244,432,188,000 games of Catan - Numberphile
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- เผยแพร่เมื่อ 4 ธ.ค. 2023
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It's always been about the cones
there was once a thing called 'Multicatan' online playable on their site and it was PERFECT even tho it was a Seafarer (only) variant without Knights&Cities or basegame-only feature, the ultimate holy grail would be finding the algorythm for generating the map in that...
The longest road you can build in Catan is 15 units long.
The mathematical reasoning behind this is that each player only gets 15 road pieces to build with...
Yeah. The road pieces are the icing on the cake of Catan. They are way more devastating than the robber.
Yes, it isn't that hard to determine, is it?
fascinating how they thought of everything when creating the game.
Funny... xD
It depends on the layout though. If one plays for road, one will strategize to place the towns where one will not be boxed in, I think that is what the lad meant.
"Why have we rolled 5 '12's already but no '7's?!" ~ Typical Catan game
Because your mom
To remove the bandit from the 12 everyone's built on, presumably.
I literally won the catan tournament with these exact rolls
Every time
Meanwhile whenever I play 3 out of the first 5 rolls are 7.
This guy is definitely part of the "hexagons are the bestagons" crew
Except when he keeps calling them "squares".
Squares with attitude!
They are the worst people. The type of weirdos to support BLM or wear two different earrings. Weak jaws usually.
shame, pentagons are so much better. ;P
@@chigginheadD They don't tessellate.
Catan also normalized the phrase "I have wood for sheep."
i believe Scotland normalized that centuries ago
@@tiagopaim3060you’re thinking of the Middle East bud
Does ANYBODY have wood for me?
@@tiagopaim3060 You're thinking of the Welsh!
The best part is that this made it into a skit on episode of "The Big Band Theory".
I love how this guy explained the rules as if there is a single numberphile fan who has not played Catan
* _looks around, blushes, raises hand_ *
@HasekuraIsuna * virtually lends you game and three friend npc's *
I hadn't played it until a couple of months ago.
I actually haven't played it. Mostly a lack of opportunity tbh, I don't have many friends.
@@fdagpigj Don't worry, it breaks friendship worse than Monopoly.
I think the thing I love most about Catan is how the incentives shift throughout. Early on you're incentivised to trade and play nicely to gain points, but by the end your incentives are to try and screw over the leading player and it just leads to a wonderfully satisfying system.
Never played, how do you win? never mentions in the video 🙂
@@SK1BBEit’s a number of points someone gets by doing things with resources. Building structures and having the longest road and largest army will give you points.
@@seanmichaelbradySet before game by the players or is it the same number of points every time set by the rule book?
@@SK1BBEit's set by the rules, but you can play by house rules if you want.
You just described monopoly.
One thing I love about Catan is that everyone will call the resource different things. Even though there are official names in the rules.
They changed what they are called in the rules many years ago because no one liked the names that used to be in there. So there are multiple official names for resources.
Heh, "sheep" vs. "wool" and "ore" vs. "iron"...
It's obviously sheep, wood, wheat, brick and rock.
Not rock, ore!
We always called it stick, mud, sheep, wheat, rock.
This game taught me that not all results on two 6-sided die are equal and that opened my eyes for a lot of mathematics, so I am very thankful for it. RIP Klaus Teuber, one of the very greatest.
I bought a 12-sided die, so there is less chance of rolling 7. But we did not like that, because it could last a very long time before one was able to free their occupied hexagon. 2 dice is better.
@@coreC..so you leveled the chances for every resource to 1/12th and was surprised it got even duller? 😅
See also: The game _Can't Stop_.
@@coreC..You literally eliminated one of the most important aspects of the game by using a 12 sided dice💀 let me guess, you‘re an american?
@@anshxyz. It was only for trying out such a die. We played once with it. After one game we found out immediately how that plays.
And i only wanted to make some comment about such a die, because it exists.
We play with 2 dice, always.
At least we tried out the 12-sided die. Have You?
I am no american, but i am still a victim of their politics.
Very bold to call it the best board game ever. I’m sure that’s not going to ignite any kind of discussions at all
Good thing YT comments are always calm and well-reasoned, eh?
He also added *"I think, possibly* the best game ever".
Defo gonna have to disagree on that one.
It is the best board game ever, trust me bro
whaddya think is a better board game?
I knew that the dots represented how likely a certain number is to be rolled, but somehow I never realized that it's literally just the number of ways you can roll it!
A close friend has a saying: for each 1000 Catan persons that we introduce to board gaming, we get at least 1 Twilight Imperium IV player. Welcome to the hobby.
I didn’t climb all the way to Twilight Imperium, but I went on to love board games, with Root settling on as my favorite of all time.
Catan is for sure a great gateway drug into boardgaming. Scythe is probably my favourite right now.
I remember when Catan was my favorite game. Now it's next on the chopping block when I run out of shelf space in a few weeks.
@@knaughta.masso-kist5944 never played scythe, however i have looked at the artwork and the vibe is _impeccable_
@@linkypete I would take it if you‘re throwing it away😅
Who'd have guessed Brady would make the Catan video before Grey?
