Infinite Chess | Infinite Series
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- Опубликовано: 1 мар 2017
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How long will it take to win a game of chess on an infinite chessboard?
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Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Sources and further references:
Joel Hamkins Website:: jdh.hamkins.org/tag/infinite-c...
Two Hamkins papers about infinite chess:: arxiv.org/abs/1302.4377 and arxiv.org/abs/1510.08155
A ton of links about games:: www.ics.uci.edu/~eppstein/cgt/
XKCD Optimal Tic-Tac-Toe:: xkcd.com/832/
Comments answered by Kelsey:
• Splitting Rent with Tr...
• Splitting Rent with Tr...
• Splitting Rent with Tr...
Imagine your queen just get SNIPED out of no where by a rook.
haha
Lol
infinite chess is fun. It can be scary cuz most people who play it like being snipers
Is this gonna be the real Tennison Gambit- ICBM Version?
"Yeah, ill play infinite chess with you, let me put my pieces on the edg-"
-e, what, did you expect something different?
I guess Pawns can't become Queens in infinite chess :(
Olivier L. Applin they could after ω1 moves, but that's one more move then infinity....
edit, yeah I guess not, that's just a further expansion on the board/moves so that just makes it harder to become a queen
maybe we can choose to define a larger promotion zone similar to how shogi or xiangqi works.
Jay Koerner it can't be harder. it will be impossible on infinite board. promotion is possible only when the pawn reaches to the last rank.
M.K.D. I believe it's only to the 8th rank, I don't know if it says final
M.K.D. Promotion is a chess rule that a pawn that reaches its eighth rank is immediately changed into the player's choice of a queen, knight, rook, or bishop of the same color.[1]
Slight correction: a game value of omega^2 does not mean that there are two instances where black can delay the inevitable by an arbitrary amount of moves - that would be omega * 2. Instead it means that black can delay check mate by an arbitrary amount arbitrarily many times.
Oops! You're totally right and I misspoke. Thanks for mentioning that! Hopefully there will be a future episode that addresses these chess games with higher ordinal doomsday clocks.
PBS Infinite Series that's what I love about you folks ...you are human and make mistakes and you don't mind correcting your own.
orbital1337 this came to me as well. Pin this comment please.
Thanks Tomi! I think that's one of the great things about math. It's so difficult that we all get things wrong, often. It teaches you to own your mistakes and just embrace it. :)
I am really fond of this new channel. I was waiting for a channel on mathematics and PBS Infinite Series is absolutely wonderful. Thanks a lot!
"Let's also assume we're playing with some rules that exclude ties."
Avid chess player here: that would be absurd.
Shogi comes close.
Anish Giri has left the chat lol
Me: checkmate
Opponent:w-where?!
Me:*points somewhere off In the distance* I put a bishop there a while ago, so I win
lol
That's some Japanese anime move right there.
@@MB-fp9lq As opposed to a British anime?
@@redforest9269 its better than korean anime
That's how 5D Chess plays out 😆
"And, Commander, it is possible to commit no mistakes and still lose."
- Jean-Luc Picard
love the fact that this channel targets an audience with at least some level of exposure to mathematics beyond the standard highschool and basic university level curriculum (zfc and set theory are mentioned without delving into a long tangent explaining what they are). please continue this
So many chess nerds in the comments. I'm a huge fan of this.
PBS Digital Studios Not really a lof of them actually
I just realized the title's a pun, validate my exquisiteness.
What about the Go fans? Any material on that soon, maybe?
PBS Digital Studios try infinite horde chess the games are fixed
or you should look at caplanca chess variation
Aaand that bishop is still going . . .
The bishop is rumored to still be continuing it's quest to infinity diagonally...
No more back rank mates
I read that as Anand
It is not possible to have 3 rooks because you cant promote pawns if the board is infinite
Daniel Dagan
Special promotion squares
You aren’t limited to the standard number of each piece, as she said
6:16 "There can be any number of any of the other pieces."
Promotion occurs on the 8th rank, so if we keep the same rules, it would still be possible.
Make a video about Ramanujan's infinite sums! It would be really interesting.
watch 3Blue1Brown videos , they are great.
