[Retros] The Longest Chess Game

[Rol, Guus]

Hi Franco,

I have seen this question and the answer - under the assumption of
automatic 50 moves termination - to it before, I think on the Internet
(or possibly in die Schwalbe). A quick Internet-search today delivered
these results:

(A) "The longest Chess game theoretically possible is 5,949 moves"
supposedly explained on chessposter.com (members only)
(B)(In a mathematical discussion forum):
There is a very interesting short note entitled "A mathematician
gives an hour to chess" by Donald MacMurray in the book
"The best of Chess Life and Review", Volume 1, 1933-1960.
The article is entertaining in itself, and comes up with the
bound of 6237 moves.
(C)(In the same mathematical discussion forum):
Given the 50-move rule, it is fairly easy to calculate an upper bound on
the
maximum game length. Every sequence of 50 moves (by each side) must
contain
either a pawn move or a capture; let's call such moves critical. Each
pawn
can move at most 6 times, thus the maximum number of pawn moves in a
game is
96, and there are at most 30 captures in a game, thus in any game there
are
at most 126 critical moves. Also, to permit each pawn to advance to the
8th
rank, the opposing pawns in each file must get out of each other's way,
which can only occur by a pawn capture. If a pawn captures another pawn,
then the remaining moves of the captured pawn are lost, and there are at
least 2 such moves, since a pawn capture can take place on at most the
6th
rank. Thus, we want the pawn to capture a piece, which costs only one
critical move, since the capture is also a pawn move. Consider for the
moment just the a- and b-files. One sequence of moves to unblock the 4
pawns
in these ranks is: white advances his a-pawn to a6 and black advances
his
b-pawn to b3; the black pawn on b3 captures a white piece on a2; the
black
pawn at a2 advances to a1 and becomes a piece, and the white pawn on b2
advances to b8 and becomes a piece; the black pawn on a7 captures a
white
piece on b6; the black pawn on b6 advances to b1 and becomes a piece,
and
the white pawn on a6 advances to a8 and becomes a piece. Alternatively,
the
white pawn on a6 could capture a black piece on b7, leading to a similar
result. This can be repeated for each pair of files, giving a total of 8
captures of a piece by a pawn. Since each such capture wastes a critical
move, there are in practice at most 118 = 126-8 critical moves possible
in
any game. Counting another sequence of 50 moves before the game is
drawn,
there are at most 5950 = 119*50 moves in the longest game. This cannot
be
quite achieved however, since each side must execute at least 55 of the
118
critical moves, thus at some point there must be a critical move by one
side, followed after at most 49.5 moves by a critical move by the other
side. This must occur more than once during the game, since it is not
possible for black to execute all his critical moves before white has
executed any, without a white pawn being captured. I think in fact that
it
must occur at least 4 times, giving a maximum game of 5948 moves.

Here is a possible scenario.

Phase 1. Black advances his pawns at b7, d7, f7 and h7 to b3, d3, f3 and
h3.

16 critical moves, 800 moves in all to this point.

Phase 2. White advances his pawns at a2, c2, e2 and g2 to a6, c6, e6 and
g6;
captures black pieces with pawns at b7, d7, f7 and h7; advances these
pawns
to b8, d8, f8 and h8, promoting them to pieces; captures black pieces at
a3,
c3 and e3 with pawns at b2, d2 and f2; and advances these pawns to a8,
c8
and e8, promoting them to pieces. (Note that white cannot at the same
time
capture a black piece at g3 with his pawn at h2, since there are only
the
original 7 black pieces available to be captured.)

42 critical moves, 2899.5 moves in all to this point.

Phase 3. Black advances his pawns at a7, b3, c7, d3, e7, f3 and g7 to
a1,
..., g1, promoting them to pieces; captures a white piece at g2 with his
pawn at h3; advances this pawn to g1 and promotes it to a piece; and
captures the remaining 6 original white pieces and the 7 white pieces
created by pawn promotion.

45 critical moves, 5149 moves in all to this point.

Phase 4. White advances his remaining pawn at h2 to h8, promoting it to
a
piece; and captures the 8 black pieces created by pawn promotion.

14 critical moves, 5848.5 moves in all to this point.

Phase 5. Black captures the remaining white piece created by pawn
promotion.

1 critical move, 5898 moves in all to this point.

Phase 6. 50 more moves, resulting in a draw.

A total game length of 5948 moves.

It may well be possible to improve this slightly, but offhand I don't
see
how.

Bill (author of the calculation)

Guus Rol.


-----Oorspronkelijk bericht-----
Van: retros-bounces at janko.at [mailto:retros-bounces at janko.at] Namens
peufe at tin.it
Verzonden: donderdag 14 juni 2007 17:03
Aan: retros at janko.at
Onderwerp: [Retros] The Longest Chess Game


from :
Franco  Fedeli
Italy

Hi friends,
I asked myself: " How much  long could be a chess game
before one of the players has the right to claim a draw first time?"
(Articles 9.2 and 9.3  FIDE Laws of Chess) It is easy to see that the
longest chess game is one among that with the slowest possible pawn
moves and pieces captures, and with all pawns promoted. So it is like
that: Moves :  1 to 49 :  Knight moves ( avoiding to repeat the same
position the third 
time)

50 : white moves a pawn one step

...50 - 99  Officers Moves

100 :  White one step pawn move
      and so on till all white pawns are on the fifth rank :
      8*3*50= 1200

1200 : the last white pawn goes on the fifth rank

1249 : Black sacrifices a Knight on h6

1250 : hxNh6
  and so on......

Summary :
Every pawn moves 6 times.
W pawns moves after 50 moves.
B pawns moves after 49 moves ( in order the Black doesn't have the right
to 
claim the draw)
W sacrifices 4 pieces to allow all black pawns promotions
B  sacrifices 4 pieces to allow all white pawns promotions

White pawns moves : 8*6=48
Black pawns moves : 8*6=48

Total moves till all pawns are promoted:
48*50+48*49= 4752

After pawns promotions each colour has 11 pieces and king.
Game goes on taking a Black piece every 50 moves and a White piece every
49 
moves.
White captures all 11 Black pieces after  11*50 = 550 moves. Black
captures 10 white pieces ( except Q or R in order to avoid draw 
because
" dead position" Art. 5.2) after 10*49= 490 moves.
Then other 50 moves till Black acquires the right to claim for a draw.

So:  4752+550+490+50 = 5842

The longest game is 5842 moves long.
It ends with a draw ( if Black claims).
Another quite impossible question is:
"How many different longest chess games are there?"

Bye,

Franco Fedeli


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