This is more an algorithm question than Perl per se, but I consider it to be an interesting problem, so...

The basic form of Round Robin tournament schedule with all two-way matches is a reasonably well understood problem, so I won't go into a lot of detail here, but in general the constraints are that each team should quiz each other team at most (ideally, exactly) once, that every team plays the same total number of matches, that no team may play more than one match in any given round, that no team may play against itself, and that when team A plays team B, team B also plays team A at the same time. This is a solved problem.

sub roundrobin {
  my ($n, $rounds, $bye) = @_;
  if ($n % 2) {
    $rounds = $n; $bye = 1;
    # There will be $rounds rounds with exactly one bye for each team.
  } else {
    $rounds = $n - 1; $bye = 0;
    # There will be $rounds rounds, and every team will quiz every rou
+nd.
  }

  my $m = $n; ++$m if $n % 2; # $m is the lowest even number greater t
+han or equal to $n
  my %s = (teams => $n, rounds => $rounds, bye => $bye); # The working
+ schedule.

  # Fill in the working schedule with a nice diagonal pattern:
  for my $round (0 .. ($rounds - 1)) {
    for my $i (0 .. ($round-1)) {
      $s{table}->[$round]->[$i] = (($rounds + $round - $i + 1) + $m) %
+ $m;
    }
    for my $i ($round .. ($n-1)) {
      $s{table}->[$round]->[$i] = (($rounds + $round - $i) + $m) % $m;
    }
  }

  use Data::Dumper; print Dumper(+{tentative => \%s});

  # Now, do knight-like moves with the 0 (bye) in the first row.
  # Every time the 0 lands on a number, put that number in the first c
+olumn:
  my $byeround = 0;
  for my $i (reverse (1 .. ($m - 2))) {
    $byeround = (($byeround - 2) + $rounds) % $rounds;
    $s{table}->[$byeround]->[0] = $s{table}->[$byeround]->[$i] || 'Oop
+sie';
    $s{table}->[$byeround]->[$i] = 0;
  }

  # If m != n, then remove team n from all the games, and replace with
+ -1 (which means a bye, i.e., the team sits out that round):
  if (not $m == $n) {
    for my $i (0 .. $rounds-1) {
      $s{table}->[$i]->[$i] = -1;
    }
  }
  return \%s;

}
[download]

However, the quizzing program that I'm involved in doesn't just have two-way matches. There are also three-way quizzes. There are several reasons why the three-way quizzes are necessary:

• A quiz rally must take place at a single location in a single day, so it is desirable to have 6-8 matches irrespective of the number of teams (not counting the semi-finals and finals, which match off the four teams with the highest scores for the day in a small single-elimination tournament at the end of the day). Three-way quizzing allows each team to quiz more other teams in the same number of matches than would otherwise be possible.
• If there are more than about six teams in a given division (based on age and skill level; currently we have three such divisions in the North Central Ohio district; it is normal to have anywhere from 3 to 12 or so teams in a division), it becomes difficult to find enough quizmasters and rooms to have all teams quizzing in two-way matches at the same time. Three-way quizzes allow half again as many teams to quiz at once.
• There are three-way quizzes at the national level, so the district would like to have them in order to better prepare the best quizzers so that the district team will be able to do well at nationals.
• It's tradition.

So when constructing schedules for this type of quizzing, the constraints are a bit different...

• There must be 6-8 rounds of quizzing.
• No team may quiz against itself.
• No team may quiz more than one match in any given round.
• Every team must have the same number of two-way quizzes, the same number of three-way quizzes, and the same number of byes (i.e., sit-out rounds) as every other team. The number of byes per team should be kept to a minimum: 0 or 1 if possible, no more than 2 at most.
• Each team must quiz each other team at most once (unless there are too few teams for this, in which case each team must quiz each other team at least t and no more than t+1 times for some value of t to be determined based on how large it needs to be to get a good number of quizzes)
• The number of rooms needed should be reasonable: no more than r rooms for up to r*3 teams if possible, and absolutely no more than r+1 rooms for r*3+1 teams.
• No room may have more than one quiz taking place in it during any given round: a room may have a two-way quiz with two teams, a three-way quiz with three teams, or no quiz. Every quiz must take place in a room.
• Whenever any team A has a two-way quiz against another team B, team B must also have at the same time a two-way quiz against team A, in the same room.
• Whenever any team A has a three-way quiz against two other teams B and C, team B must also have a three-way quiz against teams A and C, and team C must also have a three-way quiz against teams A and B, at the same time, in the same room.
• The room where Nathan is acting as quizmaster must be located as close to the kitchen area as possible.

I've been looking at this problem off and on for a couple of years, but I've never gotten anywhere very useful. Some numbers of teams (e.g., 9) are very easy to do by hand, but others are not, so I'd really like to have an algorithm for doing this. Ideas? What approach should I take? Should I predetermine how many two-way and three-way quizzes there should be, based on the number of teams, or should I approach it from a different direction?