We're going to talk about sudoku
and in particular something called the Phistomefel Ring.
- (Brady: Phistomefel ring?) Well it's named after a German sudoku constructor
called Phistomefel who discovered it a couple of years ago and it's a property that exists
in all sudoku puzzles but not many people know about it and it's it's got a quite a
lovely proof and it might blow your mind. - (It might, all right. Is it)
(use- like is this like a a secret that will help people solve sudokus?)
- It can definitely help help you solve some sudokus and yeah maybe- shall I show you?
So maybe while I'm doing this I'll tell you- I'm sure everybody watching Numberphile knows
the rules of sudoku; the idea of sudoku is you have to put the numbers 1 to 9 just once each
into every row, every column, and into every 3x3 box. That is our sudoku. And what I'm going to show
you now is something called set equivalence theory. So we know that in this sudoku, if we complete
it correctly, there'll be the numbers 1 to 9 once each in every row, column, and box. So I'm now
going to highlight some cells in red: so I'm going to highlight row 3, row 7, this box this
whole box here, and box 6. And the interesting thing about those those four things I just
highlighted is because of the properties of sudoku we actually know- we don't know what the
disposition of the digits will be within those red cells but we do know exactly what those
red cells contain altogether. Because it's four complete sets of the digits 1 to 9; one for
this row, one for this row, one for this box, and one for this box. So altogether we know that the
red cells here contain four sets of the digits 1 to 9. Now I'm going to switch to green and
I'm going to highlight a different four sets of the digits 1 to 0. So I'm going to highlight
the whole - oh can we see that that's green? So I'm going to highlight the whole of column 1, the
whole of column 2, the whole of column 8 and the whole of column 9. Okay so now let's describe
the the the green set of digits, that's also four sets of the digits 1 to 9; one for each of
the columns. So at this point we know that the set of the green cells if you like, and the set
of the red cells, are identical - each equal to four sets of the digits 1 to 9. Now the trick to
understanding the Phistomefel ring is to focus on one of these cells that has two colours in it. So
let's look at this cell here and imagine that we removed it from both the green set and the red
set. So let's let's imagine that we we just take this out - I'll make it blue. So I'm removing this
from both the green set and the red set. What can we say about the remaining cells now in green
and red? And the thing we can say is that they still contain the same set of digits. We don't
know now exactly what that set of digits is, because we don't know what digit will fill in
row 3 column 1 in the final solution, but we do know that the red set and the green set are
still equivalent, they're still equal. And we can do that for every single- every single cell in
this puzzle that contains both colours. We can remove it from both the green set and the red set.
Let's now focus on what we've got left now. So we know at this point that the green and the red
set sets are still equivalent, and if we look at the red set you can see 16 cells here ringing
the central box of the sudoku. Now this is the Phistomefel ring, and we know that that is exactly
the same digits as appear in the four corners of the sudoku - these four 2x2s in the corners. And
you can try this on any sudoku you solve on your train home, fill it in and then ring this central
box and compare those digits to the corners 2x2s and you will find that this works. It works
for every sudoku and it's- I think it's quite magical. (One of the questions that pops into
my head is, could you make any other shapes or) (configurations on a grid that have this property?)
- Brady, you are brilliant. Yes, you can indeed um and in fact there's a famous Dutch constructor
called Aad van de Wetering who- he found another trick. He found that there's a square in the top left
of a sudoku and a square in the bottom right of the sudoku that have a relationship using exactly
this trick. And we've even seen puzzles now where you can use this feature to create the difference
of squares equation in sudoku form - it sounds completely crazy but you can see it on the screen
here. And you can create some mathematics out of the geometry of sudoku that actually can yield
to some really beautiful results. So- and it's very very much in in sort of really advanced sudoku
solving, this equivalence of different parts of the geometry of the grid is important and can be-
it can crack some fiendish puzzles. check the links on screen and below to see another video about that mathematical sudoku that Simon just mentioned. And to see Simon and his offsider Mark solving all
sorts of puzzles check out their amazing YouTube channel: Cracking The Cryptic.
Again links below.
Thanks for watching this video, and thank you to our Patreon supporters. You're seeing some of
their names on the screen at the moment and you can also see me signing these unique primes in
the post. I'll be mailing these out to most of our supporters later this month. What prime number
will you get? I might even send twin primes to some people...