We don't just shuffle and place tiles blindly but the numbers and ports too. We even had some extra numbers from expansions that we sometimes took out 2 and 12 and player who rolled it would either get 1 of any resource or a dev card. I also appreciate the peices being made of wood instead of plastic
One property all hexagonal numbers have is that they’re the difference of 2 consecutive cubes. Using that, we can write x^3 - (x - 1)^3, and expand the terms algebraically to eventually get 3x^2 - 3x + 1
So, H(n) = 3n^2 - 3n + 1
Not sure how I could have solved it, but I put on the first numbers in the OEIS and got the formula lol😅
I concur.
Each hexagon "ring" of size n for n>1, has 6 sides of length n, but then we counted the 6 corners twice, so each "ring" has 6n-6 = 6*(n-1) hexagons. Which is 6,12,18,24,30,... for n=2,3,4,5,6... (and n=1 has 1 hexagon).
So we can get the total number as the sum of each "ring" up to n and using the formula: 1+2+3+4+5+...+x = x*(x+1)/2:
Total # of hexagons: 1 + 6*1 + 6*2 + 6*3 + .... +6*(n-1) = 1 + 6*(1+2+3+...+(n-1)) = 1 + 6*(n-1)*n/2 = 3*n*(n-1) + 1
That's numberwang!
Now you're just making stuff up
I don’t think the combinatorics of the numbering pattern were correctly described. The number tiles are indeed laid out in a spiral, but the random location of the desert (which is skipped) creates a “glitch” that considerably increases the possible numberings. In the animation showing the (supposed) six numberings, you can see that most of them have a non-desert hex without a number.
Edit: Actually, now I think I'm wrong. Each of the possible permutations of hexes places the desert in some location, and _given_ that placement, there are only six possible disc spirals (following the standard placement rules). I was distracted by the fact that the animation doesn't shift the numbers according to the desert's location. While there are indeed more than six numbering patterns - 6*19, I believe, discounting symmetry (one for each desert position along the spiral) - that extra factor of 19 is already accounted for in the oringal calculation of hex permutations.
@@aytcs
Not completely at random. The red numbers can't be placed directly next to each other. But yeah, there are way more number layouts than he suggested.
I think the location of the numbers don't matter at all. Take a look at a board with only 2 different regions.
You can have 2 games on them.
Now have one field heads and the other tails. Can you still switch them and create more games this way??
@@aytcs the tiles are already at random. What you want is multiply by 19 factorial and divide by 19 factorial again. That is why i already included the 2 tile board with 2 numbers as example. And you can expand this example to 3, 4, 5 all the waybup to 19 tiles and numbers.
@@aytcs can you give an example of what you mean???
@@aytcs ah, it seems you don't understand completely either what I meant.
If they shuffle the tiles, that is the same as shuffling the numbers.
But are you perhaps referring to assymetrical effects? If they occur, 'where and when'? Not just a "they do happen".
How many tiles do we need for that?
What are the assymetrical combinations of the tiles and numbers combined?
If you say they can happen, would love to see it happen too.
Some friends and I did wonder what would happen if we introduced a toss-a-coin element into chess. So what we would do is we would name two moves we wanted to play and depending on whether the coin went heads or tails we had to play the move assigned to that side of the coin. It was actually really interesting to have to adapt your strategy to incorporate the smallest element of chance, and I would recommend it for the curious to try out as well.
I've tried using dice as a countdown. Basically, move a piece and then roll a d4 for pawns, d6 for knights/bishops, d8 for rooks and d12 for queen. The number that comes up is the number of turns you have to wait before being able to move that piece again. Creates very interesting gameplay at the beginning. But at the end, the game slows down to crawl (because all you do is move the King until the other pieces cool down). You can fix that by removing the cooldown limit on pawns and dropping all other cooldowns down by one dice
Wouldn't the number of possible games also be reduced by the symmetry of the board? If I just flip the board, I have essentially the same game, but the number of possible board states would be halved.
The numbers on the tiles.
If you pause at 8:36, you'll notice that the borders of the map (with the seas and the ports) are not symmetrical. For instance, the central column has a port at the bottom and nothing at the top. At least, no axial symmetry. I can't tell for sure about a radial symmetry.
@@Krokrodyl There is a symmetry for 60° rotations :D
@@julienl.5045 Not when you account for the different ports around the board.
@@julienl.5045 there isn’t, because the 2:1 ports all trade for different resources. So which resources you can get on those ports makes a huge difference to the game
i love catan, i've played it a lot with my family. i never knew that the numbers were in fixed places? we randomized them as well as the board. that probably adds a lot to the number of "games" in a box of catan.
The base game rules give two setup options, one with fixed numbers placed in a spiral order (this is why the number tokens have letters on the back, to be placed in alphabetical order), and a “variant for experience players” where you randomize all the number tokens, except that you can’t have two red numbers touching.
@@anora8973 We thanks the manufacturer for making the game, but we will take it from here. Random everything let's go!!!
