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The Longest Possible Chess Game

Chess is a game of small advantages and large numbers.  For example, according to the U.S. Census Bureau population clock (http://www.census.gov/ipc/www/popclockworld.html) there were 6,637,552,653 people living on the earth on December 14, 2007, as I write this.

Suppose that each of these persons was engaged in a game of chess with an alien from another world, and they played bullet chess with only one minute on each…um…being’s clock to finish each game, and each game took the maximum two minutes to complete.  Further suppose that these beings played continuously without rest.  Under these circumstances there would be a total of 4,779,037,910,160 games played per day.  After one non-leap year they would have played 1,744,348,837,208,400 games, and after a billion years they would have played 1.74434883721 x 10^24 games.

That may sound like a large number of games, but even if we increased their playing speed to an entire game in only one second, the Chess Gods would laugh at the small number of games thus presented to them for analysis.

The number based on one second games is vanishingly small compared to the number of possible games in chess even if we limit those games to just 40 moves. 

And games can last longer than 40 moves.  I’ve played some games myself that approached 100 moves against my PDA.  Recently, I’ve begun wondering just how long the longest possible chess game could be. 

Within a single google I was able to find that the longest recorded game of chess in history was an epic played in Belgrade in 1989.  The Belgrade Marathon was a contest between Ivan Nicolic and Goran Arsovic that lasted over 20 hours and ended in a draw after 269 moves due to the so-called “50 Move Rule”, which I will discuss in a moment.

Here is that amazing game. 

 



That is a long game indeed, but it is far from the longest possible game, which is what we are trying to determine.  Is it 1,000 moves?  10,000 moves? 

 The site chess-poster.com http://www.chess-poster.com/english/notes_and_facts/did_you_know.htm  claims that the longest possible game consists of 5,949 moves, but did not offer any details regarding the computation.  Since I have not seen this particular number documented elsewhere, I decided to compute it for myself to either verify or refute the claim.

The first question to ask is whether there are any limiting factors that prevent a game from continuing ad infinitum.  For this, we turn to two authoritative bodies, Fédération Internationale des Échecs (FIDÉ) and the U.S. Chess Federation (USCF).  According to the FIDÉ official handbook rules:  http://www.fide.com/official/handbook.asp?level=EE101

“The game may be drawn if each player has made at least the last 50 consecutive moves without the movement of any pawn and without any capture. (See Article 9.3)

9.3   
The game is drawn, upon a correct claim by the player having the move, if
a. he writes his move on his scoresheet, and declares to the arbiter his intention to make this move which shall result in the last 50 moves having been made by each player without the movement of any pawn and without any capture, or
b. the last 50 consecutive moves have been made by each player without the movement of any pawn and without any capture.”

This is the well-known 50 move rule in competition chess.  Let us compare that definition against its counterpart at the USCF.

(http://main.uschess.org/content/view/7324)

“50 Move Rule-
If no pawns or pieces have been traded for more than 50 moves, a game is determined to be a draw. This rule requires that the player who makes the claim write down his moves.”

These two definitions differ.  In the USCF version there is no reference to the absence of a pawn move, so presumably the 50 moves could include a pawn move.  And a careful reading of both versions reveals that the game is not required to be drawn, only that it may be at the discretion of the side to move.  Let us be merciful on the poor players, however, and assume that any extremely long game would indeed result in a draw under the 50 move rule.  Otherwise, we have nothing further to discuss here regarding the longest possible game.

With respect to the issue of a pawn move taking place within the 50 moves I have to rule against my own countrymen and side with the FIDÉ version of the rule, which is much more clearly written, and which prohibits pawn moves during the 50.

So here is our approach, roughly speaking.  (We will see a modification to this below.)  We need to compute 49.5 full moves, where a full move is a move by each side, followed by either a pawn move or a capture, and then repeat this process until it is no longer possible.  The maximum number of pawn moves is 6, at which point it will promote to another piece.  Since there are 16 pawns it appears that we could have 6x16 = 96 total pawn moves.  However, before all 6 moves for a pawn can occur the opposing pawn in front of it must be dealt with, as the following diagram shows.