Bibek Panthi I know! But I want more! I think I might be addicted to maths...
Hi Joel! :)
J. H. I know I am addicted to the maths... I just want another video about prime numbers
J. H. for sure
Whenever infinite ordinals pop up somewhere, it makes the topic omega times more interesting!
Still not as complicated as U N D E R W A T E R C H E S S
oh no
Agadmator's fans here?
Yes sir
Hello everyone.
@@papayaspice1155 Sorry About That
So yeah.
@2C (02) Chan Kwan Yu No thanks, I just want to enjoy the show.
A) Is chess determined?
B) Nope, you can draw by repeating moves
A) ok, ok, but let's assume you can't repeat
B) Well, you might not have enough material to win the game
A) Ok, ok, but let's make it a determined game.
B) Err... ok?
A) So now, Chess is determined, so is infinite chess. Proof
B) ¯\_(ツ)_/¯
There's also a way to draw the game, where one player doesn't have sufficient material, and the other player could mate but his time runs out
Armageddon chess is determined, since a draw is a loss for one of the players.
6:10 , 9:38 just leaving those here
You can have a stalemate by not having a legal move while you still have all of your pieces
you can have draws, but i think it was meant to mean that you can force a win, provided you played perfect...
same with the piles example, where the first person to move could loose but never has to, privided perfect play...
it even makes sense by intuition, if you think about it: white always moves first, and there is nothing that provides a counterbalance to this first move advantage, so it would only be logical that white can win any game, if they play perfectly.
you might have to concede the 50 moves rule, that could be true
I read the second paper by Hamkins and it's incredibly interesting. I encourage everyone else to do the same. It's definately worth the time and effort!
You claimed that chess is a determined game and argued via Zermelo's Theorem. But chess games could possibly end in a tie (stalemate), so how do you get to your conclusion? Is chess really a determined game?
At 0:57, she says that we'll modify the rules of chess so that it can't end in a tie. Here's one way to do that:
One way to think about chess is that the last person to make a legal move wins (except when there's a tie). If I move such that you are in check and can't get out of check, there is no legal move for you to make, so you lose. With draws, it's a draw if we revisit the same position too many times or if we reach a state where one of us isn't in check but has no legal moves. Here's how I'd turn chess into a game satisfying the assumptions for Zermelo's theorem:
It is illegal to make a move that ends with you in check. It is illegal to make a move that ends with the board in a position it's been in twice before. It's illegal to make a move that would result in the past 50 moves having no captures and no pawn movement. If it is your turn and you have no legal moves, you lose.
The difference between this and chess is that we pick a winner whenever there would normally be a draw. If you cause a stalemate, you win. If you cause a threefold repetition or make the 50th move without a capture or pawn move, you lose. It's still enough like chess that all the other assumptions are maintained. The strategy would be different, but it's close enough for the game theorists.
To answer "Is chess really a determined game?", I'll go with "probably not". Chess is a big game, and we don't know if there is a winning strategy. Lots of people (myself included) guess that it is possible to force a tie, but it is not possible to force a win. Since the definition of a determined game is that one player has a winning strategy (can force a win), chess probably fails. However, I think it is a strictly determined game, in that there probably exists strategy pairs for White and Black where, if each player knows the other's strategy, it is impossible for them to change their strategy such that they improve the outcome. I can't remember the name of the theorem, but I'm pretty certain there's one that says chanceless games with perfect information always have this property.
Ah, I missed her mentioning that. Yes, the changes you suppose would make it possible to apply Zermelo's Theorem, but if you translate the statement to the real chess again, it would mean that either can white or black can force at least a tie!
"At 0:57, she says that we'll modify the rules of chess so that it can't end in a tie." I don't think that's what she says.