Loved how Magnus played his troll opening with the dice example
I also love the house rules that come from this. Like our group plays “no 7s in the first turn”. Or, the placement of numbers dont follow the rules. Even further, ports and such. So many options
Another amazing video CPGgrey! Glad that Catan video finnaly sees the light of day 😊
Nice joke, Tim!
I call my go to strategy corn lording (I first thought wheat was corn), you play to only collect corn and hog all the corn spots to corner the market when everyone starts building cities. It rarely works but when it does it’s hilarious.
CORNering the market!
@@Alkoluegenial Out! 👉
I havent laughed so hard all day 😂
All hail the Cornlord! The dawn of corn! May he corner us all!
Hence the saying "no wheat = defeat"
my brother does this, to everyone's frustration lol. he calls it the wheat king strat
I have been playing board games for decades with thousands of players and I just realized I have never ever heard someone refer to Catan as the greatest game. This man is a unicorn
Yeah I don't think anyone actually thinks that lol
It’s an outstanding board game.
A better description would be that Colon of Catan is one of the initiator of the modern era of board games
It goes without saying
There's a 4 player variant called 'hand and brain' where on each side you have one person choose the piece and the other person choose where to move it. This would obviously not be as bad as dice choosing the piece but that can give you an idea of how big of a handicap it is when there's a weak player choosing the pieces and a strong player choosing where to move it.
Swap the roles, let the stronger player pick the peace, it should be somewhat saner, but the stronger player will have to avoid pieces where the poor player will guess the wrong move, so perhaps going for the second best move will be the better option. E.g. knight to c3 is the "best" move, but the "hand" is just as likely to move to a3, so maybe pawn at c2 or d2, because pushing either is not a bad idea? Or start with pawn at a2, to potentially block a3 as an option, before going for the knight.
My parents bought this game decades ago, we didn't have the nice puzzle-like outer ring, our ocean was also made up of hexagons. Also, I believe that the one desert tile has to be in the center. Our number plates also don't have the dots on them, we had to figure the probabilities out ourselves.
I don't know if it gets mentioned, but at the start, when you place you 2 free houses, the number plates are upside down, so you don't know what numbers you'll get.
That last rule sounds like something your family made up. That really simplifies much of the strategy in the start and leaves more to chance.
@Benjamin-mq6hu
Well, our version has capital letters on the backside of the numbers, so you could learn them.
Edit: I've just looked at the old book of rules. The letters on the backside are there to lay them alphabetically in a spiral. After that, they are turned over, and then the houses get placed.
We've always played it wrong, for over 25 years.
@@Matty0311MMSeh, there isnt a wrong way to play
Ours indicates probability by the number's font size and a red colour for the 6s.
@Frrk
Also for the 8s (at least in ours).
Calling ANY board game "the best" will get you very angry stares from any game designer. XD
However, I loved how the speaker dodged at the last second to go on record saying "I need wood" and went for "I need... Logs". Smart move, the internet is a bad place.
one of the classic moves in catan, long before the internet took hold, was to need wood for sheep.
@@kenzieduckmoo : exactly. It's such a classic that was even used in an episode of The Big Bang Theory, of course with Sheldon not getting why everyone was laughing at his honest request.
@@kenzieduckmoo Or to _have_ wood for sheep.
@@MattiaBulgarelli Let's not forget "Cones of Dunshire" from Parks and Rec.
Marcus didn't account for the rotational symmetry of the board game. If you rotate the game board by a 1/6th turn, that doesn't really affect the game, since the relative positions of the tiles are the same. However, the way Marcus counts the different games counts each of the 6 possible 1/6th turns as a different game.
The final total should be 1/6th of the reported total.
@@beanshadow008 I thought that in the beginning too, but the available ports are not invariant to rotation.
Settlers of Catan was the darling of the board game connoisseurs twenty years ago but its design shows its age and has dropped a lot in ranking. Trouble is, it has exponential growth and if the dice don't give you resources early, you can't build or trade and won't catch up. You might be better off with Ticket to Ride, Bohnanza, Space Base, Azul, or Wingspan. Throw Monopoly in the bin as an archaic relic.
Hmmm.... kind of like real life, then.
You can always score points (and do some other actions like taking all resource cards of one type from every player, moving the bandit to a different spot, etc.) by buying development cards if you have enough resource cards and are stuck with only a few settlements/cities. Ticket to ride is a fine but simpler game than Catan and players don't talk as much. Never played the others but would love to try Wingspan.
My biggest issue with Catan is that it's too luck based. Any game the requires a dice roll every turn is going to have a heavy element of luck.
Absolutely. A great game of its time, but one that has not really aged particularly good. All around, no matter if aestherics, design, mechanics, there are tons of games that do pretty much everything better than Catan does. Having dice ross is fine but there is just so little player agency regarding how you can change outcomes in your favour. Personally there is just no board game that comes close to the variety and addictiveness of Dominion.
Just one disagreement: Monopoly is a disease made boardgame. And noone should play it except as a lesson on how to not design a board game. Which... kinda... was the idea of Monopoly in the first place.
