Hi, everyone!
Recently, I made a quick video with my friend, Katie Steckles,
about a game with a winning strategy.
Well, actually, I think there's more to say about this game.
It's called Wythoff's Game,
and as a quick recap
you play it on a chessboard,
and there are two players.
You take it in turns
to move a piece.
And that piece can move any number of squares
to the left,
any number of squares down,
and any number of squares on a down-left diagonal.
And the winner is the one
who moves the piece
to the bottom left square.
And I asked you
"What are the losing squares?"
Well, let's take a look at this, because it actually
involves some surprising maths.
Let's start with a 3 x 3 grid,
and work our way backwards.
And we'll call the goal (0,0).
Then any square in direct line of sight
of the goal, we'll call a 'winning square.'
So you can reach the goal
from a winning square.
Any other square
we'll call a 'losing square.'
You can reach winning squares,
but you can't reach the goal itself.
So, in this coordinate system,
the losing squares are (1,2) and (2,1).
Now let's work backwards to a 6 x 6 grid.
Any square in direct line of sight of the goal,
or of a losing square, is a winning square.
That means you can move directly to the goal
or you can move your opponent
into a losing position.
If we carry on, we can do the full chessboard
and we see that the losing squares
are (1,2), (3,5), (4,7), and their reflections,
which means the two coordinates are swapped.
And if we carry on,
we can see that the losing squares
form two very definite lines.
And the gradient of those lines
are the Golden Ratio, and it's reciprocal.
WHAAAAAAAT???
Yeah.
If we just look at the losing squares above the main diagonal,
this is what we get.
Just remember that you get their reflections for free.
Now there are a couple of things to notice here.
The first thing to notice
is that for the nth losing square,
the difference in the coordinates is n.
So, for example,
for the third losing square
the coordinates are (4,7),
and the difference is 7 minus 4, and that's 3.
The second thing to notice is
every positive integer
appears once, and once only,
either as an X coordinate
or a Y coordinate.
And using those two rules
we can generate all the losing squares one at a time.
For example, if we want to work out what the fourth
losing square is,
we use the smallest positive integer
that hasn't appeared yet,
which is 6 in this case.
We know the difference in coordinates
has to be four,
so the fourth losing square
has coordinates (6,10).
So Willem Wythoff was a Dutch mathematician,
and what he did was to find a formula for the coordinates.
That means you don't have to keep generating
the losing squares one at a time.
He did this in 1907.
And the formula he found for the nth losing square
was this.
So here phi stands for the Golden Ratio,
which is 1.618.
And the first coordinate
is n times the Golden Ratio,
and then rounded down to the previous integer.
And the second coordinate is n times the Golden Ratio squared,
also rounded down to the previous integer.
What Wythoff showed was that
the Golden Ratio was the only number you could use
in this way that had the desired properties.
Those properties are
that each number appears once, and once only,
and the difference in the coordinates is n.
And that's it! But with just one more thing to say.
This version of the game that I showed you
with the chessboard
was actually invented much later,
and independently, by a different mathematician, called Rufus Isaacs.
Wythoff's description was actually very different.
He described the game
as having two piles of counters.
Let's say a red pile of counters,
and a blue pile, and you can have any number of counters in each pile,
and then two players take it in turns
to remove the counters.
They can either take as many red counters as they want,
they can take as many blue counters as they want,
or they can remove an equal number
of red and blue counters at the same time.
And the winner is the one who takes the last counter.
Some of you may recognize this as being
a lot like the game "Nim."
But this is equivalent to the game that I showed you.
And I will let you think about
why those two games are equivalent.
And so, for now,
if you have been,
thanks for watching.