I fell in love with mathematics at
school when my
teacher explained to me that what this
subject is all about
is actually something he liked to call
'The Art of the Shortcut.'
So what I wanted to do was to give you
this perspective that my teacher gave me
about why mathematics is so amazing;
because it's basically
the best collection of shortcuts that
we've gathered over our kind of
two thousand, three thousand, years of
exploring mathematics.
I think another reason I fell in love
with maths is because I loved solving
puzzles,
the idea of having a puzzle - and
there's clearly a very stupid way to do
puzzles quite often -
but the real joy is finding that clever
way of thinking about the problem,
what I call the shortcut, and then the
whole thing just falls apart and you go
oh that's-
oh I understand this now.
So here's the little puzzle and and
maybe we'll solve this at the
the end of the video so you're a
grocer and you've got one of those
scales,
balance scales, so you can put things on
either side. And you want to be able to
weigh things from 1 kilogram up to 40
kilograms, just whole
number units. What is the smallest number
of weights
you can get away with? So you know what's
the shortcut for the grocer
in not having to buy, you know, weights
from 1 to 40. That's- obviously you could
do 40;
but what is the smallest number you can
get away with?
(Brady: So we have to be able to cover every)
(integer presumably not- not-)
(not 0.7?) - No I don't want 0.7. So I just-
you want from 1, 2,
3 all the way up to 40. And so remember
you can put
weights on either side; you're going to
be using that fact to sort of cut down
the number you need. Actually the
problem that caught my imagination when
I was at school,
and it's a problem- you've probably
done a Numberphile on it,
that moment when Gauss at school was
asked to add the numbers up from
1 to a 100. And the story goes is
the teacher who challenged
the class with this problem was hoping
to get a little bit of rest because he
knew this would take a long time;
1 plus 2 plus 3. And sure enough
all the other students
started doing their arithmetic. But
almost before
the teacher could finish with the
challenge Gauss had put down his
blackboard with the answer on. You know,
how did he manage to find a shortcut to
doing this huge amount of work of adding
up numbers?
And you know the lovely shortcut is he
spotted a pattern, which is if you add
the numbers
1 plus 100, 2 plus 99,
3 plus
98 - they're all 101 and you've
got 50 of these pairs and so
you've just got 50 lots of 101, 5050.
And I think this beautifully illustrated
for me the power of
mathematical thought as a shortcut. First
of all you you've just
cut out just having to do a huge slog of
hard work,
1 up to 100, but the beauty of
this shortcut again is
that it it works however big the number.
If the teacher had said count up to a
million
the same trick works. And I think for me
that was
just so exciting. I think it sort of
appealed to me as a lazy
teenager, that I didn't want to do a lot
of hard work,
the idea of learning this kind of skill set
of shortcuts. And that's actually what
I've been
doing over the last few months is
sort of gathering in a book
the idea of different shortcuts that we
have in mathematics. And I think one of
them is pattern searching which is the one
that Gauss was using. But there's a kind
of funny
paradox here, because although I think I
actually
was drawn to shortcuts because I was a
bit lazy
at school and I liked the idea that if I
thought cleverly I could get through to
the answer really quickly and then I
could go and play football;
actually in my professional life as a
mathematician
finding shortcuts, finding those
shortcuts,
can take a very long time. So you're
kind of stuck in front of this wall, you
just don't know what to do.
But that's actually the joy and it taps
into that idea of a puzzle, the best
puzzles are the one that you wrestle
with for ages and then
you get it. So I think the shortcuts that
we've come up with
throughout the history of mathematics,
one has to remember that
mathematicians have spent a long time
coming up with these shortcuts that now
we teach at school and university - we've
made the tunnel
through the mountain and that's what
I hope, you know, we
we communicate. But I think it's a sort
of new perspective, I don't think people
think of mathematics as kind of
like the art of the shortcut.
(I imagine a lot of the things I learned)
(at high school when I was learning)
(mathematics)
(that are considered, you know, the tools)
(of mathematics are in fact shortcuts.)
(They've just become so standardised that)
(we just think that's what mathematics is.)