I haven't player The Settlers for years. There are simply better games nowadays.
"I've got a lot of sheep, but I need a lot of...of...of...logs."
That, ladies and gentlemen, is somebody knowing he is on camera.
The main problem with Catan is that you can end up in a situation where there is an hour left to play but there is no way for you to win (due to how you got blocked/positioned).
But an entire game should only take 45 mins…
@@cdub9923 Its a 60 to 120min game on the box, in reality its at least 120min often closer to 3 hours. Because people talk, laugh, drink, and dont make their moves instantaneously, most dont think about their own turn on other people's turn.
Then you need to practice your first settlements positioning, or table talk so you don't get plowed.
i just got recommended your 301 views video and i am incredibly impressed by the fact that you are still uploading to this day.
Love this game. Love this channel. Perfect combo.
What an absolute beauty of a game this is. It's a shame I've had no one that had any interest in this to play at home...
Doesn't the number of boards need to be divided by 6 to account for rotational symmetry? I can create a board and rotate it by 60 degrees to produce another 'board' that would be counted as a separate permutation under the method given, but still would not change the gameplay configuration.
Correct, though with how the numbers are there, you still end up with 6 different boards. However multiplying by 6 is definitely wrong because all 6 rotations are already included in the 19! possibilities. Weird mistake for a mathematician to make.
@@viperhd70 I suspect you haven't considered the location of the ports in your comments. If the ports don't move, then there is no 6-way rotational symmetry and rotating the numbers around does actually change the game and increase the combinations. Perhaps the mathematicians were correct after all.
It may not appeal to mathematicians, but my kids and I have found we like making custom Catan boards and rules. Sometimes I'll set up the board more like an organic island, with coves, inlets and peninsulas. Then we randomly put the numbers on the board, just making sure that the good and bad numbers are well spread out and that no 2 identical numbers are touching. Then you place down the ports, deliberately placing ports on the best number for their associated resource. Starting rule change: you have to start on the coast, so no one is going to be able to have more than 4 different resources, but most, if not all players are going to start with a good port.
We have a couple of variations, one using a 12-sided die.
I am the logical one, and my wife is so not logical. But dang, is she lucky. My daughter and I actually think she controls the dice! In any case, it works out to about even. We play a lot of games over the winter.
I would like to know more about your customizations.
@@fk319fk What do you do if you roll a 1 using your 12-sided die? There's no 1 on the board.
Our version has hexes for the water too, and we mixed them in as well. Really changed how we thought of resource production and space
When you factor in how the numbers can be spread over the board and how the ports are laid out, you get a much bigger number of possible games. I wish they would have calculated that.
The numbers are all set in a predefined order, so the order they appear cannot change if that's what you mean. Other than the desert piece which offsets the numbering
@@irok1 I distinctively remember the numbers having letters on their backs that had to do with how the numbers could be spread out and it was not only one way.
@@irok1I always put them around randomly. Same with the ports and hexes. It makes the games more unique, although sometimes but rarely worse balanced.
You can start at any coastline tile then place them in an alphabetical spiral. So there are 11 options for each of the 12 cases with a desert on the coast and 12 options for each of the 7 cases with a desert on the interior. But some of those are equivalent to others (for example, if the desert is in the center, there are only 2 unique placements because each side is equivalent-- but if you consider ports I think it is 4 unique placements ). Far fewer than with the random number placement rules (red numbers can't touch--6, 8)
@@gavinorth7294 same, we always just threw them out on see where the chips laid. maybe move some of the 8s and 6s around to not make TOO OP a placement
I feel this is going to summon CGP Grey to try and make his Catan video again
Its not just about getting the maximum dots though. Some resources are more valuable for certrain strategies, so it may be worth it to take a probability hit to gain a key resource.
I remember rushing for cities to be a very powerful strategy.
They neglected to mention, the way you win Catan is by being the first player to earn 10 points. Each settlement is worth 1 point, each city (upgraded settlement) is worth 2 points. Longest Road is worth 2 points, biggest army (collecting the most Knight cards) is worth 2 points and there are some development cards worth victory points.
6:32
The formula for the total number of exagons on a Catan board with n layers is 1+6*(Sn)
Where Sn = n(n+1)/2 (which is the sum of every integer from 1 to n)
With the central square being layer 0 and each next one being one more
So for example the third layer, which is not calculated in the video, would be:
1+6*(3(3+1)/2) = 1+6*(12/2) = 1+6*6 = 1+36 = 37
The next one would be 61, then 91, and so on, with each gap being the next multiple of 6
It's the first time I ever read about this kind of sum and I had to just to get this to make sense because I was trying stuff for a while and nothing worked after the third layer :P
Always a great time to learn!
I worked this out today as well. n(n+1)/2 is the triangular number formula so there's six triangles spiraling off from a central cell.
