Round Robin Math
So I haven't posted in a while but right now I have some downtime and the internet so here goes... I have three new games that I should post this week. This post is a question to readers of this blog.
Is there anyway other than brute force to determine the end scenarios of a round robin in process?
I ask this question because I am playing a 12 player round robin at Reno Chess Club right now. If you click results you can see the cross table. I have a perfect score and I do not intend to start losing next game, however I was wondering if I had clinched a spot in the next stage of this tournament. I was able to find one scenario where I would not finish in the top 4 of my group:
IF:
I lose to Sheryka, Hong, and Alsasua.
Alsasua beats Hong, loses to Rand, and draws Fleming.
Hong and Rand draw.
AND Fleming beats Sheryka.
THEN the standings would look like this:
1.Rand 9
2.Alsasua 9
3.Fleming 8.5
4.Hong 8
5.Gafni 8
and I would be eliminated due to my loss to Hong.
I was wondering if there are any other combinations and just if anyone knows cool things about the math to round robins in general.
Is there anyway other than brute force to determine the end scenarios of a round robin in process?
I ask this question because I am playing a 12 player round robin at Reno Chess Club right now. If you click results you can see the cross table. I have a perfect score and I do not intend to start losing next game, however I was wondering if I had clinched a spot in the next stage of this tournament. I was able to find one scenario where I would not finish in the top 4 of my group:
IF:
I lose to Sheryka, Hong, and Alsasua.
Alsasua beats Hong, loses to Rand, and draws Fleming.
Hong and Rand draw.
AND Fleming beats Sheryka.
THEN the standings would look like this:
1.Rand 9
2.Alsasua 9
3.Fleming 8.5
4.Hong 8
5.Gafni 8
and I would be eliminated due to my loss to Hong.
I was wondering if there are any other combinations and just if anyone knows cool things about the math to round robins in general.
I don't have anything to offer on the math, though likely someone really good with combinatorics would find an algorithm more efficient than brute force. It seems fairly complicated, probably better to work on chess than to actually work through the permutations.
Your final three games seem to be with tough opponents, so it should be a good battle. Good luck!
I can see you losing only to Alsasua at this point. Hong and Sheryka would have to put forth an unusual effort in order to win. That leaves you with a score of ten, certainly good enough to qualify.
Fleming is an unknown with many games to make up, but I don't think he will score better than eight.
Rand will certainly post 7 1/2, but could lose both his games to Hong and Alsasua, normally tough opponents.
Hong himself may score no better than 7 1/2 as he has Rand, Gafni and Alsasua to play. No easy road there either.
It looks to me like it will ultimately be Alsasua and Gafni at the 1 and 2 spots with scores better than a 9 1/2. The other two might qualify with a modest score of 7 1/2.
With Fischer and Sheryka eliminated (neither of them can post a score better than 7 or 5 1/2), and Fleming being the only player who 'could' post a score of 8 (he'd have to win the rest of his remaining games save one), that leaves only Alsasua, Gafni, Hong and Rand.
That's how it's probably going to play out. But Fleming shows the magic number--"8". No one can be happy until they score at least 8 and have tie breaks over Fleming. Hong just lost to Fleming, so "8" is not a good score for him. He needs 8 1/2. I doubt Fleming will score 8, but it is possible. The others have to keep that mind.
Here's my guesses and their scores:
1. Alsasua-10 1/2
2. Gafni-10
3. Fleming-8
4. Hong-7 1/2
4. Rand-7 1/2
Fleming is very likely to score 8, now that I look at it more closely, his game with Sheryka being all important!
At the moment, it looks like either Hong or Rand will not make it. That game is really important and it is coming up soon! That game is huge!
In order for both Hong and Rand to make it, Fleming has to lose two of his tough games remaining, but since he also has the easier part of his schedule coming up, he will post at "Least" a 7, and probably will at least win 1 of his remaining games among the top players left, most likely against Sheryka who's not playing well.
In short, there is no scenario where you do not qualify as there is no score of "8" that would defeat you on tie breaks even if you only posted a score of 8 yourself.
That is the likely scenario. It changes big time if you dump all of your games and Hong, Rand and Fleming put up scores of 8 1/2 or 9, but that is very unlikely and all the more so since Hong and Rand still must play each other.