# Infinite Chess | Impossible Puzzles on An Infinite Board

hello everyone and welcome to a video that is completely different than all of my other videos why is it completely different you ask well it’s because this video uses an infinite chess board and to really show you what I mean by an infinite chess board first we’re gonna use this chess board to show to show just you know a very simple puzzle so here in this puzzle it’s black to move its like to move but the real question is in how many moves does white win so as you can see it’s not a very complicated position black black King is pretty much stuck here and if he’s not very careful white will simply play Queen here and this will be checkmate moving the rock for example here will result in Queen here checkmate so black pretty much has very few options if he goes something like rock here then you get rocky for check King moves and Queen Queen to see for this will be checkmate so this is checkmate in two moves this is in in the event that black actually plays a roof to d6 if black plays rook to d7 then white checkmate in three moves so here it is check King moves check again King moves and rookie six this will be checkmate another thing black can play instead of rook to d6 or rook to d7 black can go rook to d8 if black plays rook to b8 then you get rocky 4 check King moves Quincy four check King moves brookey six check King moves and Quinta c6 and this will be checkmate so it’s not a very complicated puzzle but it’s not a very complicated puzzle on a on a finite board what if we make the board infinite so as I don’t really have the option of making the board infinite I can add a couple of layers here so here we can add a layer down there and a layer up there so they’re not the board is infinite it goes all the way up and all the way down so now the d8 is not the final square there’s also d9b 10 the 11 to 12 D 13 and I will put a link in the description below where you can watch a very nice video by the PBS infinite series youtube channel they really made a nice video where they use this puzzle although they used it without without to this King so I added this to make it a really a chess puzzle and they they’re mostly interested in it you know for mathematical purposes but I it seemed very interesting to me a subscriber Jevon Kearse send it to me over twitter so I watched their video and it was very nice I do recommend it for all of you to watch it but it’s very interesting because they say now that if black decides to move this rook for example an infinite number of squares up in how many moves does white win so intro video like I said do check it out it’s really interesting white will eventually win because black can like move this rook all the way up like okay I can’t go to d9 because my interface doesn’t allow me but if it could it could go maybe to D D 7 millions or something white would still have a win but it will just take him a whole whole other moves so that’s actually not true because they’re only interested in this for some mathematical purposes if black went for example to D 60 was that it would actually be a draw due to the 50 move rule because no piece would be captured and no pawn would be moved so after 50 moves of simply moving Queen and rook trying to capture King it would it would simply not work so how do we how do you do this it’s a it’s very interesting I mean that’s that’s what got me interested so what what I was thinking is could there maybe be could there maybe be a position where if you use an infinite board of course where white would be winning on a finite board but if we used an infinite board that it wasn’t it will not be a draw like in this case but it would actually favor black so that’s something something that I started thinking about and I really couldn’t couldn’t find any positions like that but maybe you know there are a lot of you maybe I mean maybe you can find such a position and that would that would really be interesting so a losing position but if you make the board in then it’s like the other guys winning so that would be very interesting so yeah basically that’s that’s just what I wanted to show you as this is the first time I actually heard of infinite chess thought it was quite a quite an interesting concept so I mean I I do hope you enjoyed it and I would like to thank Beal all I want for a contribution to my channel thank you a lot I really appreciate it and I really do suggest you watch the video in the description it’s not very long really it’s like 12 minutes maybe 19 I’m not sure either 12 or 19 so do check it out and it really talks a lot about infinite chests and about infinity in general in other games so if you enjoy things like that maybe maybe a little little bit of math do check it out so yeah as usual you can check to my previous videos here thank you for watching and I do hope you enjoyed this video as it is a bit different than all of my other videos but you know it’s always good to mix things up a little bit so thank you all and I will see you soon

## 100 thoughts on “Infinite Chess | Impossible Puzzles on An Infinite Board”

1. sixillusion says:

I think black is winning if you are playing on a klein bottle. For a klein bottle, you need to identify the top edge of a8 with the bottom edge of h1, the top edge of b8 with the bottom edge of g1, and etc… For instance, rook to a5 is check. I believe you can capture the queen after three moves.

2. Pete Chan says:

cant white just move to d2 if black rook decides to go more than a few spaces up the d file and fork the peices?