She seems to me to say "Chess is a determined game, let's look at that." I think it would be much clearer if she said, _Chess is __*__not_*_ a determined game but we are going to consider a variant, called "Finite Chess" which is played using rules linked to in the description._
Pi Fisher Strategy pairs is an interesting idea. In determined games it doesn't matter if you know what your opponents moves will be. If you are the determined winner you only need to know the response to make to your opponents actual move, as opposed to knowing the winning response to all potential moves. What kind of chanceless perfect information games are there where knowing your opponents strategy can change the outcome of the game?
if it can't end in a tie, it's not a finite game. A perpetual would last forever
7.40 slight correction:
If black decides to move a pawn, it would be mate in one. The queen could just move up one square.
Also if black just moves the middle rook one to the right white can't win in six, or even more turns. (At least from what I've played through)
Other han that, Great video, thanks.
I loved this episode! Very clearly and articulately spoken, not too complicated to understand, but also not questioning the viewer's intelligence. Keep up the good work! As for a comment on the video, the only finite doomsday clock setup would have to come in some sort of boxed in situation I imagine, where white physically creates a finite space with the pieces that black cannot delay doomsday from.
0:24 my brain hurts looking at that chessboard. It's set up incorrectly. Why is it so hard for people to get something as simple as the correct starting position right?
6:18
No there can't be. You can't have a king and bishop on each side. That'd result in insufficient material to checkmate. Did I miss something?
The situation at 7:00 can be easily prevented by simply moving the black rook to the right instead of up. If white proceeds according to the stated moves, then the black rook can capture the white rook and prevent the checkmate.
Now instead White moves the Rook 1 square left. King moves. White Queen one square forward, mate.
I support any type of educational videos, like these! Keep up The good work, and push in as much content as possible
Respect
why can't the black move the midlle rook one space left insted of up?
physics pony White's queen would move one square up and to the right for checkmate.
I see it now.
Move the white rook up to the king and black will have no way to kill it (it's protected) while being dead next turn. If you go right, just do the same from the other side, but move 1 tile. Dead next turn while protected.
Could someone explain white's winning move if black goes rook to the right? I'm seeing left clearly, but right has eluded me so far (I'm not much of a chess player, though).
If you move the black rook to the right, the white rook moves one to the left, the king is forced to move one to the right, the queen moves one up placing the king in checkmate. (Took me a while to find that one, guess I'm not that good at chess either :P)
Will there be a video explaining more about the fact that a game's determinacy can depend on the chosen set theory axioms? (perhaps there's one already?)
how have I not heard of this channel before? it's awesome!
*YOOO, MY PROF HAMKINS IS MENTIONED!!!*
Also, incredible how Chess can portray Ordinals, wow...
There is a typo at the bottom left of the game tree Alice v bob where bob takes 2 marbles from 1 marble
Like so they can correct this error
There are a bunch of errors on that tree, I have half a mind to scan the correct one I sketched out and email it to them.
Matt Kilgore also the chess posición is not a forced mate
O Fenómeno yes it is
(1)
|
[2]
Bobs
wins
I played two infinite games of chess this morning. I'm still playing one of them.
loving this channel
I've tried coming up with a Turing-Complete setup using chess rules. I couldn't do it, but with a couple accomodations (most likely an infinitely repeated grid of walls and pieces), it looks doable.
Promising setups for Turing-Complete chess:
- Given an infinitely repeated grid, it might be possible to build a cellular automata such as rule 110, though it might require multiple kings.
- If you can build some kind of path with forced king movement, path crossings and one-way path joinings, then with an infinitely repeated grid, you can build a 2-counter Minsky counter machine, using the x y position of the king within the grid to store information. (Snakes and Ladders with an infinitely repeated grid or many other infinite patterns is also Turing-Complete)
- Building a more traditional Turing machine with a proper tape is more complex since it requires some sort of resettable 1-bit non-destructive memory cell... I'm not sure it's possible, but if so it requires a 1-dimensional infinitely repeated section still.
- The holy grail would be to encode information in the position of some infinitely movable pieces to build a Minsky counter machine, since you would then only need a finite number of pieces/walls, but that sounds very difficult and probably requires a lot of other compromises (fairy chess pieces, weird rules, who knows).
If they ever stop making these videos, I'm suing PBS for false advertising :P.
Has anyone explored a version of chess where the pieces' coordinates are real numbers instead of integers?
That would be really interesting.
