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0:00
This is the game of Dobble, a card game for apparently negative six
0:06
players. I have no idea how that works, but I do know how the
0:12
game works and it's super interesting. So the way you play Dobble is you draw
0:19
cards from the deck. Give me a second.
0:24
Right, so you deal two cards at random. I've just picked two
0:31
arbitrary cards and you've got to spot which is the matching symbol
0:37
that's on both of them and you are correct. It is the anchor.
0:42
Not gonna lie, took me longer than I expected. But that's the challenge, right? You've
0:45
got to be able to quickly spot which icon is the same on both of
0:49
them and this time it's the dog. There you go. How's that? The question now
0:53
is, how do you make a set of cards such that any two cards you
0:59
pick are guaranteed to have one and only one symbol in
1:05
common? For example, the
1:11
hand. Before we start though, we have to rule out the bad ways
1:17
of doing this. So for example, if you wanted to make a version of this
1:21
game with seven, As I was saying, if you want to make
1:27
what I'm now calling in general a mono-matching game, because there's always exactly one match
1:33
between any two cards, you could do that some bad ways. So if you have
1:37
seven blank cards, you could just put the same symbol on all of them and
1:42
call it a day. That technically fulfills the requirements. I mean, you could pad it
1:46
out with other just distracting symbols, but all you've really got now
1:52
is a glorified version of Find the Atom, which to be honest isn't every spotting
1:58
a thing game just a version of Find the Atom and or atoms. The other
2:04
extreme is to just have a unique symbol for every single pair of
2:10
cards. And so seven choose two is 21. And so for each pair of cards,
2:15
you give it a unique one of 21 different symbols and technically that works. That
2:21
is a very nice mono matching game. If you did that for double there are
2:26
55 cards in a double deck so 55 choose 2 is 1485. You
2:31
would need about one and a half thousand different symbols. They actually do it with
2:36
57. So the question now becomes not just how can you make a
2:42
game like double it's how can you make a game like double well.
2:56
This video came about because my friend and fellow YouTuber, Steve Mould, challenged me to
3:01
explain the maths of Dobble. I challenge you to explain the maths of Dobble. See?
3:06
We're actually both nearing a million subscribers, and so we've got this reciprocal challenge thing
3:11
going on until we get to a million subs. So thank you very much to
3:15
Steve, and thanks to my sponsors for this video, KiwiCo. They make these kind of
3:21
construction kits where it inspires curiosity in young people and indeed
3:27
old people alike. So at the end of the video there'll be more about this
3:31
and you can see me try to build some headphones. Off you go. Slick!
3:38
Okay let's start by coming up with actually an optimal way to make a seven
3:43
card game of mono match. And optimal can be defined as uses the
3:49
minimum number of symbols but yet is the maximum amount of fun to play, which
3:54
means the symbols that match need to be sufficiently surprising or unpredictable for
4:00
each pair of cards. So we've got our seven cards and the bad way of
4:05
doing this was just to have individual links between every pair of cards and each
4:09
link, those lines, represents a symbol that will go on those cards, but like we
4:15
said before, we don't want to use 21 symbols. We can actually start to rearrange
4:19
the cards to use symbols on more than two cards. You can actually get
4:25
three cards to a line, and if you're very careful at rearranging them, you can
4:29
get it down to seven lines. So you can have seven card mono match using
4:35
only seven symbols. And the structure we have discovered is actually a mathematical object called
4:40
the phanoplane. There you go, plane, because it kind of comes out of geometry, meaning
4:45
a surface, sort of. And the phanoplane is actually an example of a
4:51
finite projective plane. And in terms of finite geometry, we mean kind of like discrete.
4:57
So there are only certain points where you can draw a line, which in our
5:01
case is where the cards are. There's only a finite number of cards. But because
5:06
we now know the maths involved, we can scale up to a bigger version of
5:11
a finite projective plane and see if we can crack the underlying logic behind double.
5:17
I'm going to start by laying out 49 of the cards in a 7x7 grid
5:23
which makes it a bit easier to see the structure of where the lines are.
5:27
So this top row here you can see they've all got that lightning bolt symbol.
