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Impossible scorelines

Yesterday I was watching Exeter Chiefs vs Saracens in the premiership final. During the first half I was thinking about scorelines and how they are constructed and I thought that you could come up with some interesting activities around scorelines.

My first thought was “what scorelines are impossible?” – in Rugby Union there are a variety of ways to score, you can kick a penalty goal or a drop goal for 3 points each, you can score a try for 5 points and if you score a try you get a chance at kicking a conversion for an extra 2 points. From this we can see obviously that 1, 2 and 4 are impossible but I wondered briefly if any others were. I don’t think there are as you can make a difference of 1 between an unconverted try and 2 penalties, however that’s not really a strong proof. I may think about how to prove, or disprove, it later.

I then thought about the 4 4s challenge, and the variety of related challenges based around the year etc. I thought this might be interesting to attempt with rugby scores. It would be nice to investigate how many ways there are to make each score too, and to see if there were any patterns to it.

My thoughts turned to rugby league, the scores in that are 1 for a drop goal, 2 for a kicked goal and 4 for a try, thus all scores are possible, but it still might work for a 4 4s type challenge or an investigation into how many ways each score can be made.

I considered other sports too, football would of course be pointless, basketball would provide a simpler version which could be good for embedding the 2 and 3 times tables and that was as far as I managed.

Have you considered any of these activities or similar? Do you know of any other sports with interesting scoring systems that could be investigated? I’d love to hear in the comments or on social media if you have.

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  1. May 29, 2016 at 4:24 pm

    Interesting thoughts! For rugby union you can argue as follows: 5, 6 and 7 points are all possible. Then since +3 is always possible, you can therefore score any larger integer. Not sure about the how many ways part yet, though đŸ™‚

    • May 29, 2016 at 4:27 pm

      Aye, that is a succinct argument, thanks. Much more coherent than mine!

  2. June 5, 2016 at 6:08 pm

    Relevant here: Frobenius Problem, aka https://en.wikipedia.org/wiki/Coin_problem
    The Chicken McNuggets example is nice too:, here: http://www.ams.org/samplings/feature-column/fc-2013-08 and here: https://cs.uwaterloo.ca/~shallit/Talks/frob6.pdf

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