That’s why if you buy two games you have enough pieces to play with a third layer. 19x2=38. Don’t use the second desert. Four is hard though you can always make your own pieces but to play with five layers you need to buy five game sets, you’ll end up with the same amount again. (4/5 deserts unused)
Taking into account the 6 different orientations of the board is not enough. The board is also symmetrical and can be mirrored on 6 different axis (3 diagonals and 3 laying on edges). Also you can mirror on combinations of those symmetries. So the amount of possible games is even less than that.
But then you dismiss the different ports around the board.
Until you multiply by the various ways the numbers can be arranged...
I can relate so much to the sentiment of playing games:
-hating chance, where chance is almost the sole determinant of the outcome: basically wasting time.
-as a kid I had some games people didn’t like to play with me, because I used good strategies.
One of my favourite memories is playing Nine Men’s Morris with my Grandma.
If a player only has 3 pieces left, he can jump with his pieces, but I boxed her with 4 pieces in so she couldn’t make a legal move. That was more fun, then to deal with jumping pieces.
There was a very high probability to achieve that outcome. -That was fun (for me).
No matter what the subject is, I always get some sort of eureka moment from Numberphile videos. Love love love that. Today for some reason it finally clicked why you divide by 3! to get rid of the %age of permutations that are technically the same when you have 3 of the same tiles.
Important to note that a lot of people also randomly generate the layout of numbers, instead of stacking them counter-clockwise. This will also add a bunch of new games.
I literally never realized that there WAS an official numbering order for anything but the starter learning game that they give in the book.
We played that game for longer than 15 years and never once we used any official layout. We are stacking them counter clockwise... but on their white side, so they are randomized. Sometimes we even do "fog of war" games, where numbers STAY hidden until you get a road near them. That makes for a very very interesting game where you have to strategize on the fly.
@@GeneralPenemonto Yeah we never use the official stacking either, but I know it exists.
Interestingly, in another version of Catan where you are in space, they do actually use a Fog of War thing so if you pass by a star system you can check the numbers, and if you don't like them you can fly by. That Fog of War idea sounds pretty cool! I might snag that from you next time I play...
12:27 Magnus would absolutely obliterate you, me, and most people in general.
Maybe, but youve got to consider, if you take one of his defended pieces, he cant recapture unless allowed to do so with the dice. Id still favoir magnus, but its not as easy as it sounds
Yeah I feel like the best strategy would be to play super aggressively. Not much a grandmaster can do once you got all their pieces.
@@toadounetlovesyou I think the exact idea of a GM would be to trade of as many pieces as soon as possible with the idea of reaching a king and pawn(s) endgame, which you're losing every single time
@@matejstankovic9843 If the relevant super GM strategized, they'd probably come up with that plan and if they did, their opponent (you/me/etc) would lose every time as you said.
I definitely think Magnus would win against DrHaran_Oam if he's rated 137. If the rules allow for skipped turns or illegal moves though then anyone should be able to win with the simple strategy of attacking the king and a little luck.
With n layers, there is a very simple observation for the amount of fields placeable in a hexagonal shape. A hexagon has 6 sides onto which to lay your other hexagons. Now, if every hexagon has a neighbour left and right, every hexagon has exactly one independent edge and 1 edge that shares a corner between the neighbouring hexagon to the outside. This means: For every new layer we get exactly the previous amount of hexagons of the previous layers (shared corners) and 6 additional hexagons at the independent edges. So on every level, we get the previous amount of hexagons + 6. The only thing to do here is to just sum up for all the layers. Like this, we get the summation: 1+ 6+12+18+24+30+36... total hexagons. We start at 1 hexagon in the middle, so 1+ Sum from k = 0 to n layers k*6, which is equal to: 3n^2 + 3 n + 1. The reformulation from the Sum expression to the polynom is for you. :D
244,432,188,000 is only for the hexagons with the terrain and as someone else pointed out - there are other symmetries to the board so only considering the hexagons, I think the number is less (I might be wrong).
BUT...
considering that the game also has ports, each terrain has even more variants depending on how the ports are arranged.
And here it gets wild:
Some game variants have letters on the back of the numbers, so you don`t play with the same number order every time.
You choose different letters, order them from A to Z and you put them on the board either starting from the edge or from the center(just like you fill the hexagons).
And considering that there are more letters than hexagons then you can choose just some of the letters, not all.
Also, there are multiple choices for one letter - eg: you can choose A, Aa, or Ab and so on.
So yeah, you have way way way better chances to win the lottery than to setup the same Catan game twice :)
As a competitive catan player and TH-camr, I LOVE to see this. Thanks for sharing this beautiful game.
Hey Drew.
Dandy Crew woop woop 🎉
The man i was looking for :D
Originally I thought he simulated this number of games to finally tell us the best strategy
This was so fun, I hope you have Marcus do a LOT MORE games!
Thank you! Where does playing chess rank in ALL the games Marcus plays??
That's only counting the board configuration for 3-4 players, but there are extra hexes to form a bigger board for 5-6 players, and IIRC one smaller pattern for a 2 player game as well.
Catan for 5-6 is horrible, Teuber knew that, but the publisher has to milk the cow every way possible, so they released that.