5:33
We've then got a consistent symbol across every other single horizontal
5:39
row. We can then be nice and systematic with the columns and the diagonal. So
5:43
if you start for each of the ones in the top row and come directly
5:47
down, you will get all of the columns. You can then, instead of going directly
5:52
down, if you go one across, you'll get the diagonal. So there's the main diagonal
5:57
here. Every single one of those has got the car in it. If you start
6:01
on this card, you've got your main diagonal coming down here, the maple leaves. But
6:06
when you hit the edge, it wraps around and there's our bonus maple leaf. And
6:11
then likewise you've got the four leaf clovers coming down here and the last bit
6:14
over there. And so we have all the diagonals which are one across each time.
6:19
Then instead of just going down one across one, you go down one across two.
6:23
So there you go, carrot to carrot, down one across two, carrot, down one across
6:27
two, carrot, down one, wraps around across two, there's the carrot. And so we can
6:32
put in all of those parallel diagonal lines. And then likewise, across three, across four,
6:39
across five and across six. So if we count them we've got the horizontal
6:45
lines, we've then got seven straight downs or diagonals
6:51
which is a total of eight different directions which is why every card has eight
6:56
symbols on it and this would be a perfectly playable version of double but we're
7:02
not after perfectly playable we're after all optimal. And at the moment this is what's
7:07
called an affine plane where any two points have a line that joins them but
7:13
there's no vanishing points. If we want to go to the full projective plane we
7:18
can get some extra cards for free. If we took all the horizontal parallel lines
7:24
we could say that they meet at some distant vanishing point. So parallel lines converge
7:29
infinitely far away and this contains all the symbols from
7:35
those lines. Likewise we could do that going straight down. This is the card which
7:41
has all the symbols from the vertical lines. Then we can fill in all the
7:45
extra vanishing points for all the other directions. Technically on a projective plane all the
7:50
points at infinity are themselves on a line which is perfect for this game because
7:55
we do need one more symbol on all of these. So if you pick two
8:00
cards from the vanishing points they themselves have have a symbol in common and I've
8:04
deliberately picked those to all be the terrifying clown symbol. because I wanted that to
8:11
be as far away from me as possible. And so infinitely far away, mathematically speaking,
8:16
felt about right. So that's how double works. You've got your n squared grid, which
8:21
has n plus one different directions, because there's the horizontal and then n different ways
8:27
you can go down. And on top of that, you've got your n plus one
8:32
banishing points, which is why for double, you've got 49 here plus the extra eight,
8:35
57 Hang on a minute. Whoa, whoa. If you have a look on the back
8:41
of Double, there's actually only 55 cards in a
8:47
set. So while they have used this mathematical structure, this is the proper game of
8:52
Double, right? They've not actually used all of the cards. This card doesn't exist,
8:58
and this card doesn't exist. And by doesn't exist, I mean you don't get it
9:03
with your Double set. You've got to make it yourself. So I've added in the
9:08
extra doble cards. And so there's the extra doble missing
9:13
card there. And we've got, there's the extra doble missing card,
9:20
which has got this combination of symbols. And you can work that out because you
9:23
know all the lines that go through that point. So you just put all the
9:27
missing symbols together. Um, and there you are. So, actually, if you do own a
9:32
set of Double and you do make these extra two Doeful cards, you've increased the
9:38
total amount of Spotted by 3.6% for free! Just
9:44
make them yourself! For the record, I was not able to find out why on
9:49
earth they don't include those two extra cards. There are theories that it's to do
9:54
with how you manufacture cards, but I don't buy that. And... There's another version of
10:00
Dobble. There's a junior kids version of Dobble which uses a
10:06
5x5 grid to get a projective plane which means they could have 31 cards
10:12
for their six symbols but they only include 30 of them. So there you are.
10:17
For some reason they just don't like including all the cards. Now I need to
10:22
make it very clear that that system of using a grid to generate the projective
10:28
plane for this mono match game does work for the doubles we've seen so far.