There are two ways to manage this obstacle.  The first is for one pawn to capture another, allowing the victor and its companion behind it to advance to promotional nirvana.  The pawn that previously opposed the victor on the original file is also now free to advance its full 6 moves.  If we allow the vanquished pawn to advance to its 6th rank, that pawn would have its maximum number of moves before its sacrifice.  For example if the e2 pawn moved forward 4 times to reach the e6 square, whereupon it was captured by its opponent on f7, then the remaining 3 pawns on the e and f files could continue their full 6-move journey.  Thus, among these 4 pawns there would be a total of 22 pawn moves.  If a similar set of circumstances occurred on the other 3 pairs of adjacent files, then 88 of the 96 theoretically possible pawn moves could take place.

However, the second way of dealing with the issue would allow all of those 96 moves to take place.  This could happen if the capturing pawn took a piece, rather than an enemy pawn.  Look at the simplified diagram below, which shows a position after fxe6. 

 

You can see that the white pawn on the f file now has no enemy pawn in front, so it can advance unopposed.  But what about the 3 pawns now on the e file?  This can be handled in a similar fashion.  We simply offer up a piece for the white pawn on the e file so it can capture onto the f file, and now both white pawns on the f file can advance and both black pawns on the e file can advance. Clearly we can resolve all of the opposing pawns in this same fashion, allowing all 96 pawn moves to take place.

We are now almost ready to proceed with our computation.  The game will begin with knight moves, 49 by each side and ending with a 50th knight move by white.  With black to move he must either capture or move a pawn to avoid the draw, so let us say that he plays any pawn forward one square, say e6.  This frees his queen as well as his bishop and king to move onto e7, if they so desire.  Remember, we’re only interested in moving as many times as possible, not in winning the game.

Both sides then move their pieces repeatedly until another 49.5 full moves have occurred, at which point black again either moves a pawn, or he could also capture to relieve the monotony. 

But wait!  This cannot proceed indefinitely or we will fail to achieve our goal.  Notice that Black is doing all the work on his 50th move, and if this continues unchecked (sic.) we could end up with the position shown above in Diagram 2, and now a pawn must capture a pawn and we will lose some of our precious 96 pawn moves.

Clearly, we cannot always play the full 49.5 moves before a pawn advances or a capture occurs.  In the latter case, we would end up with all of White’s pieces being taken, and he would have only pawns remaining on the board.  In order for them to move at all, they clearly have to move sometime before Black takes his 50th move on these sets of 50.

And for both sides, some of the pawn moves need to involve piece capture rather than forward moves, as discussed previously.

So we must alter our original outline and we will say that the game begins with 49.5 knight moves, followed by a Black pawn move one square.  This is then followed by 49 full moves, whereupon White moves a pawn one square on his 100th move.  We then repeat this process, alternating between Black making a pawn move or capture on his 50th move and White doing so. 

We can now begin to sketch our computational formula as follows:

(49.5 + b) + (49 + w) + (49.5 + w) + (49 + b) + … , where b stands for a move by Black that consists of a pawn move or a capture, and w represents a pawn move or capture by White.  The ellipsis stands for continuing repetitions of exactly the same terms of the formula that are fully written out.  Notice the pattern of 49.5 moves alternating with 49 moves on adjacent terms, and each of b and w occurring twice in a full cycle.

Now we need to determine how many times this cycle can repeat itself.  As we said each b or w can be a pawn move or a capture, and we want some of the pawn moves to involve captures.  To be precise, we want exactly one such pawn capture per file as outlined above in order for all pawns to be able to promote.  Thus, it seems that we need 8 moves involving pawns capturing pieces.  This can involve either the capture of original board pieces or pieces that were created from previously promoted pawns.  It does not matter which side does the capturing, just so long as one per file happens. And since there are those 8 pieces captured by pawns, we know that after the 96 pawn moves, there will be 24 units remaining on the board. 

And following those 96 pawn moves, the remaining occurrences of b and w necessarily involve pieces capturing other pieces.  Since there are 22 pieces available for capture (omitting the Kings), we now have the necessary information to complete our formula.

There will be 96 terms of the form (N + m), where N is 49.5 in 48 of those terms, and 49 in the other 48 terms, and m will be b in 48 terms and w in 48 terms, equally distributed between the 49.5 and the 49 coefficient terms.

