I like rugby but I'm not a good player at it; there are some interesting consequences of the
rules I want to explore, and this is actually a famous piece of applied maths about a real
world sporting problem. And I'll explain some of the rules here just for clarity; uh you can
score points by putting the ball down over a line - it's called a try - very like in American football
a touchdown. Point about this though is that once you've scored a try in rugby you then also get a
chance to get some bonus points with what they call a conversion where you kick over the posts.
So full disclosure, in American football you put the ball down over the- as a touchdown, you
then get to take the kick from in front of the post. Is that right? I think that's right?
Never played American football but I'm reliably informed. In rugby you cannot do that;
you have to take the conversion kick somewhere in line with where you put the ball down. So if you
put the ball down right at the edge of the pitch, like close to the the touch line - which is quite
common because people are sort of bleeding you out there as you get tackled - you have to take a kick
really close to the touchline, but you can choose how far back to take it. That is our question: if
you have the choice, which you do every time you score a try, how far back should you take the kick?
If you are really close if you like then you can imagine the post being at a weird angle; but if you
go really far back - I mean technically I think you can go to the other end of the pitch but I don't
think it would be clever for a range of reasons. Even if you are a good kicker, unlike me. - No we can't!
(Brady: It's in!)
(First time) - The real issue is that maybe your advice would change. So let's get this one
thing out of the way; if you put the try down under the posts you have to take the kick in
line with that. And there isn't really a problem here, you just go back far enough that you can get
it over the bar; and there's no worry because the further back you go the narrower the angle looks
because you're further away. Obviously if you try and take it from the try line itself you're
going to struggle to clear the thing. But it's the first modeling thing I want to get out of the way,
is that let's just assume that under the post is not a problem, you take it close enough and
you're fine. If you put a try down to the side of the posts - and this is where I want to show
you on the diagram. So here's a picture of rugby pitch, I've got a nice 3D model, uh I think you
might recognise the software, this is GeoGebra. So this point here is where it's called the
try, I need to take a kick somewhere in line with that. And if I move this point
the line moves up. We just talked about going under the posts, no big deal. So
if you're to the side of the post you can imagine taking the kick from over here,
looks a bit difficult if I sort of zoom the view around - like it's a terrible angle. But if you take
the kick from really really far away it's arguable that maybe you've got a better angle but the range
starts to become an issue. And if you go really far away, like way down the other end of the pitch,
the goal is just a really long way away. And even if the range isn't a problem the angle
is narrowing because you're getting further away from it. So the question I have here is how do you
best get the angle of the posts to be maximized, because that's going to be what matters for your
accuracy of the kick. There are some other things, like can you clear the crossbar? We've talked about
that a bit - if you're too close it's going to be tricky. Incidentally there is- there's a five metre
line on the rugby pitches here. And I think a good rule of thumb is that at the angle most people kick
at, about 30 degrees they say, as long as you're a distance the other side of that line you're
likely to be able to clear the crossbar so we can just dodge that part of the model.
(You're saying if you're closer than that it's almost too steep?) - Yeah you're going to have to sort of
do a proper chip and actually that's really hard to give any power anyway, and it's just unnatural.
I looked up apparently 30 degrees is roughly what a good kicker will kick at, and at about 26
metres per second goes a long way- - (You mean 30 degrees up, not 30 degrees angle?) - 30 degrees elevation
yeah. Everyone thinks that 45 degrees is the best but it turns out if you as a kicker, if you
kick and aim to do 45 degrees you lose a lot of power because it's not the way your your leg
moves. So this is the real model, look at the rugby players and they kick it about 30 degrees.
And that also is going to solve a problem with range, is that if they do that and they're
professionals they can kick a long way, like maybe beyond 30 metres. Which means even if you went to
the halfway line there's still a chance that you could get it long enough if you could- I'm aware
that I should probably go and try this but uh I don't think I could do that. But that means the
modeling is really how do you get the best angle? And the extremes are - I know you like thinking
about extremes - if you're really close to the try line the angle is going to be really narrow, we saw
on there, and if you're really far away it's also going to be narrow. Somewhere in the middle is
an optimum, and that sounds like a task for maths. So that's our job. The first thing to do is to draw
some diagrams, so I'm going to show on here a angle. Here is the angle, they call it the angle subtended;
um most people have never heard that word unless they remember some circle theorem from school, but
it's the angle that the posts make to your point here. And you see that actually the measurement
here, if I'm really close to the try location, the angle's down to three degrees. That's a super
narrow angle, and if you're really far away it goes down to single digits as well but
somewhere in the middle I'm into double digits, 10.11 degrees angle - that's looking like a best
place. But where is the best place? And does it change when you move your try location around? Yes.
