Rugby scoring math challenge

[Cabbage]

If you simply want to count the number of different ways to arrive at a different score, generating functions are a handy way of handling the data.

Take a power series for each method of scoring points. Let d = drop goal, t = try, and c = conversion; these score 3, 5, and 2 points, respectively. A conversion cannot be scored without a try, so let me amend that by letting c stand for a try with conversion, for 7 points, while t stands for try without conversion, for 5 points. Our respective power series are then:

1 + d[sup]3[/sup] + d[sup]6[/sup] + d[sup]9[/sup] + d[sup]12[/sup] + d[sup]15[/sup] + …

1 + t[sup]5[/sup] + t[sup]10[/sup] + t[sup]15[/sup] + t[sup]20[/sup] + t[sup]25[/sup] + …

1 + c[sup]7[/sup] + c[sup]14[/sup] + c[sup]21[/sup] + c[sup]28[/sup] + c[sup]35[/sup] + …

I’m not at all familiar with rugby, so I won’t enter this part of the debate, but if you wish to also include a 3 point penalty § as being distinct from a 3 point drop goal, we can also include:

1 + p[sup]3[/sup] + p[sup]6[/sup] + p[sup]9[/sup] + p[sup]12[/sup] + p[sup]15[/sup] + …

Now, generating functions can be useful for many things. For example, let’s say we want to know how many different ways there are of scoring n points. Take all four of the above power series, change all the variables to “x”, and multiply them together:

1 + 2x^3 + x^5 + 3x^6 + x^7 + 2x^8 + 4x^9 + 3x^10 + 3x^11 + 6x^12 + 5x^13 + 5x^14 + 9x^15 + 7x^16 + 8x^17 + 12x^18 + 102^19 + 12x^20 + 16x^21 + 14x^22 + 16x^23 + 21x^24 + 19x^25 + 21x^26 + 27x^27 + 25x^28 + 27x^29 + 34x^30 + 32x^31 + 34x^32 + 42x^33 + 40x^34 + 43x^35 + 51x^36 + 49x^37 + 53x^38 + 61x^39 + 60x^40 + 64x^41 + 73x^42 + 72x^43 + 76x^44 + 87x^45 + 85x^46 + 90x^47 + 102x^48 + 100x^49 + 106x^50 + 118x^51 + 117x^52 + 123x^53 + 136x^54 + 136x^55 + 142x^56 + 156x^57 + 156x^58 + 163x^59 + 178x^60 + 178x^61 + 186x^62 + 202x^63 + 202x^64 + 211x^65 + 228x^66 + 228x^67 + 238x^68 + 256x^69 + 257x^70 + 267x^71 + 286x^72 + 288x^73 + 298x^74 + 319x^75 + 321x^76 + 332x^77 + 354x^78 + 356x^79 + 369x^80 + 391x^81 + 394x^82 + 408x^83 + 431x^84 + 435x^85 + 449x^86 + 474x^87 + 478x^88 + 493x^89 + 520x^90 + 524x^91 + 540x^92 + 568x^93 + 573x^94 + 590x^95 + 619x^96 + 625x^97 + 643x^98 + 673x^99 + 680*x^100 + …

The exponent gives the number of points scored, and the corresponding coefficient gives the number of distinct ways of scoring that many points. For example, there is 1 way of scoring 7 points, 6 ways of scoring 12 points, 106 ways of scoring 50 points, and 680 ways of scoring 100 points.

On the other hand, if we drop the distinction between drop goals and penalties, we get:

