Query with each value only used once MySql - php

Question

I've got a problem!
I've got a simple table which contains player_1, player_2, and played.
+---------+----------+----------+
| player1 | player2 | played |
+---------+----------+----------+
| 1 | 2 | 1 |
| 1 | 3 | 2 |
| 1 | 4 | 1 |
| 1 | 5 | 0 |
| 1 | 6 | 0 |
| 2 | 3 | 3 |
| 2 | 4 | 2 |
| 2 | 5 | 1 |
| 2 | 6 | 3 |
| 3 | 4 | 1 |
| 3 | 5 | 2 |
| 3 | 6 | 0 |
| 4 | 5 | 3 |
| 4 | 6 | 1 |
| 5 | 6 | 3 |
+---------+----------+----------+
and so on... There are 78 records, because there are 13 players in total.
I would like a result that shows me which 'matches' have been least played. So, e.g. a person was sick and couldn't play last time, the played games between that person and other players are lower. This game is more likely to be in the result.
But in each result, which shows me 6 matches (1 player has some time off), a player can be displayed only once! However, the least played games have to be in the result too. (well, the last part is easy, ORDER BY played ASC, LIMIT 6)...
How do I get each 13 players (well, without the one) to be in the result?
A nice result would be (well, now I'd like only 3 matches, because in this example are only 6 players)...
Every player plays a game in this round, and it is selected by ordening the least played games. The rest is just being filled in.
+---------+----------+----------+
| player1 | player2 | played |
+---------+----------+----------+
| 1 | 5 | 0 |
| 2 | 4 | 1 |
| 3 | 6 | 0 |
+---------+----------+----------+

Answer 1

You can think of this as a Graph Problem.
Let's say each player is a node, and every record in a table represents an arc, a weighted arc with weight of 'played', The number of games played between the 2 players connected with the arc.
Now, what you're looking for is an algorithm to find the 6 matches least played (if I understand correctly, the 6 matches to play in the next round) providing that every player plays only once a round.
I suggest using a greedy algorithm:
1. Sort arcs ascending.
2. While there are still arcs left.
2.1 Remove first arc and add to round_games.
2.2 Remove all arcs with any participating nodes in the previous arc.