10:33
However, it's not the only way to go about it. Let's say you wanted to
10:38
make a 21 card mono match game. So you start with your 4x4
10:45
grid, you've then got 17, 18, 19, 20 and 21 cards
10:52
if you include the vanishing points that are infinitely far away. The problem now is
10:58
If you start putting in your lines, and we'll start with the top left corner
11:03
here, you do your horizontal line, so you have the same symbol across all of
11:09
those. You then go and do, let's say, the straight down line, so
11:15
you do the same symbol on all of those, and the diagonal one. Same symbol
11:20
on all of those. You're thinking, oh this is going really well. This first card
11:25
filling in nicely. But then when you try to do the next line, which is
11:30
down one and across two, you go here. Then you go down one and across
11:35
two and it takes you off the side and you come back on here. And
11:39
you've now overlapped. Huh. that's not good. What this means is
11:45
this card and this card are going to have two different symbols that
11:51
match. They're going to have the yellow symbol and the green symbol in common. And
11:56
it's because our system of doing the horizontal line first and then doing all of
12:01
the diagonals simply doesn't generate a nice projective plane in this
12:07
situation. That only works if you're using a prime
12:13
numbered sized grid to generate your game, which happens to be the case of this
12:19
and, I'll get it, the kids' version of, um, Some quick side
12:24
facts, double is based on a grid of n equals seven, double junior a grid
12:29
of n equals five, and we call these numbers the order of the resulting finite
12:34
projective plane. What I was just trying to draw would have been a finite projective
12:38
plane of order n equals four, and while the technique I was using to
12:44
generate them does not work in this case, it is actually possible. Here's a picture
12:49
I found on Wolfram of a finite projective plane of of order.
12:55
The question now is for which orders can you actually get a finite projective
13:01
plane by whatever method? And it turns out you can definitely get one if your
13:06
order is a power of a prime number. And we currently don't know about other
13:12
cases. No one has ever found a finite projective plane which has an order which
13:17
is not a power of a prime, but no one has ever managed to prove
13:22
that they don't exist. It's one of the most important open questions in combinatorics
13:28
and indeed mono matching games. So there you are, some cutting-edge mathematics. So
13:34
just because the maths works in one specific situation doesn't mean it works in general
13:39
and actually it's not even the only way to understand the maths of Dobble. I
13:45
say the maths of Dobble like it's one set definite thing like there's the game
13:50
and and there's the maths of the game, but no, mathematics is just a way
13:54
of exploring and expressing and exploiting patterns and there's always more than one
14:00
way to describe something. So you can't just say this is a finite projective plane,
14:06
even though you can understand the logic behind it as a finite projective plane
14:12
there are other ways of doing it. So I first came across Double in 2012.
14:18
Someone called Martin Whitworth saw it in a small shop, bought it and they reverse
14:24
engineered the mathematics. This is before it blew up and became a massive game and
14:28
became renamed as Spotted in America. What on America? Say what you see there. This
14:34
is back when it was quite a niche game and Martin got obsessed with working
14:39
out how it functioned and they came and did a presentation at the 2012 Math
14:44
Jam Weekend which is where people get together and share cool bits of mathematics and
14:48
even then after they pointed out the projective plane stuff they said there's another way
14:53
to understand how Doppel works. So if you get all the cards
14:59
and you put them into a giant circle and then you look at the positions
15:04
of where one specific symbol is, you can arrange them in such a way
15:10
that they are at the positions 0, so the zeroth position right at the top,
15:14
1, 3, 13, 32, 36, 43 and 52. And those
15:20
positions have the interesting property that If you take the difference between any
15:26
two of them, you get a unique number that's different to all the other ones.
15:31
And in fact, you get every possible difference is in there somewhere, which has the
15:35
side effect that if you rotate those positions one position over, they will only
15:41
overlap with the original positions in one place. Rotate again, single overlap. Rotate again,
15:47
single overlap. In fact, for the other 56 positions, there will always be one and
15:52
only one overlap. So this is another way to generate the game.
15:58
You put your icons at those positions, you rotate it around for every single icon,
16:03
job done. Not a projective plane in sight. So maths isn't like the definitive
16:09
answer, it's just a way of understanding the logic behind it. So it turns out
16:14
double has the same underlying logic as a finite projective plane, which is the same
16:19
underlying logic as something called a cyclic difference set and it's
16:25
been around as a game for a lot longer than we thought. I say
16:31
game but you can judge for yourself. This is from the American Mathematical Monthly from
16:37
May 1906 in their Diophantine Analysis
16:42
section, problem number 142. Let n be an integer greater than 1 and set
16:48
p equal to n times n minus 1 plus one, etc, etc, etc, etc. And
16:54
they're setting up the same problem. They're basically saying, how would you design double? But
16:59
back then, no messing around. Now, it's all fun and games and I've got
17:05
hilarious props and all this stuff to make maths fun. Back then, no. Let
17:11
n be an integer. That's how you get people hooked. What an opening, my goodness.