I haven't played Catan in ages. But IIRC, some of the expansions have you add tiles on your turn, or turn over tiles that have been placed in advance upside down (which is equivalent, really, assuming the area for adding the tiles has been pre-determined). Also, expansions add water tiles and boats as well as roads. And the expansions have you sometimes playing on more rectangular-ish board shapes, not always on big hexagons.
So we made a sort of mega-game with our house rules and our (or at least, my) favorite way to play used to be to place ALL the tiles into a big bag and then on your turn, you would first draw a tile and then place it anywhere adjacent to the existing board. Then you would draw a number and add it to the tile, with certain restrictions (you can't place red numbers adjacent to each other -- those are the most common numbers -- and I think we might have also not allowed the LEAST common numbers to be adjacent to each other). If you created a port opportunity (placed water next to land or land next to water), then you would roll a die to determine whether or not a port would go there (I think it was a 1/3 chance), and if you got a port, one was drawn at random.
I think we experimented with different ways to start (start with a certain number of tiles already placed at random, place starting properties on those tiles as per usual rules, and go from there... or start with just the desert tile, wait until you have a pre-determined number of tiles placed, and then stop and do property placement... or you have the option of placing one of your starting 2 properties on the tile you've just placed on any turn, one per turn, but you must place both of those properties by the end of your 5th turn (at which point, there will be about as many tiles on the board as a normal starting board would have, assuming a 4-player game).
I like the last option best, I think. It allows more strategizing. And I always loved seeing how the worlds developed and the shapes they took when players were allowed to place tiles however they liked, as they went along. You might wind up with one or two main continental clusters, and other places where there are long strings, or even loops with holes in the middle (undiscovered territories). The possibilities are truly infinite, played this way.
I love how he goes on about how they're hexagons, not squares. Then, 5 min later, he asks how many squares there are.
Another of the considerations is what the resources are used for, as the board becomes more developed the emphasis is more on building cities for some strategies than it is on roads and villages, or hoarding cards.
Anyway, it can get pretty competitive... Monopoly has a reputation for starting fights but even that has more of a dice-made-me-do-it than Catan has :)
Thanks for mentioning the game of go at 11:03!
The number is MUCH, MUCH higher, because the players can place the 18 NUMBER CIRCLES any way you want. Where you place those numbers has a HUGE influence on the game. This would multiply the 244 billion in the title by 18! / 16 (because numbers 3 through 11 have duplicates). How you rotate the collection of hexagons (6 different ways) multiplies this number by 6, and you can rearrange the 6 ocean borders in 6 factorial ways -- multiplying it further. I come up with a final value of 4.2 x 10^29 permutations (ignoring the issue of if a number circle of value 8 lands on a mountain hexagon, and the other 8 also lands on a mountain hexagon ... too hard for my math abilities).
The number circles have to be placed in a specific order so the guy in the video in right
@@LightPinkno, that's just a suggested order for a balanced board.
Both are you are wrong. Predetermined numbers just for beginners. Played at least a thousand games with dozens of players. No one ever used the predetermined. Always went random with numbers. There are several rules with that. An 8 and 6 can’t be adjacent. The 2 and 12 can’t be adjacent. And the same number can’t be adjacent. Changes the math ( to above my level to figure )
@@terpfan2279 there are online Catan board generators that generate fair random boards. The possibilities are endless!
we need a video on the dice chess!~ ❤
We played Monopoly only once with my friends and swore never to touch it again but Catan we always had a lot of fun, we never felt upset after a session. We changed one rule, and that is the length of the longest road to the point when it wasn't possible to put anymore pieces on the board, otherwise it was ending too quickly.
At last an advert (for Jane Street) which is relevant and interesting, and designed to go with the video it accompanies. Were I younger I’d have been clicking through.
Catan is the gateway drug to board games beyond Monopoly.
It opens you to a world of games that are, IMO better than itself.
Catan isn't bad by any means, but not the best either.
I started with catan and got hooked. before I knew it I was playing Illuminati and Secret Hitler alone in the dark.
bring back Iron Dragon!
My favorite board game will forever be chess, but Catan holds a special place in my heart. There is just nothing that‘s more fun to play together in a group especially with people who don‘t play board games because it‘s surprisingly intuitive.
You can tell he's a genuine fan because the art on that copy is from like 10 years ago! The nostalgia is crazy
Astounding,I understood that.
Now my head hurts. ❤️
I’d love a colab with Magnus!
That would be awesome! There are many cheating suggestions in chess where a number is secretly sent to a player just to indicate the best type of piece to move. But, if the number is random, I think it will be nearly impossible for even Magnus to win.
love this board game breakdown!
Would really love to hear a breakdown of, "diplomacy".
My family always does full random placement of tiles and numbers, with a few optional sanity-checks at the end to avoid a repeat of the Legendary Brick Game, wherein the brick-producing tiles got distributed with some very poor roll probabilities and then got settled at opposite ends (which means you only get two settlements per tile rather than three, for those of you who haven't played the game). Bricks, it turns out, are necessary in order to take most of the interesting strategic actions in Catan; while that game was _fun_ (and definitely memorable), the sheer slog of reaching a board state where normal play even became possible was not an experience any of us have been eager to repeat.