Then how would pieces capture other pieces? Like, if your rook is on (3.1, 8.2), does my move to (2.9, 8.1) capture it because it's close, or miss it?
Well, you could say that moving to strictly less than radius 1 of another piece captures it, so that normal integer moves come out the same while all situations are determined.
The bishops would become considerably stronger, about as strong as rooks. A normal light-squared bishop can't reach any dark squares, but a real-valued light-square bishop could simply move halfway along the diagonal - more precisely, 1/sqrt(2) units - and be able to attack dark squares.
Tehom, the details would be up to the person defining the rules; if a given set of rules resulted in uninteresting gameplay, the mathematician studying them could simply note that and move on to some other rules. I do not pretend to have fully fleshed out this idea.
Though I admit I am curious about the consequences of "exact match" rules for capture. It's true that a bishop would have much more trouble capturing other pieces, but they would also be harder to capture in turn. I wonder if there would be a fair way to extend the movement rules for pawns and knights.
Assuming the exact match rules, the biggest problem would be the knights, pawns, and king. These pieces are defined to move not an arbitrary number of spaces, but a defined integer number of spaces. The problem with that is that these pieces become weak. The queen, bishops and rooks can all move any real number of spaces. This means that there are an infinite number of positions that they could be in (big infinity, not wimpy infinity). Not simply board positions like (3.1, 2.3), but positions like (pi/sqrt(3), e^sqrt(2)). It would be basically impossible to set up a capture of one of the free moving pieces except by waiting for the piece to land on an integer number spot (something that wouldn't be required to capture an integer piece). This is because any piece could always take advantage of the infinite degrees of freedom available. In fact, the optimal strategy for this game would be to rush the queen onto one of the diagonals that lead to the king. Once that happens, the king could be perpetually checked until either a) the player makes a mistake and loses, b) the first player allows a stalemate by repetition, or c) the first player disengages in order to prevent a stalemate. (I am not sure which is the most likely outcome.)
We could extend the real movement to all pieces by allowing the king, pawn, and knight to move as they normally do, except that the moves can be scaled by a number between 0 and 1. This would allow real movements among all pieces, but would only make the games impossible to win as every piece falls through the infinite folds of the continuum. The king cant be mated (no way to force him into a position he cannot escape from by simply moving an arbitrary unit up or left etc...) and so every game would end due to one of the three stalemate rules. If we replace those stalemate rules with only the requirement that in perfect play, it is possible to capture the king in a finite number of moves, then the game would still be a stalemate. It would be easy to show that there exists a shrinking series formed as the king moves from square A to B that cannot be intercepted by any piece moving linearly. This would apply to all pieces including the knight.
In short, exact match is uninteresting and should probably go back to the drawing board. I don't think that ranged capture is more interesting either. It seems to me that ranged capture would simply quantize the board. If the capture range is r (in some fractional unit of a square), then the chess board could be redivided as a new board of NxN where N = 8/r. Play would effectively be the same as a regular integer chess board with NxN size. That is to say, everything we know about finite chess would still be applicable in this example.
+Adam Billman You could fix that by allowing the pawns and king to move any length less or equal to 1,
and allow the knights to move any length less or equal to sqrt(5), with one of their coordinate deltas being double the other coordinate delta (Δx=±2Δy or Δy=±2Δx; (Δx)²+(Δy)² ≤ 5)
Actually, chess has ties, so it might or might not be determined, which is part of why no one knows the "optimal" strategy.
White bishop: catch ya later bro *slides off into infinity*
Why are there three black rooks?
A pawn promoted by reaching the end of the infinite board lel.
black must have thrown double 4s and played their get-out-of-jail-free card
You should see the number of pawns in the ω⁴ game.
jdh.hamkins.org/a-position-in-infinite-chess-with-game-value-omega-to-the-4/
Affirmative action.
tscoffey1 I was going to offer a genuine explanation, but that's just hilarious.
Here from Agadmator :)
There's a small problem with the infinite scenario as presented. There is no White King. Therefore White has either already lost or it is placed somewhere else on the board. Which itself presents a bit of a problem.