1 + x^3 + x^5 + x^6 + x^7 + x^8 + x^9 + 2x^10 + x^11 + 2x^12 + 2x^13 + 2x^14 + 3x^15 + 2x^16 + 3x^17 + 3x^18 + 3x^19 + 4x^20 + 4x^21 + 4x^22 + 4x^23 + 5x^24 + 5x^25 + 5x^26 + 6x^27 + 6x^28 + 6x^29 + 7x^30 + 7x^31 + 7x^32 + 8x^33 + 8x^34 + 9x^35 + 9x^36 + 9x^37 + 10x^38 + 10x^39 + 11x^40 + 11x^41 + 12x^42 + 12x^43 + 12x^44 + 14x^45 + 13x^46 + 14x^47 + 15x^48 + 15x^49 + 16x^50 + 16x^51 + 17x^52 + 17x^53 + 18x^54 + 19x^55 + 19x^56 + 20x^57 + 20x^58 + 21x^59 + 22x^60 + 22x^61 + 23x^62 + 24x^63 + 24x^64 + 25x^65 + 26x^66 + 26x^67 + 27x^68 + 28x^69 + 29x^70 + 29x^71 + 30x^72 + 31x^73 + 31x^74 + 33x^75 + 33x^76 + 34x^77 + 35x^78 + 35x^79 + 37x^80 + 37x^81 + 38x^82 + 39x^83 + 40x^84 + 41x^85 + 41x^86 + 43x^87 + 43x^88 + 44x^89 + 46x^90 + 46x^91 + 47x^92 + 48x^93 + 49x^94 + 50x^95 + 51x^96 + 52x^97 + 53x^98 + 54x^99 + 55x^100 + …

[Boo_Boo_Foo]

@Malacandra

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.

That’s some pretty broad brush-strokes you’re painting there. You’ll find your assertion is based on one rather ugly headline by just one newspaper after the England vs Western Samoa game (which was unexpectedly close).

However, amongst rugby fans in the general Australian population, there was nothing but respect and admiration for how professional and well drilled the England team were. Every rugby fan I know happily conceded that England were the team to beat, the whole way through the tournament, and that further, the best team won on the night. But I see such magnanimity is rarely acknowledged if a better beatup story is waiting in the wings.

I love the way armchair experts refer to themselves and their team as “we”, as if they were on the pitch themselves doing the real work.

[LSLGuy]

As a 'merkin who’s always loved rugby on the rare occasions I can see any on what passes for sports on TV here in the States, I just have one question …

How come a sport with continuous action and reasonably frequent scoring and genuine mud & violence & injury is NOT all the rage in America and instead we get to watch hundreds of hours of sissy basketball, while Soccer (rest-of-world says “football”) slowly gains ground as well??

Blood, mud & action; what’s not for an American to love???

More play happens in 10 minutes of rugby than in a season of Soccer.

Anyone have a theory why Rugby or Aussie rules football can’t get going here in the good 'ol USA???

[Boo_Boo_Foo]

@LSLGuy

How come a sport with continuous action and reasonably frequent scoring and genuine mud & violence & injury is NOT all the rage in America and instead we get to watch hundreds of hours of sissy basketball, while Soccer (rest-of-world says “football”) slowly gains ground as well??

Blood, mud & action; what’s not for an American to love???

More play happens in 10 minutes of rugby than in a season of Soccer.

Anyone have a theory why Rugby or Aussie rules football can’t get going here in the good 'ol USA???

Well, you know… it’s worth noting that American Football is a direct descendant of Rugby. I’ve seen documentaries on the evolution of American Football, and up until 1890 it was 100% Rugby and it was primarily played in Ivy League schools etc. But over a 30 year period, the game evolved away from Rugby into the set-piece format that American Football is today.

But there are oodles of left-over clues which are worth noticing. For example, the line of scrimmage in American Football which is set for each “play” - well that used to be the Rugby lineout which was held on the sideline. But it seems that a lot of American Rugby pitches were very muddy on the sidelines in winter so the lineout was moved to the centre of the field where it was dryer.

Also, the size of the fields are identical - or so they were up until 20 years ago when Rugby pitches increased in size from 100 yards to 100 meters.

And the concept of a conversion after a touchdown? Straight from Rugby.

All I’m saying here is that oddly, Rugby is soooooo close to American Football that I rather think it’d be quite a threat to the extablished powers that be I suspect. After all, there’s some big money at stake.

Interestingly, there’s another version of Rugby called Rugby Leaque which is even closer to American Football. Sadly, it’s not played anywhere near as universally as Rugby Union, but it’s an amazing game with some wonderful athletes.