And I don't think it's obvious. So I- if you interview rugby players about where they do it
I am sure that they are solving this problem every day without doing a hefty amount of maths. But if
you got down to it, and this is now big business in sport, getting those margins optimised; even
if there's a place that you should take it from and then you adjust for your range and the wind
distance and how tired you're feeling it would be good knowledge and I think we should solve
it. Let me draw a rugby pitch; those are- this is my idealised goal posts and let's say we're working
here and you score try there, this is the line that you are allowed to take a kick from. And just
to clarify, let's say you take the kick from here then this angle is what we're trying to maximise.
Call it theta, let's call it theta. Really close in, the angle is going to be tiny; really far away
the angle gets smaller as well - there must be some optimal thing, how do we find it? Now you
could do this the long way. I recommend it as an exercise for any interested parties, you'd
need some calculus because we're optimising, we're maximising something, and it gets super messy so
don't do it unless you're brave. There's a better way - and this is quite common in mathematics, is
that sometimes you just need to draw a better picture. There's a few things you should know,
this is the distance between the posts, actually I think it's 5.6 metres. If you change the rules
of rugby though we can still work with d. And the distance we score the try at I'm going to call x
and that's to the middle of the post. So it feels like a good place to measure where you score the
try. So that's our variable x, and what we're trying to decide is this distance here, let's
call it y. - (How far should you pace back with the ball and take your kick.) - Yeah so mathematically
we're trying to find a function uh of x - so y equals something to do with x - practically you
said exactly the same thing, which is how far should you go back from where you put it down.
Any three points you put on paper, or anywhere else, you can draw a circle through, a unique circle
through three points. And there are three points we care about here, which is quite important: the
left post, the right post, and your kick position that you may or may not choose. Wherever you choose
there is a circle that goes through those three. There's a thing that people learn in schools about
circles, with a whole bunch of circle theorems. Circles have beautiful properties and one of
those properties is in play right here because this circle crosses the blue dotted line there,
that's why I went through that, but it also crosses here. And if I draw the other angle from there
we've got in the UK, a GCSE level circle theorem going on here, we've got a segment
um or rather a chord across a circle, the segments the other side of it, subtending an angle on the
edge of the circle. And the theorem says that angles subtended off the same chord are the same. Now this
is nothing to do with where you've kicked from, but it's showing us that if you chose to kick from
there you would get the same result. Which means whatever distance you choose there is another
place where you get the same angle. Maybe this is not a surprise; close in it's going to be
narrow, and really far away is going to be narrow. So you're seeing those pairings and the circle is
backing us up. But here's another observation: so if the circle is really big you can have a long
thin angle coming out here and the angle is going to be small; if the circle is small then the angle is
going to be bigger. What we want to do is minimise this circle, and you might begin to see from this
diagram how you could do that. It's easier to see if we make it move, so let me show you on on the
moving version here. Turn on the circle here - that's the same circle you can see that I drew and you
see that it's drawn both angles I've got. So let's make these angles even more separate, you can see
if the angle is really small the sort of pairing angle is further away. I mean I don't want to
drop the spoiler in here but maybe Brady you have an instinct about how to make that angle bigger? Like
what what would you want to be true about these two angles for that to be as big as possible?
(Well I feel like it should be the point where they're on top of each other, where the circle)
(touches the line, yeah there.) - Yeah bingo. I mean certainly if you look at the numbers we're at
nine degrees there, it's going down that way and it goes up back nine then it goes down again that way.
(There must be one angle that doesn't appear twice or that's the same angle to us?) - And it's exactly
what you said about touching, uh technical maths word, tangent, that where that line is a tangent
to the circle is going to be where the top two versions of the angle coincide and we've got the
biggest possible outcome. And that means we've got a really simple geometry problem to solve,
and that's it. So let's solve it. It's quite hard to actually draw this because it's not
obvious to me where the circle is that you said needs to just touch it, but it's got to be sort
of just touching here, so eyeballing it some- something like that? Sort of touching isn't it.
Yeah, it's close enough for Numberphile sketches. What- have I said this before on Numberphile? Maths is the
art of reasoning from dad drawings. So the angle here, the the point that we're after, is the angle
where there's the point where both of these angles have coincided. This has got to be just
touching which also means we get a- another circle theorem kicking in which mean that the radius
of the circle is at right angles to the tangent. That's another circle theorem you learn age
16 roughly in the UK. And there's one other fact which is now true that wasn't true before which
is that that distance x is now the radius of the circle. Which means any other line from the
centre to the edge is also a radius and also x so in particular that line there is x. And if I
draw one extra line- we mentioned before this this centre of the post is really kind of the
centre of everything so if that's on here you realise that this is a right angle triangle.
And we know some things: we know this distance is d over 2 - in actual practice it's 2.8 metres.
