Let's forget all the pieces on the
chessboard except one; I've always wanted
to do this - oh yeah. We're going to focus on
the knight. For those unfamiliar with the
knight it moves in a weird way, it's like
an L-shape: two squares by one or one by
two. And it can do this in any direction
meaning that a knight here for example
can move to any of these squares. What I
want to discuss today is a puzzle called
the Knight's Tour, and it involves moving
a knight around a chessboard using that
move but landing on each square once and
only once. You start here, there-
Now this is a puzzle that's intrigued
mathematicians for centuries, including
none other than Leonard Euler. You could
try it yourself or if you can't be
bothered keep watching and you'll see
here the knight's tour is indeed possible;
and this one is one of the first ones
discovered. Here it is again but this
time I've numbered the moves.
You can follow it all the way through.
Now you can probably see, the final
square here, number 64, can be joined with
the opening square number 1 using one
of the knight's L-shaped moves. This means
this is sort of a loopable configuration
and we call this a closed tour.
Now here's another knight's tour:
As we watch you'll see when we get to the end
the endpoint here does not connect with
the start, and this is called an open
tour. Now closed or open you'd think this
would be something that's pretty rare
right? Well it's not. That closed tour,
the first one we showed, you there are 26
trillion different ways to do it on an 8
by 8 chessboard. The open tour, how many
of them do you think there are? Well it
turns out we don't know how many there
are, and that's pretty mind-boggling. But
if you think that's cool,
here's something even cooler. Now you
might recall from previous Numberphile
videos something called a Magic Square;
little boxes like this where each row
column and diagonal add up to the same
number. So can we perform a knight's tour
where those move numbers create a Magic
Square here on an 8 by 8 chessboard? Well,
have a look at this one:
Seems pretty normal. And here are the
numbers and have a look: each column here
adds up to 260, as does each of the rows.
And as if that's not enough have a look
at this: each of these quadrants, each of
the rows and columns in each of those
quadrants sums to 130, which is
interesting. And as if that's not enough
each of these 2x2 sections some to 130.
But alas this is not a truly magical
square, because the diagonals do not
add up to the required 260. This
square is what we call semi magical. So
for a long time this was bit of a
mystery, can you make a truly magical
knight's tour where the diagonals will also
give you the sums of the rows and the
columns? And the answer is - no you can't.
It actually took a lot of computing time
but it's been done, they've actually
crunched the numbers, done them all and
found out there is no totally magical
knight's tour that can be performed on an
8x8 chess board. But they did find
140 different semi-magical
tours where the rows and the columns
added up. There are some brilliant
articles and research papers about the
knight's tour, it's a really interesting
and active area of research in
mathematics and computer science - I'll
put some links in the video description.
Now while I start picking up these
pieces I will tell you we're going to
have some more videos about chess and
the numbers and mathematics behind it
coming very soon; so subscribe to
Numberphile, that's the best way to keep
track of it, or follow us on Facebook or
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the scenes.