17:17
This is exactly the same problem. They're basically asking people to come up with a
17:21
cyclical different set and it was solved a little bit later in May 1907.
17:27
Someone wrote in with the answer and there it is, there's the table. So the
17:30
case for eight, which is the equivalent to our 7x7 grid plus the 7 plus
17:36
the 1, which is what we got for double. Their numbering system is just
17:41
offset by 1 when they're looking at these groups and they're the values 0, 1,
17:46
3, 13, 32, 36, 43, 52. They cracked it! So exactly
17:52
the same bit of mathematics had already been discovered in a totally different situation. And
17:57
you don't even need any of that mathematics to make a mono match game. I
18:02
actually came across this game which has exactly the same gameplay
18:08
but completely different mathematics going on behind the scene. This is the earlier
18:15
example of a mono match game. Catch the match. Catch the Match is
18:20
absolutely a mono match game, the same as Double, but it works slightly differently. So
18:26
there's only 15 cards and all of the cards, they all have the same pictures.
18:31
So you can see every single card has the same things on it, but the
18:35
colours change. So to play Catch the Match, you've got to spot the
18:41
same thing on both cards with the same colouring. Now, I'd never come across this
18:47
game before. No one I spoke to has seen it before. To be fair, I
18:50
only spoke to mathematicians. But I was like, okay, I've got to work out how
18:55
this game functions. So I sent it to Martin. I figured they would enjoy looking
18:59
into another version of this. And they cracked it. So all of the different icons
19:03
have eight colorings each, one of which is unique on a single card, and the
19:09
other seven are on two cards. They're the ones. that match and if you arrange
19:14
the cards in a circle you can join up the ones that have the same
19:17
colouring for each of the different pictures and you get this fantastic kind of a
19:20
pattern if you overlay them all at once you get a complete graph you get
19:25
every single link so what they've done with catch the match is the version I
19:30
was very dismissive of earlier where you just have a unique picture for every single
19:36
pairing except they've hidden it by clumping them into just 15
19:43
different pictures and then each picture has different colorings but each coloring of each picture
19:49
is a unique pair. I'm not sure if that's genius or lazy. If step one
19:53
in maths is understand the pattern, step two is exploit the pattern or think how
19:59
can I mess around with this? And so that's what mathematicians did next. Someone called
20:04
Martin Harris realized that you could take the idea of your cyclic difference
20:10
set and write some code to find numbers with the slightly different property that instead
20:15
of every single difference appearing once, every single difference appeared twice. They were then able
20:22
to find such a set and generate Double-Double. So their version of Double-Double or
20:28
Duo Match, I guess in my terminology, every single pair of cards
20:34
has two cards that match. Amazing. I'll link to it below. Christian Lawson Perfect
20:40
realized that if you took the finite projective plane behind Classic Double, not only
20:46
are there 57 cards but there are 57 of the symbols
20:52
and on the projective plane you can flip between edges and nodes and so they
20:57
realized once you've got 57 double cards you could make another set of meta double
21:03
that has pictures of double cards on them and why stop there you can have
21:09
meta meta double where each of the 57 cards has eight of the meta double
21:14
cards which each feature eight double cards that all feature eight symbols and of course
21:19
Christian has made a website where you can go and generate these yourselves to any
21:23
level of recursion you want. It is DrosterDouble. Now with DoubleDouble and
21:29
DrosterDouble I thought I better get involved. I'm gonna make my own ridiculous version and
21:34
I'm gonna go get this very carefully for myself because you see I went big.
21:41
Behold my invention MonoMatch Myriad.
21:48
I took 10,303 symbols and generated 10,303
21:56
playing cards which each have 102 symbols on them and if you
22:02
take any two let's grab this one here car oh I've only printed out 101
22:10
because I thought any more than that would be bordering on ridiculous. But if you
22:13
take any two of these, you can try and spot the symbol which is the
22:19
same on both. There you are. I'll give you a moment. See if you can
22:22
spot it.
22:31
Did you spot it? It turns out it is. Let me see. Actually, I've gotten
22:35
pretty good at this game. I think it's... the happy little... oh it is, it's
22:41
a happy little house. So there's a happy little house there and there's a happy
22:45
little house... where... ah there it is down there, right? And so there you are,
22:50
that's the match. In fact if I go back to the previous shot, there it
22:53
is, right? And you might wonder how I got so good at that. Funny story.