3. Zain Salah says:

If it was infinite how they got the third rock? I mean how they push the pawns to the end of the bored and if it was in infinite how they can got the third rock

4. Wolf Fang says:

The key for finding a puzzle where White wins on all Finite (8xn, where n is an integer) and loses on the infinite is to identify the difference: In the finite case a move such as Rock to hn is possible (moving to the top edge).

This can be incorporated into a puzzle if you need to limit the amount of squares on the g-file attacking the rook.

Imagine an 8xn board (where n > 22 )
https://lichess.org/editor/8/6qR/8/2p1P1p1/4P1P1/R1pNP1P1/R1PpP3/RR1B1k1K_w_Q_-
And there are Black Queens on g14 and g21.

In the finite case:
1. R h7hn!! This threatens Rfn+ and eventually wins (Hopefully anyway, I may have missed some cases.

1. … Q g21gn 2. R a3an
2. … Q g14g(n-1) 3. R anxgn and wins
2. … Q gnf(n-1) 3. R anfn Q g14f14 4. R a2an This prepares Rxf(n-1) leading to mate.

In the infinite case:
1. R h7hN (Where N is a large integer)
1. … Q g21g(N+1)!! 2. R a3aN Q g14gN 3. R aNxgN Qg7xgN and black goes on to win.

5. Victor Salinas says:

The game would never end because Infinite is not a number it's a concept so the rook would move up forever.

6. Lucas Castro says:

Wtf! Yesterday I watched some vídeos in this channel and now they are here!

7. kasparov9 says:

Kasparov vs Kramnik 1994…

http://www.chessgames.com/perl/chessgame?gid=1070747

8. J D says:

Sorry Agadmator, but your explanation betrays a poor understanding of the definition of infinity. Infinity is that which cannot be reached; subtracting any value from infinity yields infinity again. If black moves the d5 rook to d(infinity), he can never be mated because any distance the rook or queen travels up will not block the king against the rook. Things moving an infinite distance away become unreachable instantly, even to other things which are moving an infinite distance away.

9. One Shot Man says:

throughout the entire video the one question in my head was "but if the board is infinite, then the rook will never stop moving up the board so the game cannot proceed" xD lol i know i have a wacky brain

10. Virio Guido Stipa says:

Fantastic!

11. DFTA (Don't Feed The Animals) says:

When he said D12, did anyone else think of Eminem?

12. Ryan Levine says:

Im confused. How does an infinite board allow a pawn to promote to rook, giving Black the needed third rock for this puzzle to be possible?

13. Brandon Nobles says:

You can't make the board infinite?! Unsubscribed

14. srj34 says:

I don't know the solution, but Carlsen would win this position with either color.

15. comedy dance with pratik Tayade says:

I got a question
If black move it's rook to infinity then you can give queen to d3 check. And then win the rook on f5 and then win other rook as well. Then go for a checkmate. Not using that pattern which is on the board but instead other patterns. Could it be possible?

16. Virio Guido Stipa says:

On infinite chess board, no pawn could queen, that is slightly classist.

17. IronDuke says:

That's what he gets for promoting to a rook instead of a queen

18. _teheheelover _ says:

What happens if u play took e5

19. Teo García Villa says:

#suggestion Wesley So vs Garry Kasparov in Ultimate Blitz Challenge 2016 – Game 10. In my opinion one of the most beautiful blitz games ever played on history of chess. I would love to see that game and I'm sure everyone else will enjoy it aswell! Can I get a shotout for the suggestion? Sorry for my english, Argentinian fan here 😂. Love your vids!

20. mAsane masAn says:

What if black play rook to b3 ??????

21. Márcio Tondin says:

It was in this position that white resigned the game.

22. thom1218 says:

Look – if the guy playing black can all of a sudden make the board "infinite" – then he's already in a "winning" position in my book, and my next question to him would be: can you do something similar with the 3 dollars I have in my wallet?

23. Shyam Das says:

#suggestion Please do a video on Sam Lloyd's famous puzzle : The Love Chase.

