Let’s put a handful

of chess pieces down on an infinite chess board. All the pieces move according

to their usual rules, but there are no bounds

on how far they can move. Who will win, and

in how many moves? To understand infinite chess,

let’s start with finite chess. Chess is the kind of

game mathematicians like. For one, it’s a game of

perfect information, which means that everything is out

in the open to be analyzed, unlike poker, where

relevant information– the other player’s hand– is hidden. Chess doesn’t

involve randomness. Games of chance, like

those that involve dice, are often harder to analyze, and

you can’t guarantee an answer. Chess only has two

players, and provided you play with some

standard rules about not repeating

positions, each game must end in a finite

number of moves. Let’s also assume we’re

playing with some rules that exclude ties. That’ll make things simpler. Mathematicians have

many favorite examples of these kinds of games. Chess, checkers, Go, dots

and boxes, and many more. Check out the links in the

description to learn more about these mathy games. Chess, even in its finite

version, is complicated. Let’s start with a simpler game. A single pile version of Nim,

which I’ll call pile game. Start with a big

pile of marbles. Alice and Bob alternate

taking 1, 2, or 3 marbles from the pile, and the person

who takes the last marble wins. Let’s say they start with 5

marbles and Alice takes 3. Bob could take 1, but then

he’d lose, because Alice will take the final 1. A better idea is for

him to take the final 2. Then he’ll win. This game seems a little boring,

but the strategy behind it is pretty cool. If they start with

n marbles, what’s the best strategy for Alice? What about Bob? If you haven’t seen

this problem before, I strongly recommend playing

with different numbers of starting marbles and

trying to find a strategy. We can make a game

tree for our pile game, which is sort of like a

diagram of all possible games. If we start with

5 marbles, Alice can pick either 1,

2, or 3 marbles. Then they’ll be 4, 3,

or 2 left, respectively. Then we map Bob’s

choices in the same way. Then Alice’s, then

Bob’s, and then Alice’s. Each time someone takes the

last marble, we mark their win. Notice that Alice has a

winning strategy, which means that she has a

method of playing that guarantees she will win. First, she picks one marble. Then Bob has three

choices, take 1, 2, or 3. Regardless of what

he does, she can respond in a way that

guarantees her win. By definition, at most

one player in a game can have a winning strategy. Not both. So if both players are playing

optimally, Alice will win. But how long will it

take for Alice to win? Let’s define a doomsday clock. The doomsday clock

tells us how many moves it will be until

the inevitable winner, Alice, in this case, wins. By the way, that’s

not a math term. Mathematician Joel Hamkins

calls this the game value, and in chess they

call this mate-in-n. Let’s look at a

game with 5 marbles. In the beginning, the

doomsday clock is 3. Alice can force

a win in 3 moves. After she moves, the

doomsday clock is 2. No matter what Bob does,

he’ll lose in 2 moves. After he goes, the clock

is 1, meaning Alice wins after she makes her turn. OK. Here’s an important

definition for us. A game is determined if one

player has a winning strategy. And here’s an important theorem

known as Zermelo’s Theorem. Every game of the type

we’ve been talking about– games with perfect

information, no randomness, two alternating players,

no ties, that end in a finite number of moves– these kind of games

are determined. You can actually

use the game tree to prove Zermelo’s theorem by

backtracking through the tree. I’ll let you fill in the

details of the proof, or give an entirely

different proof. Let us know what you’re

thinking in the comments. Why is this such a big deal? Well, for one it

implies that chess, the normal, finite

version, is determined. There is either a

strategy for White or a strategy for Black

that will guarantee a win. Of course, nobody knows

this strategy, mostly because the game

tree is really huge. Now, let’s move into the

world of infinite games. Infinite games are

ones that could last infinitely many moves. Our big theorem

about finite games was that they are determined. Now, the analogous question is,

are infinite games determined. Does one player have

a winning strategy? Well, just like pretty

much every other time we step from the finite

world to the infinite world, things get weirder. Whether or not all infinite

games are determined is a complicated issue

depending, surprisingly, on what set theory

axioms you choose. But within the

standard system, ZFC, there are some types of infinite

games which are determined and there are some games

which are not determined. What matters for our purposes

is that infinite chess is determined. There is a winning strategy. So from any starting position,