As for Australian Rules? Great game. Unbelievable athleticism involved in that game. Most centre-forwards run the equivalent of a half-marathon per match, AND manage to take NFL standard hits all the way through. Some real motherfucker men play that game. Big, muscular, tall, and fast. It’s probably the game which would get an easier acceptance into American loungerooms I suspect - if only a concerted effort was made to explain the rules regularly. The rules can be a tad confusing.

[Thudlow_Boink]

7 points to Cabbage!

[Malacandra]

@Achilles

@Malacandra

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.

• There certainly will be some tries - now that Wilko is facked.

You’re missing the point. Grayson can slot 'em near on as well as Wilko. Before Jonny appeared on the scene, Grayson was homing in on Andrew’s record at a rate of knots. He’ll be there by the end of the season. The coming raftload of tries isn’t going to be dictated by a lack of penalty-kicking talent. Also, when England and Ireland faced off for the Slam last season, Wilkinson was fit and well - and England had far more tries on the board than Ireland did.

@Achilles

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

It’s not that point that I was making, but that when you have a phenomenal asset on your side, you use it as often as possible, and if the other lot don’t like it, sod 'em. Wales weren’t ashamed to.

@Achilles

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

Well, you need to strike a balance. At the moment it pays the defending side to kill the ball - which is what a lot of penalties are awarded for: illegal and negative defence, rather than technicalities - and concede a penalty goal. If it saves a try under the posts then it’s not until they concede the third such penalty that their spoiling strategy is showing a loss. Should that attitude really be given any further encouragement? Bottom line: if you don’t want penalty goals against you, don’t cheat.

@BooBooFoo

@Malacandra

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.

That’s some pretty broad brush-strokes you’re painting there. You’ll find your assertion is based on one rather ugly headline by just one newspaper after the England vs Western Samoa game (which was unexpectedly close).
<some fair and approving stuff about England snipped>

It sure was - tho’ they’re called “Samoa” now, and they’re nearer to being your neighbours than ours, so shame on you … and we did manage to stick four tries on 'em in the end. I heard the first ten minutes of that match on the car radio and then had to go in to church, which didn’t make for settled nerves over the next couple of hours.

Anyway, hearty apologies, I’ve gone and perpetrated the Misleading Vividness fallacy - a few isolated reports about the famous Wilko photograph, Tefu claiming that England should have had the Samoa result overturned for technically having sixteen men on the field for half a minute to no practical effect, Grant Batty whining about the drop-goal being overvalued and the famous Internet poll that we successfully hijacked - have gone and led me to think that ANZACs generally were talking about nothing else. I ought to know better. I do know better: that it’s much nearer to the truth to say that, while newspapers may be run by arseheads the world over, the typical Aussie sports fan knows a shitload about the game and, for that matter, is just as quick to give his own side a roasting if they deserve it as he is to rubbish the Poms. My bad. End of hijack.

[Achilles]

@LSLGuy

As a 'merkin who’s always loved rugby on the rare occasions I can see any on what passes for sports on TV here in the States, I just have one question …

How come a sport with continuous action and reasonably frequent scoring and genuine mud & violence & injury is NOT all the rage in America and instead we get to watch hundreds of hours of sissy basketball, while Soccer (rest-of-world says “football”) slowly gains ground as well??

Blood, mud & action; what’s not for an American to love???

More play happens in 10 minutes of rugby than in a season of Soccer.

Anyone have a theory why Rugby or Aussie rules football can’t get going here in the good 'ol USA???

It’s also a lot to do with where you’re from, who your friends are. At least here in Ireland, it is. We have 4 “major” (and I use the word in it’s widest possible sense) sports - Gaelic Football, Hurling, Association Football (Soccer), and Rugby Football. The problem is that these sports are often seen as “country”, “poor”, “posh”, “irish”, “english” and nearly every other prejudicial term you can think of, by some of the players of the other sports. This leads to a situation where sport cross-over is well nigh impossible (except between hurling & gaelic football, where cross-over is the rule rather than the exception).

IMO that’s why there is comparitively little rugby in the US. Also, don’t forget that the US aquitted themselves quite satisfactorily in RWC 2003 (considering that it’s probably about the 7th sport in popularity, after American Football, Baseball, Basketball, Ice Hockey, Tennis and Golf).