occur in the first round.  By the time it gets
to the second round, the number of teams who are still alive in the
tournament should be a power of two.

Thus this interpretation is correct:
 * 1 vs. bye
 * 2 vs. bye
 * 3 vs. bye
 * 4 vs. 5

And that's what http://www.crowsdarts.com/brackets/playoff-chart.html
produces.

Brock Lee



Daniel Finnie wrote:
> I assumed the extended bracket for 2 and 3 was done on purpose by the
> program to have the output look nicer.
>
> 1 vs. bye
> 4 vs. 5
> 2 vs. 3
> This has 4 games, a power of 2.
>
> 1 vs. bye
> 2 vs. bye
> 3 vs. bye
> 4 vs. 5
> This has 2 games, a power of 2, but doing it this way seems stupid,
> especially when for other problems it seems to give solutions with the
> least amount of byes, or the highest power of 2 possible games.
>
> Basically, either way you interpret the online version, I think
> something's wrong.
> 
> Dan