Triangle colours

[MartinCarpenter]

Since we're posting combinatoric based problems, here's the obvious one derived from triangles in the 4NCL.

Now it is very easy to balance the colours to 3 each in the matches, but you do get some double colours. The question which intrigues me is if its possible to optimise the colours, ie:
a) The minimal number of double colours for people (I think probably 6)
b) Ideally also only sets of double colours which can be worked round by a smart board order.

In terms of B, the issue that board order flexibility is quite cut down by splitting the fixtures so everyone plays both teams. So, with the existing fixtures, A2 can play as A4 but not as A1 or A3.

Thus, if you have to have double colours, better if A2 is double white and A4 double black. Nearly all these teams can legally move people from board 2 to 4. (Or 3 to 5.). Maybe even better if the fixtures could be rearranged to make it only +-1 board.

The obvious, 'clean', solution of giving all of team C white on day 1, black on 2 then having A white vs B on day 1 and black on day 2 gives all of team A double colours and all non avoidable. Probably better than status quo but not ideal!

[Joey Stewart]

I guess it is true what they say - there is no straight way to do a threesome.

[David Williams]

AvB AvC BvC

Round 1 A1 v B1 A2 v C1 C2 v B2
Round 2 B2 v A2 C2 v A1 B1 v C1

Round 1 B4 v A3 A4 v C4 B3 v C3
Round 2 A4 v B3 C3 v A3 C4 v B4

Round 1 B6 v A6 C5 v A5 C6 v B5
Round 2 A5 v B5 A6 v C6 B6 v C5

C2, A4 and B6 have two whites. C1, A3 and B5 have two blacks. Apart from that . . . ?

(Having posted, all my nice spacing disappeared. But it should read as if there were three columns, so that you can easily see the pairings in the three individual matches.)

[Stewart Reuben]

Surely the only correct way to run competitive chess is for each player to play his opponent twice, once with white and once with black.
Had they gone down that route in the 19th century, that might be the way we play chess today. As it is double game tournaments are rare.

[MartinCarpenter]

That proposed set of fixtures/colours look pretty ideal to me Well found/worked out!

3-3 balance in the matches, almost certainly the minimal number of double colours. All of those double colours are very easy for the teams to avoid if they want to anyway. To get a white/black balance for all their players, Team A just has to swap a3 and a4 over from day 1 to 2. Both playing team B on day 1 and C on 2 so safe to do that.

So, I guess the remaining question is: can we please try to arrange to use these fixtures/colours in the triangles at some point?

Ok, not for the next round I guess as teams selected/organised around the existing ones. I can't think why not for the second round, or next season at the latest.

Its not the end of the world of course, but these really are plain superior to the existing arrangements with no real drawbacks.