one player in infinite chess will always have a

winning strategy. Let’s get more precise about the

infinite chess we’re playing. Both White and Black have, at

most, one king on the board, and, as always, the goal is to

capture the other side’s king. Normal checkmate. And importantly, that

checkmate will always happen at some finite

time during the game. There can be any number of

any of the other pieces, including possibly

infinitely many, starting off in any arrangement. The pieces move normally,

but they’re unbounded. For example, the king can

hop one in any direction, and the bishop can move

diagonally as far as it wants. Knowing that infinite

chess is determined, we can ask about

winning strategies from different

starting positions. Is White or Black guaranteed

to win from a given starting position? And how long does it

take to force a win? That is, what’s our

doomsday clock say? Here’s a starting

position where White is guaranteed to win in six moves. Black’s king is stuck, and so

it needs to move the rook up. White moves its rook to

keep the king in check. Black moves its king, but White

can follow with its queen. The white queen

and rook continue to squeeze the black

king up to the barricade until it’s stuck and

forced into checkmate. So on the first move,

what’s the doomsday clock? Well, Black has a few options. It can move the rook like

it did and lose in 6 moves, or do something silly like move

one of the pawns, in which case White will win in 2 moves. Well, in this situation, we say

that the doomsday clock is 6, because that’s the

longest Black can hold out before its king is captured. It’s the maximum time

before the game ends. But can the doomsday

clock be infinite? In the Hydra episode, we

discussed infinite ordinals, infinite counting numbers. After all of the

natural numbers, 0, 1, 2, 3, 4, and so on,

we have the first infinite ordinal, omega. Then comes omega plus 1,

omega plus 2, and so on. Then omega times 2, omega

times 2 plus 1, and so on. Then omega times

3, omega times 4, then omega to the

omega, and omega to the omega to the

omega, and so on. Let’s look at this

starting position. It’s like before,

except there is no line of pawns blocking the rook. Notice that the

same thing happens. The black rook moves up

to make way for the king, and the white queen

and rook continue to chase the black king up until

it gets trapped by the rook. The only difference

between this and before is that now the black

rook can move as far as it wants on the first turn. The further the rook

moves away, the longer it will take before the

white queen and rook trap the black king against it. So what’s the doomsday clock

say at the initial position? Well, the further the black

rook moves on its first turn, the longer the game

lasts, and the black rook can move an arbitrarily large,

but finite, number of moves. So White will eventually, after

a huge, finite number of moves, capture the black king. Since there’s no limit to how

far away the rook can move, there’s no limit to the number

of moves the game could last. The doomsday clock is the

maximum length a game can last. It must be bigger than

all the finite numbers. It must be the first

infinite ordinal, omega. All the games are finite,

but unbounded in length, so the doomsday

clock is infinite. Hamkins and his

collaborators have developed an incredible amount

of infinite chess mathematics. They came up with

the chess position we just looked at, and

have also demonstrated a starting position

in infinite chess where the doomsday

clock is omega squared. This is because there are two

separate times in which Black can delay its inevitable death

by an arbitrarily large number of moves. There’s even examples

of infinite chess with doomsday clock omega

squared times 4 and omega to the 4. We’ve linked to these

results and a bunch more in the description. There are so many cool

ideas behind infinite chess that we could make several

more episodes on it. Can you construct an

interesting starting position in infinite chess? Does someone have

a winning strategy? Let us know in the comments. See you next week

on Infinite Series. Hello. I wanted to respond to some

of the comments about our Fair Division episode. So Bjorn and several others

made an interesting point. In real life, you might not

always pick the free room. If there’s another

room, and it’s cheap, and it’s way better

than the free room, you’d probably pick that one. But in order to get all the

math to work out so nicely, we needed to make

some assumptions about our mathematical model,

and one of the assumptions we made inside of our model was

that you would always choose a free room to a non-free room. All right. Steve’s Mathy Stuff answered

our mini-challenge showing that there’s an odd number

of fully labeled rooms. Basically, there are an odd

number of rooms which you can access from the outside,

and there are an even number of rooms which you cannot

access from the outside. Since an odd number plus an

even number is an odd number, that means there’s an

odd number in total of fully labeled rooms. You should check out his

comments for more details. Finally, elevating moment

asked an intriguing question. Can you rig the system? Can you lie about

your room references in order to get

a better outcome? Maybe. Lying might also get

you a worse outcome. You might end up

with a room you hate. I’d be really curious

to see if one of you can come up with

an example where lying improves your outcome,

or maybe it makes it worse. All right. See you next week. [MUSIC PLAYING]

Determined games have no ties. One minute later: chess is a determined game.