Yes I mean- so I think, you know, algebra
is a shortcut to not having to do
individual calculations on each numbers,
it's spotting the underlying pattern. But
I think some of my
favourite shortcuts, ones we've made
Numberphiles about, but
one of my favourites is Euler's kind of
solution to why the bridges of
Königsberg
can't be crossed. So you know, we're
all brought up on this story as a kind
of mathematical
fairy tale. The seven bridges crossing
the river Preger;
the residents in Königsberg, Sunday
afternoon, they try and cross it.
And again here's the hard work; how would
you show that you can't
cross these? Well you've got to do all
the possibilities, that's going to take
forever. Well not forever but a long time.
The beauty of
Euler's shortcut is once he's drawn the
network and understood
that each node in this network of
bridges,
there can be either zero or two places
with an odd number of bridges coming out.
If there are two that's the beginning
and the end of the journey, but
everywhere else
you're going in and out on different
bridges so you've got an even number.
That's inspired because, you know, now I
can draw an incredibly complicated
network but using this shortcut all I
need to do is to count
the number of vertices with odd number
bridges coming out of it and I know whether
it is navigable or not. And it's an issue;
I mean Brady you you live near
Bristol so
you got a lot of bridges around
Bristol and in fact somebody's analysed,
I think it's in the region like 40-50
bridges.
Now that's quite a complicated network but
he was able to spot,
using this Euler shortcut, okay this is
possible.
And so he actually did it, I think it
took over a day to travel all of these bridges
but he knew at the outset that he'd be
he'd be able to do it. There's a nice
little coda to this story, because I
actually went to Königsberg,
it's now called Kaliningrad, it's a small
little enclave - or exclave I think it's-
because it's separate from Russia
surrounded by other countries. But I
actually discovered there are still
seven bridges but they're arranged
in a completely different configuration.
There's a railway bridge you can cross;
there's a dual carriageway - if you draw
the diagram now
there are two places where an odd number
of bridges
come in and out. So when I went to
Kaliningrad I actually walked
the seven bridges of Kaliningrad,
although the shortcut told me
I'd never be able to do the ones at
Königsberg. - (Are there lots of tourists)
(there doing that? Is that a thing you)
(do there or is that- was it just two or)
(three mathematicians a month?)
I think that I was probably the only
one
on that day scampering around
Kaliningrad, which is quite a grim city actually;
but I loved it, it was like visiting
mathematical Disneyland.
(Professor when you- when a big new proof)
(comes out - Fermat's last theorem or some of the big ones - )
(is a common thread through those things)
(that that the genius discovered a)
(shortcut? Or sometimes is the way to a proof)
(there was no shortcut; you had to- you)
(know really)
(push through?) - This is a really
interesting question because it came up
in my analysis of this. Sort of
sometimes actually you need to take a
detour
to sort of find your way through; and
sometimes that detour
will be as exciting as the shortcut. And
I don't think Fermat's last theorem-
had we found something that could fit in
the margin,
of course it wouldn't have had the same
kind of value
to us that it has today because it was
such a grand challenge.
But on the other hand if you look at
that journey,
a long journey that was made over 350 years,
absolutely there are moments of
shortcuts where suddenly you realise, oh,
I found a passage from this equation of
Fermat's through to a completely different land
that, you know, the modular forms. Now I
have to understand something about
elliptic curves
and modular forms. That was the ultimate
shortcut because it was almost like
finding a tunnel to a completely
different area. And I think in in my
research that's often what I'm looking for
is those sneaky tunnels which take me
from a problem
coached in one sort of language and then
turning it into
something else where actually there's
the tools to answer the problem.
(Is there a danger to shortcuts? If you- do)
(they become a crutch? And if you always)
(take the shortcut)
(could you miss that thing that no one)
(had seen before?) - Well for me I think the
danger of shortcuts is when does it
become cutting corners?
And I think this is a real- this is a
real
challenge because very often in
mathematics what we try and do is
throw away information which isn't
important. You know,
it wasn't important the geometry of
Königsberg; what was important was the
the way it was connected together. And
you know one of the powerful
shortcuts we come out with, not just in
mathematics but
also across the sciences, is the power of
a diagram.