22:56
So... I designed the 10,303
23:02
cards. I then picked 101 to send to my friend Lisa who actually got these
23:07
made for me and she did such an amazing job. I got them back, I
23:11
pulled them out and I spotted the happy little house and then I got out
23:15
another two and I had a close look at those and I spotted the happy
23:19
little house and I got out another two and spotted the happy little
23:25
house. Turns out when I picked the 101 cards to send to
23:31
get printed when I was writing my code to generate them I had to draw
23:36
them like without because all of them have like a line around them but they
23:40
didn't want to have a line on the ones that were being printed so I
23:42
just set a thing that the first 101 was saved twice with and without the
23:47
pentagon as the outline and the other the rest are all normal but of course
23:51
they were all from the top row of the grid the 101 by 101
23:58
grid that I used to generate these and guess what symbol is on every single
24:04
card from the top row it's the happy little house so I've accidentally made spot
24:10
the atom myriad edition it's so embarrassing
24:16
and I was so close to not admitting what I had done
24:22
but you know it's on brand So there you are, I've now got a 101
24:29
card version of Spot the Happy Little House. However, I did
24:35
genuinely make all 10,303 digital versions
24:41
of these. I'm just an idiot when I sent them to get printed. So
24:47
here's what I'm gonna do. I can't make them all available for download for ridiculous
24:53
licensing and whatnot issues. However, I can get them checked. So
24:58
first of all, I'm going to do a behind the scenes on how I actually
25:01
made these. It'll be on my second channel. You can go and watch that right
25:04
now. And I'm gonna send a unique set of three digitally
25:10
to every single person who supports me on Patreon. If you support me on Patreon
25:15
at the end of April, the very last day of April, the first day of
25:18
May, I'm gonna email a unique set of three to every single person so you
25:23
can check it and verify it. It's gonna be approved by crowdfunding. So,
25:31
make sure you're spotting me on Patreon, you'll get your own unique set of three
25:34
and you can go through and you can play, you can play Monomatch Myriad the
25:39
way the creator intended. You know I might just send some of these to Steve
25:45
Mould to thank him for the fantastic idea for this video but before that I'll
25:50
get it myself. I'm gonna thank the sponsor KiwiCo who make these construction kits
25:57
designed to inspire creativity in children of all ages. And friends of mine who have
26:02
procreated have raved about these. And I've got a lot of nieces and nephews I
26:07
want to encourage to take a kind of, you know, making tinkering
26:13
approach to life. And so I got a bunch of these kits sent out and
26:18
they loved them. They were building globes, they were designing patterns. One of them made
26:23
a pizza. Kids. But yeah, they had an amazing time and they were genuinely, they
26:28
were also raving about them. So there you are. If you've got some young people
26:32
that you want to encourage their creativity, get them building things, tinkering with things, you
26:38
can get these subscription boxes. And if you go to kiwico.com slash standupmaths,
26:44
I highly recommend them. I had a great time building a set of headphones that
26:50
I got to do. And I mean, this is right up my street. I love
26:53
it. But the point is, you can use this to trick young people into becoming
26:58
engineers. So, get onto it. KiwiCo.com slash StandUpMaths. Now,
27:04
I... You know what I am? I'm gonna send... Because these are almost as much
27:08
fun. I mean, spot the house. I mean, these are great, but they can't compete.
27:13
For spot the house. So I'm gonna send some of these to Steve and see
27:18
if he can sign off on if I've met his challenge. Do go
27:24
check out Steve Mould's channel and give him a subscribe so we can get to
27:28
a million. And while I was making this video, my friend Ben Sparks mentioned some
27:33
of his mass communication students at the University of Bath were also working on a
27:38
video about Dobble for one of their projects. I watched it, it's really good. They
27:43
cover some things I don't and when they talk about the projective plane, they use
27:47
the lines to represent the cards and the nodes represent the symbols and it still
27:52
works. So check out their video and if you've got any time left over, I
27:56
guess Steve as well. Hey, smiley house. Cool.
28:17
Smiley
28:17
house.
28:25
Smiley house, they're all Smiley house. I don't think so.
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