If we use the Black King as (0, 0) on 2D Cartesian coordinates and the White King is located nearly anywhere outside the shown starting, then Black can force an indefinite number of moves by moving a rook along the x-axis. It is also possible for White to make a losing play in this situation too.
The (+x, -y) "quadrant" is a bit tricky. The other 3 are pretty straight forward. Either there is an indefinite delay or there is a loss/trade of material. But the (+x, -y) quadrant is either an indefinite delay forced by Black or an indefinite delay forced by White. If the White King moves away from x = 0, then it is forced by Black. If the White King moves toward x = 0, eventually it will be protected by the Pawn. To remedy a clear loss, Black can move the middle Rook out to put the White King in check after it moves toward x = 0. Then the Black King has an unbounded amount of room to run. Possibly better if the Black King moved back after the middle rook is moved.
Note: The Black Rooks cannot attack the White King along the y-axis. The White King could "snake" its way to the rook and take it or force it out of position. Then White would just need to revert to the previous winning strategy. Along the x-axis the attacking rook is protected by the other rooks.
So unless the White King is protected by other non-shown pieces, or it is on certain edge cases ((0, -y) comes to mind), White is not guaranteed a win.
P.S. I really love this series.
The way the king, castle and rook was moving resembled something like a glider from Conway's game of life.
Agadmator ♥
The goal of chess is to checkmate the king, and capturing the opponent’s king is impossible if all of the rules are followed.
Well, the game ends when one side can't avoid the capture of the king.
oh shit at first i was like u could just run away for ever, but then i realized the bishop, castle, and queen could make infinite moves
The Doomsday clock should be: How many moves before a pawn gets promoted in this infinite chess.
Watch the vid before commenting sheesh
sheesh, sheesh, sheesh!!
Does halfway count
8:00 isn't it mate in 1 not 2?
1 move in chess convention is two moves in game theory convention.
Moving the Rook was a dubious move by White. A better move would have been to move the Queen instead to get Checkmate in 1. Still, what a fantastic chess puzzle. Your comments and questions please.
Eh, move the rook on the left in and avoid the checkmate entirely
Naymy, agadmator will always be better than suren
If Black moves the Rook on the left then White's Rook can checkmate the King.
Thanks for the video, Kelsey.
Seeing that there are arrangements with game values of omega, omega^2, omega^3, and omega^4, it seems obvious that there exist arrangements with game values of omega^n for any natural number n, though it is not easy to find examples. Therefore, I think the next interesting thing to determine is if there are arrangements with game values of omega^omega.
starting positions with doomsday clock omega^omega would require infinite amount of chess pieces
A-M-A-Z-I-N-G video. keep up the work!
It would be much smarter for black to move the rook one space to the right.
No. No it wouldn't.
It would be much smarter for black to move the king on its side and resign.
If you move the left rook to the right, then queen moves directly in front of King. Checkmate. If you move the middle rook to the right, then white moves their rook one to the left, forcing King to go one to the right, then queen moves up two squares. Checkmate. The third rook moving right does nothing.
guys move the center rook to the right would make things better
Myrrh Manalo my man did you read my comment?
Hey, slightly correction ;) There is rule in chess which says that a game is draw if there are more than 50 moves without a pawn moving. So the doomsday clock cant be infinit
Thomas Nordman
Yes
if the board is infinite then pawns can move infinite times
a dark, twisted dreamworld. not if they’re blocked
On your board at 6:59, I don't think there's anywhere, even on an infinite board, that you can position the white king to make him immune to attack by black's rooks. At least one of the rooks will always be able to move into the same row or column as the king, placing him in check, and preventing white from moving forward with its own check. Black can then keep this going by chasing the king around the board with the rooks, and may even be eventually able to force a win; I'm not sure. But in any case, the situation is more complicated than what you've presented. It only really works if white doesn't actually *have* a king, in which case... yeah, you're right, it's hard to imagine a scenario in which they lose.
At 6:55 if white has a king, black can give check, regardless of it's position. If white does not have a king it's not chess.
It can't if it's behind a piece though.