Capture the black king? CAPTURE THE BLACK KING?? Ugh, go back to your infinitesimally small lengths and tangents to curves and never come to the wonderful world of chess. So many of you mathematicians are defiling it. Understand the brilliance of it first, then comment on it. UGH!!

Forgive my ignorance. Is infinite chess always played with both pieces near each other or normally, at opposite ends of the board; if it is the latter then the pieces will never meet to engage in the game.

How are you gonna promote a pawn into a queen or rook or bishop or knight if its an infinite board????!!!!

"you can't guess the outcome" say that to magnus carlson and watch him get offended

0 is not a natural number

But if there is no capture or pawn move in 50 moves the game is a DRAW.so infinite chess is DRAWN

I would like to see infinite chess in real life.. 🙂 When a player dies then someone takes over. Th game would last as long as humanity. And of course, if machines outlive humans, then they can continue the game.. By the way, what if in the beginning "1" and "8" rows are infinite rows apart?

0:24 BOARD SET UP INCORRECTLY. WHITE SQUARE SHOULD BE ON THE LOWER RIGHT (from the players perspective). Do you research, PBS!!

you can't capture the king in chess

Why not just move the rook to the left/right

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?)

Um, why doesn't black move the middle rook 1 space to the right?

Video starts at 5:20

There's no corners to checkmate in.

There is a simple solution that would lead to black winning or draw by no pieces taken or repetition which is the move room D4 to E4

Here from Agadmator 🙂

Agadmator ♥

After 50 moves without capturing any pice or moving a pon the result of this game would be a draw. 😉

At 8.02 0 is not a natural number but a whole number

Agadmator's fans here?

So, black has 3 rook.. but how?

Agadmator fans unite!

9:41 I feel the doomsday clock being infinite is totally wrong. Let's just imagine a function f(x)=y, where the domain are the squares you can place your rook at, and the range is the doomsday clock value. We can't just say that f(∞)=∞, because we all know that you can't just plug in infinite as our x, and have infinite as our result, because this would mean we have squares like h∞, or ∞'8 (∞' here would represent the final column) or even ∞'∞. The board is infinite, that means it doen't have a final square…

Wait! When did natural numbers start from 0?

thumbs-up if you came from agadmator

Chess supports draws, so it’s not clear that Chess has a winner. But it either have one color that always wins or it always end in draw, with perfect game of course

There's is a small problem here. The game is draw if there are no captures for more than 50 moves. So all black has to do is to move 50 squares away from it's starting square and the game is a draw.

In addition to infinite chess, there is chess where the board is infinite, but the bishop, rook, and queen are limited to moving one million squares at a time. There are different rules you can use for knights. In the 1970's, there was a chess puzzle in Scientific American using a board like this.

9:25 Look at her hands 👌

Chess can end in a tie actually

so it doesn't enter in the theorem

6:00 No. That is wrong becuse you cannot just "remove the draws".

Only winning positions are determined

you can draw in chess. Zermelo's theorem might not hold true

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

havea king, in which case… yeah, you're right, it's hard to imagine a scenario in which they lose.But chess have ties..

7:24 Rook can move to the right, denying white a ladder checkmate.

black king can take the rook if it moves up

What if the queen makes a move, not hitting anybody? She can travel as long as she wants 😂

I have an idea, on one of your moves you move your bishop/queen/rook to the further most square, this will take infinite time to do, so both you and your opponent will die at the board resulting in a draw

The goal is NOT to capture the other player's king. She completely lost me with that comment! Sorry, if she doesn't know the rules of winning, she cannot analyze the math of winning.

That chess board is setup wrong at 0:36 white/red square always is in bottom right

BOB THE BUILDERChess would get really interesting (more than what it currently is) with more than two dimensions. Infinite 4D chess anyone?

Black king captures Rook

I didn't understand a damn thing in this video yet i couldnt stop watching.

If the sides are infinite, where do you start?

LOL

is turing completness related to the question if a game is determend?

turing complete = indetermend ?

Bah.. Apply your win determinism analysis for Infinite 3 Dimensional Quantum Chess then Miss clever trousers! Fascist.

😉

Black could move it's Rook one to the right and not be trapped anymore

My friend forgot to put his King down on the board and we accidentally started playing and when we noticed…guess who won? ME!!!

6:28 that animation tho XD

So this is supposed to be

InfiniteChess, and you said the black rook can move as far as it wants on it's first move. Then why can the rook only move afinitenumber of squares then?INFINITE CHESS IS UNACCEPTABLLLLLLLLLLLLLLLLLE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

I love chess. I almost won the tournament at my school 2 years ago and I lose in the final.