If you think about physicists and
chemists their their blackboards are
covered in all these strange little
letters with lines coming out of them;
that's a molecule but
it's a diagram of a molecule. The DNA
spiral; you couldn't put all the detail
of the DNA but they
they they crystallised what was
important about this
double helix. So I think that is a really
powerful one and again
in a way networks are tapping into
topology is about connectivity, London
underground map
is not a physical map, you've thrown away
things. The challenge is making sure you don't
throw away too much. And that's
I think interesting when it comes to our
incredibly data rich world;
how do we navigate through this
incredible
jungle of numbers? And again mathematics
has come up with
some extraordinary shortcuts to to find
our way through
understanding what is happening in those
numbers. But again you need to be careful;
there's a chance that you might
have missed some signal inside of there.
There was this advert when I was a kid
about cat food, Whiskas cat food, and
it it kind of said
very grandly 8 out of 10 cats
prefer Whiskas.
And- and we had a cat and I don't
remember anybody ever coming around and
asking our cat
'oh so which cat food you eat sir?' And I
was always perplexed by this. So how many
cats did they actually ask?
There are about seven million cats in
the UK;
how many have you got to ask in order to
be able to make such a claim?
So that again is about how much can I
throw away
to really be justified in making that
claim? And again mathematics shows you that
provided your- you know 5%
error either way
we'll we'll we'll take that, but that
still isn't good enough because
well does that mean I've got to ask 95%
of that 7 million?
No but we also do this thing about- I
don't mind providing that 19 out of 20
times
I will be getting within 5%. If you
set that as your bar of your shortcut
you can get away with about 250 cats!
That's extraordinary, I mean that's the
power of a shortcut in when you're
trying to navigate data.
Suddenly I only have to talk to 250 cats
and I can know pretty accurately, you
know, 19 out of 20 times that will be
within 5% of what the actual value is.
(It's pretty amazing if you can talk to)
(even one cat to be honest but-)
I suppose you're right! - (When one hears the word)
(shortcut, one of the things that comes to)
(your head is being a bit)
(sloppy or finding a quick and dirty way)
(of doing things;)
(but we also know that mathematics is all)
(about this sort of- this rigour)
(and this perfection, this unbreakability.)
(Presumably when you use the word)
(shortcut)
(you're not using it the way I would, like)
(you know, hanging a picture a bit)
(wonky on a nail on a wall.) - Exactly I
think this is a really important point
to stress, that
I'm after something which still gets me
absolutely robustly to a solution.
I don't want something that's going to
fall to par- pieces
a little bit later. So so this is a
distinction between
shortcuts and cutting corners. But you
know there are times when it is about,
what is unnecessary work that I don't
need to do
which is not really contributing to
understanding, it's just creating a sort
of fog around here and I can
take that away. So a part of it is about
you know what can I throw away and still
I'm going to get to to the solution?
One of the rather extraordinary things
is that, you know, mathematics is very
good about talking about itself. I mean,
I've done a Numberphile about
Gödel's incompleteness theorem, the
power of mathematical equations to talk
about itself.
Mathematics is also able to analyse
itself
and tell us when there are no shortcuts.
So sometimes it's important to know, okay,
I can't get away
without doing a huge amount of work in
this particular case.
And so people have probably heard of the
Travelling Salesman problem.
This is a challenge about trying to
visit every
city in a network in the shortest
distance possible;
and we now think that there is no clever
algorithm
to being able to solve this problem and
that's one of the millennial problems.
One of our greatest challenges
is to prove there are no shortcuts. But
that's- mathematics is very good at that
in showing
why you cannot get away with doing
something quickly.
Unlike Euler's very clever trick for
Königsberg you don't have to navigate
the whole of the network to know
something's possible or impossible;
we think that in order to find the
shortest route the mathematics will
ultimately tell us
there is no shortcut. So this is
one called uh P versus NP. So P
actually for polynomial time, those are
the kind of shortcut
algorithms, those are the ones that when
you run them it doesn't matter how big
they get it still
finds you the solution quickly. So the
other problems
are ones where actually the
algorithms that we have
explode as the problem gets bigger. But
they have this quality that if you do
find the solution it's almost like a
needle in a haystack, you can't find it
but soon as you put your finger on it
wow you feel it. So so you can check
these things very quickly.
