# Real Combinatorial Problem

I found this real combinatorial problem on a puzzle group. And wanted to share it to all. Have a look

In two weeks I have the responsibility of arranging a special singles bridge tournament for 21 people. Bridge is played by 4 people at a time, 2 pairs, so there will 5 tables playing in parallell (the same boards; three duplicates), and one person sits out each board.

For various reasons (minimize change and movement is one reason) it has been decided to conduct the tournament as follows:

– in the first round player A, B, C and D sit down at a table and play boards 1, 2 and 3. A partners B against C and D when playing board 1;
A partners C (against B&D) when playing board 2, and finally A partners D (against B&C) when playing board 3. Similarly, EFGH do the same on the next table (playing the same boards 1, 2 and 3), and so does IJKL and MNOP.

– QRSTU is slightly trickier, because only 4 can play at a time.
Basically Q and R sit still and partner S, T and U who rotate. So Q&S against R&T, U sit out board 1. Q&U against R&S, T sit out board 2. Finally Q&T against R&U, S sit out board 3. Note that Q and R do _not_ partner each other in this round.

– 9 rounds of 3 boards each, so 27 boards.

I want to design the seating in round 2, 3, .., 9 so that:

– all possible player pairs partner each other at least once, and no more than twice
– all players sit out at least one board, and no more than two boards

Using inspired guesswork and some Sudoku-like skills, I am unable to get rid of the last couple of triples (players who, in my scheme, will partner each other 3 times).

Anyone (including computer programs) up for the challenge?