24. Mat P says:

Hello Agadmator! I'm from Mexico City and I really enjoy watching all your videos! I'd like to request a game that is really interesting: Khanya Mazibuko (South African 16yr old CM) VS GM Hikaru Nakamura. It's very interesting that he actually decided to sponsor him after that game. I've not yet found the match, but I'd love to see it analyzed by you… BTW, HE BEAT THE GM!!! Thank you as always for these wonderful videos!

25. Infinity Salinity says:

#suggestion

Leela vs lovlas
https://lichess.org/study/kLXrkT4M

26. Maple Tree says:

With infinite board, passed pawns loose their value as they can never promote, so say a game where it is black with a rook versus white with 6 passed pawns, finite board means white is likely going to win but on infinite black would likely have better chances

27. Ronen Zinman says:

Two principal ideas for a position that will favor one side and on an infinite board will favor the other side:
1. Queening that normally cannot be stopped, but is irrelevant on an infinite board.
2. A piece that would normally get trapped in a finite board, but can save itself in an infinite board.

28. Aaron Morefield says:

Depending on how pawn promotion works on an infinite board, the difference between obviously winning and obviously losing could be trivial

29. PissedFechtmeister says:

If it's an infinite board then pawns can never get promoted. So how did black get that third rook?

must be hard to checkmate on an infinit chess board

31. Escape Room Leander says:

White to mate in 232,655? I can barely see it when it's mate in 2.

32. Darkota122 says:

Hey short answer to your question. No its not possible for white to be winning on any finite (no matter how large) board, and black to be winning on an infinite one. In Math you could discribe it like this:
The number of ways for black to win (if white plays the best moves) have to be zero for every amount of additional squares on your board. But for the limes of your squares to infinity the number has to be higher than 0. (At least one way for black to win).
We can build a sequence of A(n) where A is the number of ways to win for black and n is the number of squares you add to your chess board. It trivial that A(n)= 0 for every n < infinity. So you can use convergence laws on a constant sequence and see that the limes -» infinity is just the constant itself. Meaning A(infinity)=0 and there are zero ways to win for black. This contradiction proves that it is not possible for white to win on an finite but black to win on an infinite board.

33. Pranav Khetan says:

I think it's your first video which got so many dislikes and so few likes

34. Eric Bender says:

I have a question about your name. Do you like comic books?

35. Josh Belan says:

Do black have a defense?

36. Josh Belan says:

What if re5?

37. Pizzous says:

Infinite Othello/Reversi would be a draw because there's no corner.

38. ytmndman says:

Try doing the Knight's Tour on an infinite chessboard, then do it over again a few more times.

39. Isanes West says:

А что на ладья Rdc5

40. YEVGEN SHEVCHENKO says:

Here is the position: https://lichess.org/analysis/8/kp4nP/8/8/8/8/8/4K3_b_-_-
It remind me about "The Defense" by Vladimir Nabokov. The main character of the novel is top chess player. In one of the games he gets into losing position and then he realizes that the knight should jump out of the board to win the game. Eventually it becomes metaphor for the suicide.

41. Kaushik Chail says:

Maths has huge application in almost every field as you can observe in chess also!!
There are tons of reason to fall in love with this subject!

42. Ong Bonga says:

1… Rd57,435,432,765,754,123,543

*offers draw

43. Razi Akhtar says:

Hey man.. do a live-stream.. its been quite some time.. a 2 hour 3-0 non cash tournament on lichess.com with your witty commentary and lively beer fest would be my preference …
#suggestion

44. איתי סיטבון says:

Black- Rook in g8 king in h8 queen in a8and pawns in h7 and g7
White- knite in h5 king in h2.
White started mate in one in regular board and he lose in infinity board

45. H3CCH3 says:

Interesting puzzle.

Note:
A really important factor in this logic is that you only expand the Y-axis indefinitely and not the X-axis, and that no one wants to draw through the 50th move rule.

If there was no 50th move rule, there wouldn't be any possibilities for change and the game would simply end in a draw or continue indefinitely, following this pattern. However, introducing this rule will actually provide black a chance to win as long as white doesn't resign through this rule, or is fast enough to get to check with a new queen before black's checkmate. Think about it, for white to avoid a draw they have to keep moving a pawn. Since the rule for promotion is still the same, https://en.wikipedia.org/wiki/Promotion_(chess), black will have three moves, without being checked by white, before the fastest white pawn can get its promotion at h8. As long as white continue this pattern of walking up the board and trying to promote the fastest pawn, black has a checkmate in three. Resulting in a win before white's promotion.