Fundamental doubt. How is it possible to make an infinite move on an infinite board? say I'm black in that position in the video, and i want to move my rook infinitely far away. but then my move will never end. Because there will always be another square beyond the square I just put the rook on, so I will keep moving the rook, and it will never be white's turn. Alternatively, if there is a square that I am forced to stop at, then the board is not infinite. How do you deal with that?
MeloDeathKT, You can't make an infinite move, that wouldn't make any sense. What you can do is move any finite distance you want, without any limit. That's why the game ends in a finite number of moves even though the doomsday clock is infinite, since as soon as you make a move the clock drops to a finite amount.
Basically, you have to pick an integer of squares to move before moving ur piece. The reason the board remains infinite is because you can choose any finite integer of squares. You are right that you can stay and think about ur move saying. "Hmm maybe I'll move it 5 spaces. Maybe I'll move it 5+1 spaces" and continue thinking forever. But eventually once you make ur move it will take white that number of spaces you chose to deliver mate.
Reddles37 that seems like a complete cop-out, and not really an infinite board. what that is, is an arbitrarily large but finite board. Limiting the range of movement to 8 squares back means the chessboard is 8x8, and limiting the range of movement to an arbitrarily large but finite number equally means that the board is arbitrarily large but finite. After all, limiting the range of movement is the DEFINITION of limiting the board. how can you call this an infinite board?
Daniel V: So you will never finish the move in the first place. Or you need an infinite amount of time to finish it?
Move to the first square in 1 second. Then the second square in half a second and the third square in a fourth of a second ... Now you completed your move in 2 seconds yay!
Agadmator fans unite!
Hello everyone.
There's a classical example of (King and) two Queens mating a King in any staring position on infinite chessboard for only 4 moves:
1. One of the Queens gives a check (not diagonal, but by file or rank).
2. The second Queen moves in a kind of manner that the King is trapped between two Queens and has only two files or ranks to move on.
3. After any King's move one of the Queens cuts him off by file or rank. Then another Queen checks by rank or file respectively.
4. After one of two King's possible moves the first Queen checkmates the King by giving diagonal check. In the final position two Queens are two squares from each other.
This is a lesson for beginners. The same technique is thought on finite - 8x8 - board.
4:07 "No ties"
There are ties in chess, so I think that disproves the whole premise of this video concerning chess being determined.
If black moves his rook to the right one square at 7:16 he is safe.
Then white wins immediatly.
Liam Calder queen moves one to the right and up, checkmate (or if you mean the other rook, the queen only moves one up without moving right)
how are you guys missing this? It is absolutely not check mate. you move the king up 1 and left 1, and you're golden.
Liam Calder NOT IF THEY MOVED THEIR ROOK THERE!!!!1111elf (on a serious note though, I said itˋs checkmate IF they move that rook to the right because then the king can't escape to there anymore
I worked it out on a chess board with a former professional and putting the middle rook one space to the right would end up in check mate, but not by moving the queen 1 right and 1 up.
Chess is determed. correct. But, it's wrong that it means black or white has a wining strategy. Chess could be like tic-tac-toe, where noone has a winningstrategy. Chess could be determed to be draw, if both players play perfect
It most likely is.
I've seen a lot of comments explaining that chess is a determined game if we apply a few restrictions, one of which being that you can't force the game to return to a state it has been in twice before. But, in infinite chess, would you be breaking this rule if you made the pieces all the same relative to each other, but shifted two squares in one direction? Do we simply change the restriction to exclude that as well? Do we say that there is no way to know, given a state, where the beginning was, so it is impossible to say whether or not a state is just shifted from the beginning state? Or do we allow this?
Interesting concept - really enjoyed the infinite + scenarios. 1. If the middle rook moves to attack the queen, then the queen moves diagonally to give checkmate. 2. If the middle rook moves to attack the rook, then the white rook moves sideways to check the black king; it must move sideways and then the white queen moves UP two squares checkmate. 3. With an infinite board there's no way to promote a pawn to make the third rook.
the definition of "mate in n" is not conventional and is very confusing for a chess player.
Right. If white has a mate in 6 moves, that conventionally means that (assuming black plays perfectly) white will move 6 times, checkmating black on the final move. Not that each player moves 3 more times.