So, not only are there millions of different board layouts in Catan - you actually do get an appreciably different game experience depending on how the board has been built. Which is _very_ cool, and one of the reasons I don't think the game gets enough credit among board-game people. Playing with a fair and balanced start makes the game more predictable than it really needs to be - boards which allow one player to place a few settlements and become an uncontested lumber baron while another has to scrabble around trying to trade Useless Rocks™ for half the game, only to become everyone's best friend at the mid-game point when those rocks start to become much more valuable... that's where Catan gets fun, IMO.
The sanity checks, for anyone interested -
1. Every resource type, _except for stone_ (due to stone being an end-game resource), _must_ have a tile with at least three pips producing it (for stone, this is optional and should be agreed upon by a majority of players), and _can_ have its highest-pip number swapped (second-highest number possible if it's one of the four-tile resources) if it has six or fewer pips altogether and a majority of players agree the swap is reasonable.
2. Any resource type which has all its tiles placed in a single contiguous region _may_ have one tile swapped for any other tile, if _any_ player asks to do this, _and_ that swap would create more than one region producing that resource.
3. If one half of the island has notably more pips than the other, up to three numbers may be swapped to the other half; this should ideally avoid breaking the first sanity-check.
Catan is certainly one of the greatest games, but it hasn't aged all to well. It's mostly determined by chance and snowballing.
It feel so much better to play with the old wooden pieces ☺️
We played it a bit differently. We mixed the lands, put them on the table.Then we mixed the number chips and placed them facing down. Now we select our first settlements before turning the number chips to face up.
This is wild
Please use the description field for a description (as well as all the other stuff you have to put in there). It's starting to look like videos are just being published for the sponsors.
We make it harder by flipping the numbers so first we chose locations then flip all the numbers to see what we got :) It makes it more realistic as it the decision is clearly by chance.
That makes the game less strategic and more random, why would anyone do that.
Also, how do you stop yourself ending up with an illegal board?
@@2LegHumanist same as in life, some are born lucky, some need to grind
@@iamsemideus
And you just play with invalid number assignments?
@@2LegHumanistwhat would be invalid number assignments?
@@not2hot99
You can't have 2 red numbers bordering each other. Against the rules. So a pure random (unchecked) layout will occasionally result in invalid boards.
fun fact about capturing in chess: that is still a rule in Japanese chess where you do actually capture and get the piece to work for you
The resemblance between this explanation and Aunty Donnas portrait of explaining a board game is brilliant.
You should check out tzolkin! Its an amazingly simple game (either put a farmer on the board, or remove it...) But the game introduces the concept of time that brings it to a high level of complexity. Its the simplest and most complex game at the same time i have. A underrated masterpiece!
The longest possible road is not so interesting of a question when you consider each player has a finite amount of roads they can place in the game...
In something like a thousand games of Catan I’ve seen a longest road of 15 maybe five or ten times? It’s incredibly unlikely to achieve in a real game.
Never heard of Catan. Looks like a fun whole family game. Hong long does a typical game last? An hour or more?
Probably 1 - 2h. I have played it around 8 times. It needs a bit of patience to get going and if possible play with people who know how to play it first. :)
It will be how long you want it to be.
Basic version with 3-4 players can last somewhere 1-3 hours. It’s an excellently fun game. I recommend getting the Cities and Knights expansion, it adds a layer of depth and strategy and alternative roads to victory not available in the original, however the games to tend to go longer. Very fun with pre-teens and teens to work on negotiation and planning skills
How do you win?
@@thenoobalmighty8790by getting a certain number of points which you get by building things
10:17 the longest road possible is 15 because you physically have 15 road pieces and can't build more when you run out. if you include the seafarers expansion it becomes 30 as the expansion counts ships towards the renamed "longest trade route" and you have 15 ship pieces in the expansion.
Actually, we still have the original version, where the see and harbour is modular, too. So even mor combinations.
Better show it to CGP Grey
Show what to him? Grey was going to make a video about Settlers of Catan for years now, maybe right from the beginning of his channel. He's very into it.
I think you can relatively easily set up a chess engine in a way where it skips all lines which start with a piece it isn't allowed to move according to the dice roll. Might be easier than arranging a game against Magnus.
In that particular instance it would probably be harder to win against Magnus than against an engine, since engines don't play for tricks and will always choose the objectively optimal move - which may not be the best move to confuse a player you know to be much, much weaker than you. The same reason I can win a game against Stockfish when I give myself rook odds, but would have practically no chance against Magnus in the same situation.
@@Commanberif you can win a game against stockfish with rook odds, that ain't full stockfish. Stockfish will beat you with higher odds than Magnus, chess engines are just that much better.
You absolutely have a point in general though, what you're talking about is why GMs can draw against surprisingly high ELO AIs. And why a low ELO player might survive longer against max Stockfish than Magnus. He'll go for a weird Gambit and checkmate you 5 moves later. Stockfish will take your soul. Slowly.