Traveling salesman problem is the
classic one but it's extraordinary how
many other problems
have this quality. And again here's a
shortcut; amazing
thing that we've discovered is that all
of these NP
problems, if you find a shortcut for one of them,
that shortcut works for all of them.
That's extraordinary so, you know,
one shortcut solves them all, sort of
kind of Tolkienesque.
So that's also, you know, the power of
showing- you know, once you solve one
problem you've solved loads of problems is
something mathematics loves.
(Is this unique to mathematics? Like does)
(does it work for chemistry and)
(physics and biology or is that- is it)
(more of a math-y thing?)
I was quite intrigued to see whether
other professions
actually had their own shortcuts and and
what sort of things-
you know, my life feels like I'm
trying always trying to find the
shortcut. What
what can you not get away with
shortcutting? So
so actually as part of this project I've
talked to a lot of people in other
professions; for example an international
cellist.
Can you get away from- I mean I play
the the cello here, you know, I was quite
intrigued can I
shortcut being able to become a really
great cellist playing those Bach suites?
So I talked to Natalie Clein, an
international cellist, and she really explained
actually well you're having to
physically change your body.
You know, as you practice you sort of-
you're trying to get that muscle memory
so you almost don't have to think.
That is ultimately a shortcut but that
requires a lot of time practicing. So I
think anything that requires
physically changing your body like
becoming an international cellist or an
athlete; it's very difficult to to
shortcut
that sort of physical change so that was interesting.
But there were other cases; for
example I talked to Brent Hoberman
who was the creator of lastminute.com
and he's done wonderful startups so I was
kind of intrigued, you know,
what's your secret? Do you have a
shortcut to startups? And
he said actually it's flying a little
bit close to the to the law;
you know, you want to break things a
little bit to- to make progress.
And actually that resonated with me with
another sort of shortcut,
which is the invention of imaginary numbers.
We kind of broke a rule there, which- you
know, squar- square numbers, they're
positive. But what an amazing shortcut
that provided and,
you know, there's a very famous quote
by the French
mathematician about, you know, the
shortest path between two real things
is to go through the complex domain. So
this is almost like a wormhole, very
often if you can
get complex numbers in there - radar for
example,
you could do it just with real numbers
but it's a huge mess.
As soon as you put it into complex
numbers the thing becomes very
straightforward.
In fact I went to visit air
traffic control
and and the people sitting there were
saying, you know, we wouldn't be able to
land these planes
in time if we hadn't used the shortcut
of breaking that rule that every number
is positive. So that was interesting,
there are kind of resonances with other
professions that are similar to to
shortcuts that I've learned
as a professional mathematician. - (I like)
(that, can't land planes without)
(complex numbers.)
I think we should solve the problem.
(Yes, go on then. There's a shortcut to solve this is there?)
There is a shortcut. Because if you try
and solve this problem,
it's quite nice because you can try out
various different numbers, you might try
powers of 2, you might try just
experimenting see what it's doing. I mean
it's interesting, I wonder how many
weights you would have got down to with
that kind of experimental
approach. Because it's quite surprising
what you can do is to get away with the
following weights: 1,
3, 9 and 27.
Four weights. I think a lot of people
will get down to five weights
but to actually see that you can go with
with four weights
is quite extraordinary. And why? Well
you've probably noticed those are powers
of 3,
so some of you might have tried
binary powers of 2, but you've got a
an extra thing you can take advantage
of here which is
these are balance scales so you can put
nothing on one side,
one weight on one side, or kind of the
negative of
that weight on the other side.
So in a way you're building a number, we
want to get the numbers from 1 to 40,
using either 0, 1, or -1. So we're actually
working with ternary numbers
and so powers of three are going to be
able to-
to weigh every number from 1 to 40.
if you enjoyed this video there's a fair
chance you'll enjoy Marcus' new book,
'Thinking Better, the Art of the Shortcut'.
It's already getting some good reviews
and I'm going to put a link to it in the
video description so you can get your
hands on a copy.
I'll also put a link to Marcus' recent
appearance on the Numberphile podcast.
...so he just took 101, times it by 50,
and he got 5050. So this story was
first written about in an obituary by
one of Carl Gauss's friends.