Conclusion:
If white plays a pawn tactic, that would make black's checkmate slower than white promotion + one more move for white, white is the winner through checking with new queen followed up with the old queen. Whereas as long as isn't the case, black will win.

46. Barbara Hajdin says:

This must be your the most stupid video ever Antonio.

47. Anup Kulkarni says:

Very Very Stupid video

48. Ilkinond says:

Ah! Three rooks means a black pawn must have traveled an infinite distance to the far rank….

49. Alexander Sanchez says:

A trivial example of a scenario like you asked for is where you have a tactic which checkmates in the corner on a finite board. If there is no perpetual and the material is uneven enough, the tactic won't work and the other side will be winning.

50. Chouche Wolf says:

I Watched her video and white would win in omega moves.

51. leendert starink says:

I think i came up with a solution to your puzzle.
It is a bit exaggerated but the idea is simple. Here is the FEN of the position
k3r3/pbQ5/N4rrr/5qqq/8/5PPP/5PK1/4rr2 w – – 0 1
Black has a big material advantage, but white can deliver smothered mate. However, smothered mate on an infinite bord is not possible, so white can no longer deliver mate and black will win the game on the infinite bord

52. LeikoLaboeProduction says:

To answer your puzzle challenge… there would have to be some position where white would be winning if he had a pawn on the 7th rank with the opportunity to queen, but with an infinite board, he would be losing.

53. Matt Mackay says:

This is stupid, if the rook can move infinite squares then white never gets their turn and if the rook could actually move infinite squares then the king can also move infinite squares, it's a draw.

54. F S says:

This must be one of your most interesting videos ever Antonio

55. Mondsüchtig says:

56. Ηρακλής Χατζηνικολάου says:

#suggestion 8 queens problem. Also a link to a numberphile related video https://www.youtube.com/watch?v=Km024eldY1A&t=642s

57. yesthatislife says:

Wtf black needs to move rook on d row, next queen d3 check and pics up the rook on d diagonal, whats the problem white wins even with infinite board easly…????

58. dumitriu laurentiu says:

if infinite rock maybe .
because rd3 and takes rock at infinit posibilities .

59. Richard Feynman says:

The normal, finite chess is alread to much, I think infinit chess is maybe hopeless…;-)

60. Łukasz Brembor says:

generally I don't like it (i mean subject) – but don't worry about it xD

61. Fandartmartiaux says:

Can't white move the queen to D6or the rook to e6 and then do something from there?

62. Evan Scarpino says:

Idk about adding pieces or whatever but ik you can't just say I move my rook to infinity it has to be finite meaning you eventually lose as black does that change?

63. Albino Deer says:

4r1k1/5ppp/8/8/8/6QQ/5PPP/6K1 b – – 0 1
In this position black is winning, but if it were infinite, white could escape and there would me no mate
It is a silly example, but shows the pattern for all other solutions 😉

64. Sybrand van der Walt says:

Really..? The solution is so easy. Keep checking the black king towards the a file. Separate the black king from the rooks and do a checkmate. Avoid the stupid infinite board going up and down (It is only there to confuse you).

65. Jesse Gimble says:

Most positions where White sacrifices a lot of material to checkmate the black king in a king hunt would fall under this category because the king could go to g9 or h9 or something like that.
Also I interpreted this position as Black can't save himself on an infinite board if we revoke the 50-move rule, because he can't actually play a rook to d-infinity. He must move it a finite amount of squares, and then white can checkmate him with that number minus four amount of moves.

66. Dexter Miranda says:

For infinite chess board, white will not able to have his turn, because the move of black rook will not end.

67. Steven Calwas says:

Regarding a losing position that flips to a winning position: How about a position where the losing player makes an infinite move which places the opposing player in zugzwang. To arrive at such a position, perhaps start with a known zugzwang position and add pieces to make it a losing position for the opponent. Then the infinite move flips the tables… to infinite zugzwang!

68. David B Jacobs says:

2:19 We learn Agad does not like math.