We can write this much out as:

24(49.5 + b) + 24(49.5 + w) + 24(49 + b) + 24(49 + w)

We then add to that formula the following additional terms needed for the remaining 22 captures:

(49.5 + b) + (49 + w) + (49.5 + w) + (49 + b) + … where each of b and w now represents a capture of an opposing piece by Black and White, respectively.  There are four terms in this expression and since we need 22 captures, we will have 5 occurrences of each term listed plus an additional occurrence of the first two terms (49.5 + b) and (49 + w).

Thus, the formula for the final 22 captures can be written as:

6(49.5 + b) + 6(49 + w) + 5(49.5 + w) + 5(49 + b).

Combining this with the previous formula we arrive at:

24(49.5 + b) + 24(49.5 + w) + 24(49 + b) + 24(49 + w) + 6(49.5 + b) + 6(49 + w) + 5(49.5 + w) + 5(49 + b), which simplifies to

30(49.5 + b) + 30(49 + w) + 29(49.5 + w) + 29(49 + b)

Since each of b and w represents a single half-move, we can combine terms of the form (49.5 + m) and we can combine terms of the form (49 + m) arriving at

59(49.5 + 0.5) + 59(49 + 0.5) = 59(50) + 59(49.5) = 2,950 + 2,920.5 = 5,870.5

Thus, the math tells us that the longest possible game is 5,870 full moves plus an additional move by White, which would be that final capture, apparently by White’s King resulting in a necessary draw, since only the Kings would be left on the board.

My answer computed here differs from the 5,950 figure that appears on chess-poster.com, which is exactly 78.5 moves larger than my figure.

One or both of us has made an error in calculation.  I am perfectly willing to believe that it is me, and I will issue a revised version of this blog along with appropriate credit given to the mathematician who corrects either myself or chess-poster.com or both.

Regardless of who is correct, the preceding discussion shows us that pawns are not only the soul of chess, but define it in a very mathematical sense as well.  And for any of you who try to actually play out a longest possible game (there will be very many such games, not just one), remember to choose your move carefully, in chess as in life.


Comments


  • 6 months ago

    iSUCKatCHESS1011234

    I just finished playing a game on chess.com that lasted more than 650 moves. The other opponent had more time than and purposely kept moving and refusing the draw so that he would win on time. Luckily, I had pre-move on, and after 600 painful clicks back and forth with my king, I won on time. I think the 50 rule thing on chess.com was glitched or something

  • 7 months ago

    JudePasulka

    my longest game was 309 moves

  • 7 months ago

    JudePasulka

    my longest game was 309 moves

  • 13 months ago

    thechessstick

    Interesting.  I gave it a cursory reading and it all looks good.  I wonder though does it take into account that before the 1st pawn move is made that the Knights can jump around for about 48 mores and return to their original squares thus adding 48 moves to this equation?

  • 14 months ago

    freaky25

    The longest game I have ever played is a 95-move 5 minute blitz game on chess.com, although I have seen 88 moves in a 1 minute game.

  • 14 months ago

    zespopolitan

    Ok well it is quite obvious if you look at it logically that the longest chess game possible is well infinite because hear me out here lest say white moves their knight out and let's say black does the same blacks idea is to copy white and white moves his knight back in to its original spot and black copies this can go on infinitely I'm not saying its a well played game on either ends put it is infinite therefore it is the longest game possible just saying sometimes when you look at things with just plain pure amazing logic you can skip complex and complicated equations

  • 3 years ago

    Flatlander

    I stand corrected, and have removed my earlier post.  My new equation is 50*(16*6+22)-2=5898.  I had overlooked the problem explained in point 1 of the previous post, made by Sonny_E.  I endorse that post as essentially accurate and sufficient to explain the solution, except where it states that just two kings on the board, “would result in a draw by 50-move rule because black cannot cause checkmate or stalemate with only a king.”  If that were the case, which it would be under some rule regimes, the result would be 5948.  But going by the imposition of an insufficient material rule, it ends the game immediately, at 5898.

  • 3 years ago

    Sonny_E

    I worked out my own answer for the longest chess game that can occur if a draw would always be claimed by the 50 move rule, and insufficient material for a checkmate causes an immediate draw.

    2 sides * (6 pawn moves to promotion * 8 pawns + 8 captures of opposing promoted pawns + 3 captures of other opposing pieces than the four that need to be captured by pawns) * 50 move rule - 4 times that the side that makes pawn moves or captures switches * (1/2 move) = 5898.