How is it not conventional and very confusing? I have to disagree. Just about every tournament chess player owns a book on mating puzzles. The book I have has 3 sections: mate in 1, mate in 2 and mate in 3 problems with solutions in the back. All the moves are forced for the losing player.
black's moves are forced, there is no perfect play, theres probably only 1 possible move, its all up to white to checkmate in the correct number of times, missing a forced mate in 3 is embarrassing unless you're in time trouble.
oakenguitar, can you please read the comment before replying?
what comment did I not read rock papa....? I just don't understand how the definition is confusing.
4:26 "let us know what you're thinking in the comments"
*wipes drool off shirt* uh, excuse me, what? I'm sorry, I was a little lost in your eyes...
Anyway, at first I was thinking that the information speed was delivered a bit too slow, but as soon as the infinite ordinals started popping up, it took just that bit more time for my layman brain to catch up, and I appreciated that a lot!
Not creepy at all.
So the doomsday clock becomes infinite when the moves can be looped into a repetitive pattern. A causality loop. The king was forced into its position and the moves were forced based on their best interests. That is amazing!
This kind of reminds me of a number to words game.
Choose any number. Spell it out and count the number of characters in it. Ex. 7 = Seven . Seven has 5 letters.
Now spell out the number. (in this case 5 as FIVE) and repeat. Five has 4 characters etc.
The Trick is that no matter what happens. You will end at loop at the number 4. Because 4=Four. has 4 letters. A loop.
I guess another loop may be a pair of numbers resulting in each other. But I personally couldn't find any. And since numbers in millions only take only 100s of characters, it decreases significantly.
Simple proof of Zermelo: a) every ending position is determined - someone has just won, so that player is guaranteed to win given perfect play on their part. b) for any position where every move leads directly to some determined position, that position is also determined - if there's a move that leads to a winning position for the current player, then this position is a current-player win; otherwise all possible moves lead to a win for the other player, so it's an other-player win.
Since the game tree is finite (and connected), every position is determined since every future position from that position leads to an ending position in a finite number of moves down every possible path of future moves, so you can backtrack from those ending positions to determine the position you're looking at. In particular, the starting position is determined.
0:30 black kings on wrong square, but tbh that's a mistake 90% of people make anyway xD
I wonder how many games of chess have been completed with the king being on the wrong square and nobody batting an eye..
is there an upper bound on omega to the "mate in n" number? Or it can get arbitrarily big on omega? Do we have to come up with another transfinite ordinal?
winning move for black is to place the rook trillions of spaces ahead and then just run out the clock by being faster at moving the pieces.
Just a small correction, there is always an undetermined game (either with or without AC). The statement ZF + "every game is determined" is inconsistent.
Quesiton: When you have a doomsday clock of OMEGA, is it considered a SUPERTASK once the piece is captured? Also consider if there are positions of OMEGA squared, OMEGA square times 4 and even OMEGA to the fourth, I would then propose that a HYPERTASK is possible. Thoughts?
Finite chess is determined. At least one player will have a winning strategy.
Infinite chess is determined. At least one player will have a non-losing strategy.
This channel has one of the best presenters for maths on youtube along with Mathologer.
dont forget 3 blue 1 brown!
I have a couple of questions.
1. What rules are used to prevent ties (for example what happens if both players only start with a king and nothing else)?
2. Why didn't black move his rook to the right in the example game. It seems that this would allow him to break the doomsday spiral?
Where can I read some publications on advanced maths? Is there any database of them by theme or something?
Dear PBS Digital Studios: How am I just finding out about this channel? You have my permission to plug yourself crazy on other PBS Digital Studio funded channels. I depend on those sort of plugs. That, and keep up the great work. PBS on RUclips is the best.
I wonder which opening word work best, like a Lopez castle quickly or something like Benoni or Kings Indian Defense/Attack
Why isn't it like this?
Black Turn 1: move the rook out of the way.
White Turn 1: move the rook right next to the king.
Black Turn 2: move the king upwards.
White Turn 2: move the queen right next to the king.
Checkmate.