This distance here is y, it's the sort of parallel version here, and we know that part of the triangle
is now x. And that's crucial because we've got a right angle triangle, we know all three sides,
Pythagoras is waiting for us. I'm just going to write down Pythagoras' theorem on this triangle,
we've got x squared equals y squared plus d over 2 squared. And we kind of want it the other way
around, we want y in terms of x so that you put the try down, you know x, how far do you go with
y? You get y equals - I'm doing it in one step and the viewers may be horrified, I'm going to
take the this one over and square root everything. So it's the square root of x squared minus d over 2
squared. That is the formula which every rugby player when they score a try does not use.
(Just to be clear here; so the distance I walk back to take my kick is where I scored the try from)
(the centre line squared, minus half the width of the goals squared, all square rooted?) - Yeah
I think that's quite tough to do in your head, even if you memorise the width of the
goals. And and I am really keen to point out before anyone's like just knocks me down for
being stupid, nobody would ever really want to do that calculation. So the art of modeling things
is to get some sort of rule of thumb which is maybe a good way to get that calculation done
approximately. And I think approximately is all we're after here. The reason they don't use it,
and this is something which I haven't heard talked about lots although I'm sure people have noticed
it, is that there's a much quicker way to get very close to that answer. The art of modeling things
is to get some sort of rule of thumb which is maybe a good way to get that calculation done
approximately and I think approximately is all we're after here because there's going to be wind
and I'm going to be tired and kicking a ball; everything's going to change, the spin of the
ball in the air, all sorts of things are going to change it but we've got a starting point.
But let's have a quick look at what it looks like if I plot it. So on my diagram here I've got 'show
locus of best spot' by which I mean as I move the try location around, the best spot location -
which I'm going to turn on as well - so the best spot you can see actually- let's just move around
so you can see. Your earlier instincts were pretty good. That's our kick spot, the blue one
is marking the best spot. So- and you can see the red circle is the one that we sketched just now,
it's the tangent circle. But if I show you how that point moves as you drag the try location around;
unsurprisingly it's getting close to the goal as you get close, but then something
funky happens there that's hard to tell. So let's just trace this point which is this,
the yellow line is the locus of the best points. It is not a straight line, it has a funky curve here.
In fact this is a hyperbola, it's a rectangular hyperbola- - (It looks straight from here.) - It is
kind of straight, if I spin this round or maybe view from the top, you can see
if I zoom out that it has this this curve as you get close. But then it does look like
it's pretty straight, and there's a reason for that, because the hyperbola has an asymptote.
You probably recognise a hyperbola and I'm sure you don't need me to point out, but let's
point it out anyway, the graph of one over x looks like this. In fact there's another one
down here. It's that sort of shape we're after, it has asymptotes going on here. In fact the
one we're looking at is a little bit more like it's um heading off in that direction
as if that's the centre of the of the posts.
But the- the point is the hyperbola has an asymptote and it is a straight line
as you go in the long distance. And here's the catch, if you look back at the formula, if x is
big here then as you increase x this d just never increases and so you could maybe arrange for x to
be super big it would just dominate this thing - and basically it's irrelevant. If you cover that
up you've got y equals the square root of x squared - otherwise known as y equals x - which
is a 45 degree line. Which means if I go back to my diagram and sketch the 45 degree line
it's pretty much that line. In fact if I show the rule of thumb which is me just plotting a
45 degree line on here, this red line running down here is pretty much exactly the same. In
fact it's overlapping for most of the pitch, the only place where it doesn't overlap and
the curve kicks in is really close to the goal and my advice is you're too close to the goal
anyway at that point. So here's the rule of thumb, and I don't actually know if rugby players are
thinking this, but 45 degrees from the centre of the goal posts. Which means pacing, which
is what you said earlier, is dead simple. If you paced out 10 paces where you put the try down,
pace out 10 paces. There's a few catches though. Because modeling is- this is assuming uh all
sorts of things like that you can kick it far enough - and I think if you start to get into the
realms of like asking me to kick it, then maybe my actual range would start to mean that
if even if that's the best angle it'd be much better for me to be closer because I'm rubbish.
What's the wind doing? And there's a big kicker which we haven't pulled down, I'm not
going to do it, but we're in a 3-dimensional world and we've been doing it as if we're just
looking down flat and actually the angle we want to measure to the posts is not flat on the ground,
it's up a bit. - (Because the crossbar?) - Because the crossbar. You've got to clear the crossbar as well.
And I think that would change the analysis. I think it does- doesn't change the outcome very much but
this is what modeling does, you start to include the complexities; life gets a lot harder and you
maybe only make a tiny incremental gain in advice. And I think at this point the inaccuracy is out on
a pitch when you're tired and the wind's blowing means that 45 degree rule of thumb is a pretty
good way forward and uh that's what I'm going to do every time I play badly rugby from now on.
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...so as I just discussed, that this should be more than twice this. And so now we can compute the the
cross ratio for these four points, and for these four points, and I should get the same answer.
So that's as quick as it can happen, so say now there's a red card we could research a fact
about that red card, say it might be the player's eighth red card, only two players have got more...