    Why not 5948 moves? 5948 being an answer given at http://www.drpribut.com/sports/chessfaq.html

    As stated by Flatlander, who also brought my attention to this thread, the last possible capture immediately ends the game, because there are only kings left. The last capture (which is of a rook or queen, so that the game wouldn't be drawn already) could be substituted with any legal move and the same number of moves would result.

    Why is it that the side that makes pawn moves or captures needs to switch at least four times? (I was confused about this for an hour myself.)

    1. Observe that black starts with the responsibility to keep the game going, on black's 50th. Only the side that has the responsibility will move pawns or capture. Black cannot make all four captures by pawns that black needs to make to be able to pass all black pawns until white moves a pawn, otherwise black can only capture two white knights by pawns (although black could capture other white pieces by knight moves.) So the responsibility must switch to white in order for all pawns to be in a position to be promoted.

    2. After white has made four captures by pawns, the responsibility must switch back to black so that black can finish putting all pawns in a position to pass. Black may then promote all 8 black pawns.

    3. The responsibility then switches a third time so that white can finish promoting all pawns. Now white can clean up all black's pieces (which may be all underpromoted to make the problem easier.)

    4. The responsibility switches a fourth time, so that black's king can miraculously mop up all of white's blundering pieces, except maybe for the last one, on black's 5898th, which would result in a draw by 50-move rule because black cannot cause checkmate or stalemate with only a king.

    Others at chess.com who have found the same result (5898 moves):

    cobra91, US, posted 10th July 2011 in a response to http://www.chess.com/forum/view/general/i-have-calculated-the-longest-possible-chess-game

    CM ilmago, Germany, posted 28th July 2011 in same thread, with a detailed decription of the four changes of sides in which capturing or pawn moves take place

    NM ozzie_c_cobblepot, US, concurred 29th July 2011 in same thread, though his earlier answer was over by 49.5

    TheGrobe, Canada, posted 29th July 2011 in same thread a description of his calculations "without reviewing anyone else's analysis"

    The earliest priority of discovery for this answer I've found: Kevin Bonham, Australia, explained the answer 5898 in a letter to Bruce Pandolfini's column on Chess Cafe, May 2001, reprinted in The Q&A Way in Chess, By Bruce Pandolfini, 2005 (preview available on Google books) and reprinted by Bonham at http://www.chesschat.org/showthread.php?p=270217 post #5

    An earlier answer made obsolete by Bonham's: "In 1966, Eero Bonsdorff, Karl Fabel, and Olvai Riihimaa gave 5899 as the maximum number of moves in a chess game" according to http://chessprogramming.wikispaces.com/Chess with a footnote "Eero Bonsdorff, Karl Fabel, Olvai Riihimaa (1966) Schach und Zahl - Unterhaltsame Schachmathematik. Seite 11-13, Walter Rau Verlag, Düsseldorf (German)"

  • 3 years ago

    JOCCLE

    First of all sorry for my poor English and congratulations for your post. This same question I did in June 2006 and 5949 was the result, not to exist in the FIDE rules no rule by insufficient material, like says Flatlander. The answer is explained in a way different from yours, but I fear that in the Spanish language!


    http://jesusorera.blogia.com/2006/070301-la-partida-de-ajedrez-mas-larga-posible.php

    Best regards

  • 3 years ago

    Flatlander

    [COMMENT DELETED]
  • 4 years ago

    vowles_23

    Looks like complex stuff...I didn't bother reading the equations :)

  • 4 years ago

    7thSense

    Wow... this is amazing. I am still looking at it today, trying to figure out the complex equations to this. Love this!

  • 4 years ago

    NM ozzie_c_cobblepot

    Kurt,

    This came up today for me. I think the answer is somewhere in between 5870.5 and 5950. The question is how often you have to make "only" 98 ply instead of the full 99 ply.

    Let's ignore the issue of the two kings moving around at the end, except to point out that the number of ply will be 99, where the player of the other king will announce his intention to claim the draw.

    If we list out all of white's moves, and all of black's moves, it is helpful for one side to make their required moves consecutively, so as to avoid "taking turns".