An infinite doomsday clock might just sound like no doomsday clock, but the rook can only move a finite distance away. So omega really means “mate in n, you choose the n”.
Watching this video fills me with determination.
The chess board is set up wrong! The white square is always the bottom right square. It's amazing to me how many times a chess board is setup wrong in movies, ads, videos ect.
Dan the insult comic dog
It's an infinite board. There is no bottom square.
There is a mistake in the tree at 2:30. At the lower left corner the "2 Bob wins" should be connected to the 2 on the left, not to the 1
We can find the winning strategy of chess using a quantum computer. How cool!
7:40 the king isn't in check, so you can move the rook right above the kind one place to the right, threatening the white rook, meaning it either has to take the black rook, which then can be taken by either of the remaining black rooks, or it can move to the left, right below the black king, where the black king can just move one space to the right, and having his/her king and all three rooks out of danger. If the rook moves in either direction to threaten the king, the king can take it without being in check.
Infinite pawns filling every square except two; one for the black king, one for the white king. These kings are two spaces apart
About the fair division question, you can make the triangle bigger and make the price of the rooms negative. So that you get paid to live in the room. Doing this will allow you to at some point say, yes that one particular room for $10 is worth less that this other room for $-10000. So, at some point, you will prefer the correct room at each edge/vertex.
So what if the optimal solution happens to be in one of the negative regions? It means that at least one the people in your group would pay you for them to live with you in a particular apartment than live by themselves. (I suspect that will not happen, so although there seems to be this weird assumption required that you will choose the free room, we can make do without it by using a bigger triangle as the feasible solution would not be around that point. )
Zermelo's theorem has "no tie" as a condition, but regular chess can end in a draw if for example only kings are left or no valid move remains. So you can't apply it here
Where do I go to find out why infinite series is a being studied and what benefits has the study produced.
I suppose that the no-repeat-positions rule in infinite chess would be expanded to say "no position may be played that is identical to any translation of a previous position".
A rule to balance this out nicely would be to stipulate that each king cannot move some arbitrary distance away from the other king
Slight correction: 7:53 ordinals are not "infinite counting numbers". Cardinals are counting numbers. Ordinals are describing the order type, that is the way the elements inside the ordinal are arranged. For example, the order type of the natural numbers is ω (omega). A property are that there is no maximum. ω+1 would be a set with order type ω with 1 extra element appended. A property is that there is a maximum. However, if you relabel the elements inside a set with order type ω+1, without regarding the order, you can biject it with a set with order type ω, so their cardinality is the same.
Is there any reason chess positions are being studied that require mate-in-N greater than ω? There's still some fundamental questions about normal chess (8x8) that have not been answered. For example, who wins with perfect play? Did we (the math world and chess theorists) give up on answering that?
In the infinite chesst example you only move the middle rook to the pawn line, but if you move the middle rook to the left or right it can save the black king or am I wrong?
Foolproof strategy: the day before your match, tell your opponent that you're severely narcoleptic. They of course will be reluctant to play you (because who wants to risk having their opponent randomly fall asleep on them, how boring it would be), which is when you introduce the house rule that slipping into unconsciousness constitutes a forfeit. Then, on the day of your match, treat your opponent to a (spiked) drink as a sign of sportsmanship. The rest follows.
Determined games have no ties. One minute later: chess is a determined game.
@4:50 Is that music from FTL: Faster Than Light?
7:40 what is stopping the middle rook from moving left or right one square?
Is it possible that you can start some series about computer science subjects like Artificial intelligence or Numerical analysis?
I thought up a 2 player game
ver 1, having six points, don't make a triangle of your own color
ver 2, having 18 points, don't make a quadrangle of your own color
ver 3, having R(5,5) points, don't make a pentangle of your own color
What will it append if for example white have a winning strategy and black non-loosing strategy? Is it a draw because chess allow it? And if it’s the case, dose the game still determine in finite and infinite games?
sorry for the bad english, i'm a french viewer
can a pawn in infinite chess move one or two spaces on its first move or only one?
I didn't understand a damn thing in this video yet i couldnt stop watching.
Waiting for the video about "Infinite Jest" :D