So the board is a flat surface or a sphere

Useless invention

More mathemagics…why do mathematicians think that their rules written into equations based upon assumptions that they pull out of their arses have anything to do with everyday reality. People win games because they see more than the opponent. Perception wins. Increase your perception. See more win more.

With that chess position, assuming the white king exists and is out of sight, black could use his three rooks to perpetually check the white king. That should be doable with care.

In the example after minute 7 she says white has a wining strategy but couldnt black just move the center rock one field to the left to then have a winning strategy

Sorry the room can move over for a trade

Don't do videos on logic and chess if you actually don't care about a correct argumentation. Maybe you should try proofing Zermelo's theorem yourself before making such a ridiculous video…

what branch of math is this stuff?

I would find more interesting a finite but continuous chess (like packman). Agadmator follower here

Since chess is always played over a finite number of moves, a limit value for the game could not be more than omega, assuming perfect play of course. It doesn't matter how many times it can delay the game, what matters is that that number is finite. Finite times finite is still finite, and so omega will do for every extended game, you simply have a finite number of moves of the whole game no matter how many times you get a delay (if finite). Omega is not a number that ever turns up in the equation, other than as the definite (up to but not equal to) limit.

Hey uhh… Why not just move the rook to the side in your example. Somebody. Plz egsplain.

Infinite Chess

Infinite Series

this was infinite

I'll just keep moving back

WHat the hell does this have to do with chess? I'm 5 minutse in and there's nothing about chesss….

This makes no sense. At 9:00, she basically explains that since the board is infinite, there's a doomsday clock. (To prevent games from going too long, of course… Right?) Except the doomsday clock is also infinite. Is chess that complicated that we're getting into like quantum physics where something is true and false at the same time? WTF am I even watching?

English Draughts, or Checkers, is a draw by perfect play by both sides. By the “one player has a winning strategy” this game would not be considered determined?!? What am I missing? Why is it not “determined” to be a draw? It fits all possible conditions but a forced win by either side. And as such, would cast doubt on define chess as determined given the above definition and Draughts creating a doubt about the assumptions made.

1. e4 e5

2. Rgdshfegjhdsduhdet75465457743378744678410948873 Qxjgthbgzubt+

Infinite board in a nutshell.

Around the 8:30 mark I was looking at it and think I see away to get out of the situation but I’m not sure I may just be blind

I objected when you said there are no ties in chess. There most certainly are.

Know what stalemate is

Its where a players king cant move without being attacked but they are not being attacked now

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?

You don't capture the king. Do some research.

Some things you said were wrong. Due more research on chess before you go on blabbering about it.

This is so confusing to me, it has nothing to do with chess if you remove ties or remove boundaries or any other rules. Also, it is not very informative to just discuss a hand full of positions, I dont see how infinite chess is easier to solve, compared to finite chess, so why looking at it in the first place? Why not look at the "real" game which almost a billion people play ^^

How can you make a entire video about winning strategy in chess and get it wrong that there doesn't have to be a winning strategy

the black rock could move left.

Chess can be drawn

This is wrong, (sorry) but I have deeply researched this subject. Theory has it that neither white NOR black has a winning strategy

at 7:40 another error not two but one move!

Things can't be both ways if. 33 is equal 1/3 and. 66 is equal to 2/3 and .99 is equal to 1 then if the first player moves the maximum number of spaces the game will never end because the number of spaces will be equal to infinity

Isn't ω+1 equal ω? Similar, 2*ω?

3 black rooks??? Wtf

Chess is my 2nd favourite favourite game after Ludo.

Wait can the rook not move to the right, and then white cannot checkmate?

wrong, you can't capture the king

Me: checkmate

Opponent:w-where?!

Me:*points somewhere off In the distance* I put a bishop there a while ago, so I win

Actually, chess has ties, so it might or might not be determined, which is part of why no one knows the "optimal" strategy.

But in chess you can draw.

Is it just me or did the Omega symbol look like a pair of butt cheeks??

Lying can indeed improve a thing, but only if accepted 100%.

why not move the rook to the left instead of up?

cries in pawnI might be stupid, but I dont understand why infinite chess must have a winner. My intuition as a chess player is that there would be much much more draws, since already in normal chess a huge portion of games are decided either via zugszwang(for which there are much less opportunities for), or worse, via queening a pawn, which is impossible in infinite chess?

That means that if you don't design a position with a winning plan,

Knights on the rim are dim.

White bishop: catch ya later bro

slides off into infinityFoolproof 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.