    Each side can make 48 pawn moves and 15 captures of the opponent, minus the 4 duplicate pawn-captures. Also, at the beginning of the game, black is "on move", since after 99 ply it will be black's move. On to an example solution.

    Black: In this first phase, black can make two of the requisite four captures, by taking white's two knights. White can avoid making pawn moves by playing Rg1-Rh1. Note that we can construct any position we like for black, in this phase. The only constraint is that there are only two pawn captures.

    White: In this phase, white can make all four requisite pawn captures, but will not be able to make all pawn-promotion moves.

    Black: Black can make the remaining two pawn captures, but will not be able to finish capturing all of white's pieces because white has not yet promoted all pawns. But black can make all of his pawn promotions, which enables white to capture all of his remaining pieces on his next turn.

    White: In this phase, white will capture all of black's remaining pieces, leaving only the black king. Also, white finishes his pawn promotions.

    Black: Here, black will capture all of the remaining white pieces.

     

    So there is a total of four ply to be removed from the 5950 number, giving 5948. The one last ply to be removed is because of the K v K position at the end, which does not add 50 moves, but only 49.5.

    So my answer ends up 5947.5

  • 4 years ago

    higher_being

    Chess is a game of mathematics of hidden patterns.The number of possible board positions(roughly of the order ~10^113) outnumber the number of atoms in the whole universe,yes u read it correctly,the universe.In game theory in computer science we always refer chess as a classic example,the permutations and combinations are so complex and large and this is precisely the reason why we never get bored of chess. 

  • 5 years ago

    ovalle1111

    That 50 move rule is dumb, I have always hated it. What if you can mate in 51 moves? Or 67, or 93? Somestimes it takes a little maneuvering and stratergizing before the pieces can be in the right place.

  • 5 years ago

    dunce

    hehe well we got 96 moves into a 15-minute game:

    http://www.chess.com/forum/view/game-showcase/craziest-game-ever

  • 5 years ago

    peterwaffles

    Insane long game! jajaja Thanks for the post!

  • 5 years ago

    ringwraith10

    Excellent post. I don't think I will ever have the patience to play a 150+ move game.

     

    Thank you!

  • 6 years ago

    estevon

    White 33rd move should of been Knight captures g2 instead of Neg3.

    white 95th move should of been Rook captures d6 instead of Rf3.

    White 166th move Kd4 was a very key positional move for white.

    The 201st move was outstanding.

    Was this game in a tournament=The very 1st game of the tournament.

    Did this game take place over 2 Days.

  • 6 years ago

    KilgoreBass

    Hey Kurt, here are some links you may be interested in.....if you Google, you will find a range of values referenced for the longest possible chess game, from 3150 (as I have read in Levy & Newborn, How Computers Play Chess, page 154), up to over 6000....this is ignoring any possible contempt for draw, of course, which can make a game last forever if the players wish it to, as the rules imply.

    this person seems to set the range, at least

    (emphasis added)

    """As a first approximation right the same for both sides, allowing as much
    as 50 * 2 * (6*8 + 15) = 6300 moves.

    I'd guess it is impossible to reach this maximum number of moves, but it
    should be possile to find a game where one side reaches this maximum
    number of moves, so truth lies somewhere between 3150 and 6300 moves."""

    http://mathforum.org/kb/message.jspa?messageID=3783005&tstart=0

    (or this link, which looks to be a compilation)

    http://www.tomshardware.com/forum/48906-13-longest-chess-game

    it seems very curious how so many people have come up with different numbers in the same ballpark, using similar logic ......hmmmmmmm.....I am now very curious who is correct!

    PS: this link supports the 5950 number, but it is the "simple calculation", maybe too simple (?)

    http://www.drpribut.com/sports/chessfaq.html

    """Subject: [24] Trivia

    How long is the longest possible chess game?
    The basic idea is a player may claim a draw if fifty moves elapse without a capture or a pawn advance. Ignoring the special cases where more than 50 moves are allowed by the rules, the answer is after Black's 5948th move, White is able to claim a draw. The simple calculation is (<Pawn_moves + - + <Drawing_interval_grace_period) * <Drawing_interval, or (16*6 + 30 - 8 + 1) * 50 = 5950; we're able to trim two moves from this total by observing that sequences of Captures/Pawn_moves must have (at least) 4 alternations between the two players."""

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