# Rugby scoring math challenge

I was watching Ireland massacre Wales at rugby yesterday and it occurred to me that at low scores it’s quite easy to work out how each team gained its points by just looking at the scores. This is because of the ways points can be scored:

A drop goal (field goal): 3 points.
A try (touchdown): 5 points.
A conversion (field goal from standing kick, only allowed after a try): 2 points.

Thus a score of 7 could only have come from a try and a conversion. A score of 8 could only have come from an unconverted try and a drop goal. However, a score of 12 could be two tries and one conversion, or 4 drop goals.

Not being terribly mathematically minded, I wonder if someone could come up with an algorithm to predict how the points were gained from any given score, and also the highest possible number for which the score can be predicted with accuracy - and if you’re feeling very brainy, how much the accuracy decays as the scores become higher.

Alright jjimm - I was watching that too. Rebirth of Welsh rugby my arse.

Anyway…

Yes you could design an algorithm to predict how the score of a particular match breaks down, but this is never going to be really accurate. Here’s why:

Teams have different styles - cf. Englands pathetic World Cup penalty kicking shit-fest and New Zealand running in a massive number of tries in virtually every game they play - it’s just a different way of playing the game.
Secondly - even boring teams like Engerland can change their bad habits and score tries, as was witnessed this weekend, once their kicker has been subsituted. An extension of this is that the style of play in every team will depend on the players picked, and just slightly less on the way those players are playing on the day, plus any injuries during the match.

What you are really asking for is a rugby simulator. They probably have one. I don’t know where you’d get one…

Hope that helps…

Additional complication: a drop goal and a penalty are two different methods of scoring with the same number of points.

Surely a penalty is just an opportunistic drop goal? They can take a penalty without kicking for the posts.

A drop goal must be dropped from the hands of the kicker, hit the floor and be kicked through the posts- it’s not just a punt- that’s what makes it so difficult. A penalty is awarded by the referee and becomes a goal when it is kicked through the posts; it is kicked from sand or a holder on the pitch. A conversion is also a goal but worth only two points.

In scores there are two ways of scoring- tries and goals, and goals are divided into three categories, conversions(2), drop goals(3) and penalty goals(3).

Nope a penalty is given for infringments offside, pulling the scrum down etc. if the kicker is within distance and doesn’t need a try to win near the end of the game he’ll go for the kick other wise they run it.

http://www.uidaho.edu/clubs/womens_rugby/RugbyRoot/rugby/Rules/LawBook/law27.html

To answer the second question, I believe 11 is the highest score with a unique solution. Any number of the form 3n, where n is 4 or more (i.e. 12, 15, 18…), can also be expressed as 3(n-4) + 5 + 7, so there would be at least two ways of achieving that score. Likewise, any number of the from 3n+1, where n is at least 3 (10, 13, 16…), can also be expressed as 3(n-3)+ 5 + 5, and finally any number of the form 3n+2, where n is at least 4 (14, 17, 20…), can also be expressed as 3(n-4)+7+7,. Since any number at all can be expressed as either 3n, 3n+1 or 3n+2, that covers all possibilities.

Oops, my answer is incomplete. Any number of the form 3n+1 can be expressed as both 3(n-2)+7 and 3(n-3)+5+5, while any number of the form 3n+2 can be expressed as both 3(n-1)+5 and 3(n-4)+7+7.

I have absolutely no idea what you’re talking about, but it looks good!

Fancy maths and all, but wrong - given the issue with drop-goals and penalties.

The truth is that there are certain scores from which one can deduce the method by which it was attained i.e.

1 - NA
2 - NA
3 - NO
4 - NA
5 - YES
6 - NO
7 - YES
8 - NO
9 et cetera NO.

Therefor 7 is the highest score from which one can deduce the method by which it was attained.

Bah. Stupid bloody game, anyway.

Anglo-bash much, Achilles? New Zealand came unstuck against Australia, who stifled their try-scoring potential and managed just one breakaway try of their own. Whereupon England gave Australia a dose of their own medicine in the final. No WC final has ever been a try-fest and this wasn’t the first to be decided by a drop-goal.

Yup. Eleven tries in two matches. There’ll be more to come, and if you think some of 'em won’t be against Ireland, you’re deluding yourself.

By Paul Grayson, who was filling Rob Andrew’s boots as a goal-kicking fly-half before Wilkinson ever appeared on the scene, and who’ll be leapfrogging Andrew as England’s second-highest point scorer by the end of the season.
Australia were whining “Is that all you’ve got?” halfway through the WC. Sorry, England these days don’t accommodatingly change their playing style just because they get called boring. We’ve wised up since the 1991 final. Check the try tally at the end of the season and then start bitching if you like - I doubt you’ll have cause - but it’s not like, say, Wales were ever reluctant to rely on Neil “Interesting” Jenkins.
Strictly speaking, a “goal” is a converted try, and replaces the try, but who pays attention to that?

• Not really, but i would really expect more from a team like England.
• There certainly will be some tries - now that Wilko is facked.

And your point about Jenkins is well taken - a penalty kicker does not a good team make.

I just wish they’d make a try 6 instead of 5. That would really push them to run their penalties.

Translated into pure math, I think the OP’s question is:

What is the smallest S for which 3a + 5b + 7c = S has a unique solution in the whole numbers (i.e. where a, b, and c are all nonnegative integers)?
(Points can only be scored 3, 5, or 7 (= 5 + 2) at a time.)

Ushram’s answer looks right to me.

It may look right - but it’s wrong. There are two different ways of scoring 3 points - therefore it is impossible to tell how any score except 5 and 7 was composed.

Sheeh. These maths boys sure are persistant…

For the sake of simplicity, can we assume a drop goal is the same as a penalty?

Surely you mean inaccuracy

Shakes fist

Why I oughta…!

Well, I meant that it was the right answer to the question as I rephrased it, without considering that there could be two different ways of getting a score of 3 (which is still an interesting question, whether or not it tells us anything about rugby). But I should have been more clear.