There are a number of Div 1 basketball teams in New Jersey: Princeton,
Rutgers, Seton Hall, Hofstra, Monmouth, and so on.  These span a range
of talent levels, but I think it would be interesting to have an event
with sixteen such teams.  (We'll import a couple from Philly if
necessary.)  I give you the New Jersey State Gauntlet:
<br>
<table>
<tr><td>
1-2</td><td>
3-6</td><td>
4-7</td><td>
8-12</td><td>
5-9</td><td>
10-13</td><td>
11-14</td><td>
15-16</td></tr><tr><td>
1-3</td><td>
2-6</td><td>
4-8</td><td>
7-12</td><td>
5-10</td><td>
9-13</td><td>
11-15</td><td>
14-16</td></tr><tr><td>
1-4</td><td>
2-7</td><td>
3-8</td><td>
6-12</td><td>
5-11</td><td>
9-14</td><td>
10-15</td><td>
13-16</td></tr><tr><td>
1-5</td><td>
2-9</td><td>
3-10</td><td>
6-13</td><td>
4-11</td><td>
7-14</td><td>
8-15</td><td>
12-16</td></tr></table>
<ul><li>
Seed the teams, the best team is 1, the worst is 16; now let them play
against each other as indicated, but if the lower seed wins, they switch
places.  (Just like you did at summer camp at the air hockey table; if
you beat number 1, you are the new number 1... for the time being.)
It works out that, even with this switching, no two teams can play
each other twice.</li>
<li>You will think it unusual that the top two play in the first round.
The point is that I stuck mostly weaker competition as far from the top
team as I could, and I don't want the second seed then to be playing all
that weaker competition stalling into the last round.  This is something
of a judgment call, and, indeed, is a reason not to really use this as
a "championship" as such; this tournament is designed so that each team
plays teams close to its level.</li>
<li><em>A miracle occurs:</em> if team A wins more games than team B, team A will end up with a
higher seed than team B.
(Try it.)  Further, the scores will follow a binomial distribution; there will
be one team each with four wins and four losses, four each with three wins 
and three losses, and six with two of each.  In particular, any team can
make it to number 1 by winning all four games.</li>
<li>An interesting observation: in each round, assuming again that it's
seeded correctly, the best team that wins beats the best team that
loses that round, the second best team that wins in that round beats
the second best team that loses in that round, etc.</li>
</ul><h3>The kicker</h3><ul>
<li>If there are no upsets, each team beats teams that have one fewer win than them, and each team loses to teams that have one more win than them at the end of the tournament.  Note that there are four teams with one loss each; this is
the number of games won by the team with four wins.  These four teams win a total of 4*3=12 games; the teams that finish with 2 wins lose a total of 6*2=12 games.  So it works out.  This kicker is, on some level, the point; I think early
season tournaments should, to whatever extent possible, leave teams with
a number of wins that is proportionate to the strength of the team, and should
ideally give the team a chance to play most of its games against teams of
similar strength.  (The mathematical coincidence is, if nothing else,
a solution in search of a problem.)</li>
</ul>
My thought is that teams could be seeded judiciously the first year, and subsequent years they might just be put in with whatever seed they finished with
the previous year.