@@DaTimmehI know how strong engines are and it's still true that they don't play for tricks. If you can avoid getting checkmated, trading down into an endgame, winning against Stockfish with rook odds is trivial. Against a GM like Magnus you won't ever get this far. Not that it is even remotely easy or even consistently doable against an engine but it's definitely possible.
And yes it was full Stockfish and no, it's not something that I can do easily. It may seem like I'm bragging but I'm not, I'm a middling club player of 1900 elo.
I managed to win one game against Stockfish with rook odds, but I lost probably dozens in the attempt and it was without time limit.
Against Magnus, there is just no chance. Humans can do one thing engines cannot: they can play in a way they know will throw you off. Engines just grind you down, but without a queenside rook that doesn't work quite as well.
Isn't there a cap on the longest road based on how many road pieces you have? There are only 15 road pieces per player, so the longest road can only be 15 pieces long.
No, you don't have to multiply by six to account for rotation, these scenarios are already encapsulated in the permutations.
Hexagons are the bestagons
It’s crazy that magnus is so good that we can’t for sure say he wouldn’t win while being hindered by random dice roll moves
He would win comfortably if he had to wait 5 moves to move the same figure again. But the randomness of a dice creates a lot of uncertainty. He would still win most games, but not all. And against other grandmasters(who also have to use a dice) the dice would be more decisive than their own thinking.
@@philw6056my intuition tells me that even a very weak chess player would crush Magnus if he had to use random dice choosing pieces.
Tactics would not work at all. Even a 2 move tactic would have a low probability that he could play it out. And the opposite is true. Tactics that shouldn’t work at all would often work. You could capture almost anything for free especially if only protected by one piece type, he’d only have a 1 in 6 shot at taking back.
He would however get stronger as his pieces types were removed, assuming he still gets to move every turn. But even with only a King and one other piece 50/50 each move would make winning an endgame very difficult.
@@lyingcat9022 You'd want to protect pieces with pawns to get the highest probability of being able to defend. However, if you had lost a bishop and then were guarding your queen with the remaining bishop, I think that it would be pretty easy for the other player to take your queen unopposed, because the odds of being able to activate that remaining bishop would be small. Yeah, it would be a rout.
@@red72pasadena If they trade Knights and it’s protected by a pawn you have a 1 in 6 chance to take back with the pawn. If it’s protected by your other Knight you still only get a 1 in 6 chance to take back.
Either way you likely give the piece for free.
Numberphile? Catan? That's the crossover I didn't know I needed!!! :D
That was v disconcerting.i listened to the first 10 minutesof this not recognising the contributor but thinking it sounded like Marcus De Sautoy and when i read the info, it is Marcus. Great to see and hear him again. Hope he's good! 😊
9:08 you also have to divide by 6... all the 6 "version" are already calculated in the initial combination.
Chess with dice sounds like a game that could be over in ten minutes or ten hours. And you don't really get to choose which one.
You could perhaps make a deck of cards with maybe 4 Pawns 2 Bishops 2 Knights 1 Rook 1 Queen.
- Initial draw 1-3 cards
- You may only move a piece by discarding the corresponding card
- King is always free to move
- You may discard any card to move a Pawn, but you may only move 1 step forward by doing so (so no capturing etc)
- After moving you draw a card
- When you are out of hand, you shuffle your discard pile and start over.
- If you lose a piece, you may remove one corresponding card permanently from your deck.
/brainstorming
Depending on the fixed harbour configuration, you may be missing out on rotational symmetries of the board. That's effectively the same game.
The borders can also be rearranged and the dice rolls (numbers) can also be on different hexes. Both of these factors also increase the number of different possible game boards. But the number of possible games also grows with the dice rolls and the people playing. It is extremely likely that no two games of Catan have EVER been alike or ever will be alike.
It's one of those games where you can feel satisfied even if you don't win it all. Great game.
Given up on Catan, the luck element is just way too strong for 1-2 games.
Now Brass: Birmingham and Lord of Waterdeep, THOSE are fantastic board games. Hell, Brass: Birmingham would make a good Numberphile video since it could be about networks or something.
I always found it pretty dull, and yes mostly lick
Love it and has returned family fun for hours
During setup, we would always stop and re-shuffle the numbers if two reds came up next to one another, because we weren't sure how to correctly randomize without re-sampling from scratch.
Catan was the best board game around only in the year it appeared. Except for the die hard Catan fans, boardgame lovers have stopped playing Catan long ago! Main reason is that - especially in the 4-player-game - it turns out that very often one player is severely disadvantaged, because he gets squished between neighbors. This will be very obvious very early in the game and the respective player will have to endure a long time of sitting at the table and not having much fun.
even more proof that hexagons are the bestagons
“Do you have wood for my sheep? Never gets old. 😂
now that you're walking on boardgames territory, i have to say i would LOVE to see a math approach on Hive (also known as the "is like chess but without a board" game).