Ask Uncle Colin: Round-Robin Progress

Dear Uncle Colin,

I lost the first game of my Big Internet Math-Off tournament - can I still win the group and qualify for the semi-finals?

- Surely Combinations Of Talent, Luck And Nous Deliver?

Hi, SCOTLAND, and thanks for your message!

Because the tie-break rules aren’t currently clear, I can’t give you a straight answer - but I can tell you some scenarios that might work.

The possible outcomes of a round-robin

In a round-robin where each result is either a win or a defeat, there are only four shapes the final table can take:

  • 3-2-1-0
  • 3-1-1-1
  • 2-2-1-1
  • 2-2-2-0

In two of those, there is a clear winner; in two, there is a tie-break.

What will knock you out

As a result, you will definitely be eliminated if either:

  • you fail to win your remaining two matches
  • the competitor who beat you wins their remaining two matches.

Let’s assume, from now on, that you do win your remaining two matches (against teams B and C), and that your conqueror (team A) loses at least one.

Possible outcomes

We’ve specified three results for the six matches (A bt. S, S bt. B and S bt. C), and added another constraint, so I think the remaining six possibilities are:

  • A bt. B, C bt. A, B bt. C (A and S have two wins, B and C one each)
  • A bt. B, C bt. A, C bt. B (A, S and C have two wins, B none)
  • B bt. A, A bt. C, B bt. C (A, S and B have two wins, C none)
  • B bt. A, A bt. C, C bt. B (A and S have two wins, B and C one each)
  • B bt. A, C bt. A, B bt. C (B and S have two wins, A and S one each)
  • B bt. A, C bt. A, C bt. B (S and C have two wins, A and B one each)

There are still possible routes to qualification, depending on how the tie-breaks resolve.

A point-difference or random tie-break

Your best hope is in the case where point difference or random choice is used to decide the winner. In these cases, two wins is always enough to give you a shot at the title.

Since it’s easier to win a coin-flip than a one-in-three choice, your ideal scenario in the random setup is the 2-2-1-1 result, which occurs when the remaining three matches are a non-transitive chain (e.g., A beats B, B beats C and C beats A.)

A head-to-head tie-break

If tie-breaks are decided according to the result of the match between the two competitors, then it’s much tougher - although there are now two scenarios in which you can win the group outright - if A were to lose both remaining matches, the table ends up 2-2-1-1 with you on 2 and the only team you haven’t beaten on 1 - so you must win.

The flip side of that is that if A wins either game, you’re out - UNLESS the team A beats loses all three games. In that case, it’s a three-way tie at the top and Christian only know what would happen then.


So it’s a long shot, but it might just work! Hope that helps,

- Uncle Colin

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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