69. David B Jacobs says:

For a position where white is winning on a finite board but black is winning on infinite board: I'd say start with one where white is about to promote their pawn. On an infinite board, promotion is impossible. Back row mates are also not a thing.

White to play normal finite board: Smothered mate in 1. Infinite board: Black wins easily.
https://lichess.org/analysis/kn2Nqq1/nn6/q3qqq1/1q3qq1/qq1q2q1/qq1qq3/qq1qqq2/qq1qqqNK_w_-_-

71. Umang Chudasma says:

Agadmator that is not possible practically bcoz if u take finite number of moves as 'x' then in infinite board it will always cover x moves irrespective of boards length mentioned as infinite and ofcourse at the x moves white is winning no chance of black

72. jonni2 says:

Really cool!!

73. Dhananjay Kavoor says:

what about Rd5 to e5 ….?

74. Dhananjay Kavoor says:

and another thing… you cannot put infinite number in real world practical applications lol…

75. Flame says:

Chess.com

76. Santiago Continenza says:

These are some crazy thoughts. I like the concept, though!

77. Charles Qualmann says:

A nice detour from the regular, Agadmator! It reminds me of your videos on the Harry Potter and House chess games 🙂

78. Ethan Cheung says:

That’s one of the interesting topic from Kelsey. Good pick!

79. Aksoan Kapoor says:

This FEN code…
7k/bR4p1/2R5/6n1/8/3q3r/6K1/5r2 w – – 0 1

The bishop, two white rooks and the black king and pawn are at the edge of the finite board. In a finite board its mate in 1 for white but on infinite board the bishop will eventually be able to block check. And the black mate the whit king.

So on a finite board white win but infinite board black win.

80. TheFirstGodfather says:

I don’t think there is any way to make it so that black is winning. Even with an infinite board, it’s forced mate in n+1

81. Lawrence Lepes says:

I've never heard of Infinite Chess before.

82. Dutchy Sweden says:

Black plays 1.Rdi

83. So ein Typ says:

Every chess puzzle that involves promoting a pawn would probably be a win for the other side with an infinite board.

84. 4bdullah says:

Why isn't the board horizontally infinite as well?

86. Dimitar Simeonov says:

wouldn't queen c8 be a checkmate after Re6 Kd7 – if the rook on D goes too far away the queen and rook can checkmate

87. Aleksandar Stojnov says:

What if rd5 move to e5? Maybe that can give a black some chance

88. Baert Bart says:

As the board is infinite, black would be moving "forever". This means white can never move. I propose you offer a draw 😉

89. KaiokenComplete says:

Is there a published solution to this?

90. Dragonfly 28 says:

Rook d5 to e5 avoids checkmate, just saying

In the puzzle on the "finite" board, what happens if black goes rook d5 to e5?

92. J7 says:

Many "trapped piece" positions may have the property you describe. Well, except that on an infinite chess board it would be very hard for either side to win. I'm not sure Q + R + K vs K would be a winnable endgame on an infinite board in all circumstances. I'm thinking something like trapping the enemy king between the rook's "wall" and the king, then sweeping in with the queen to deliver checkmate.

93. Daimon says:

Black Rook b3 allows an escape?

94. LeikoLaboeProduction says:

One simple solution to your puzzle https://imgur.com/a/iTg7PGh

95. Hovik Gasparyan says:

Agadmator, to answer your question at the end, yes it is possible in the following scenario: white is down in material, but threatening to promote to a queen on its next move and gain an advantage. By playing the same position on an infinite board, promotion would not be possible, and black would be the winning side.

96. Asit Giri says:

Do more infinite puzzle … These are interesting

97. ๖ۣۜ♥๖̶tacokitten๖̶ says:

Rd∞ achieves a draw in the puzzle for black because white will make 49 checks and on black's 50th king move he will claim a draw by 50 moves, as this is truly a "perpetual" check 🙂

98. Edgar Ampaya says:

50 move rule is when only the opponents king is only left. So white is still winning.

99. Tomas Hindrich says:

I mean you dont really need infinite chess board, just imagine connecting the 1st. row with the ninth row and a column with h column but it would need to be separeted on the start otherwise it would also break the kings rule of chess by which I mean how close can be two kings to each other

100. Joseph Donovan says:

I'm so glad rowan atkinson is teaching chess.