[Retros] Longest possible games with specific pieces moving

[Mario Richter]

Hello,

last sunday I asked:


> (1) What is the longest possible chess game, if both sides only

>     move their knights and the 50-move-rule applies?



> (2) What is the longest possible chess game, if both sides only

>     make pawn moves? (To make it clear, after a promotion, promoted

>     pawns are not allowed to move.)


The only answers given to this list were by Yefim Treger,
who stated:


> (1) ... Correct answer is 24*100+99=2499 half-moves (100% sure:).



> (2) ... My correct answer is 84 half-moves (75% sure:).





I think it's now time to show my solutions:

(1) Longest possible game with only knights moving:  2603 half-moves

   An outline of a possible game:
    50. ... Nxa1
   100. ... Nxc1
   150. ... Nxd1
   200. ... Nxf1
   250. ... Nxh1
   300. ... Nxa2
   350. ... Nxb2
   400. ... Nxd2
   450. ... Nxe2
   500. ... Nxf2
   550. ... Nxh2
   600. ... Nxc2+
   601. Nxc2 ...
   651. Nxa8 ...
   701. Nxc8 ...
   751. Nxd8 ...
   801. Nxf8 ...
   851. Nxh8 ...
   901. Nxa7 ...
   951. Nxb7 ...
  1001. Nxd7 ...
  1051. Nxe7 ...
  1101. Nxf7 ...
  1151. Nxh7 ...
  1201. Nxc7+ Nxc7
  1251. ...   Nxg2+
  1252. Nxg2  ...
  1302. Nxg7+


(2) Longest possible game with only pawns moving:

  The answer is 85 half-moves.

 An example game is e.g.

  1.e3 b6 2.e4 b5 3.e5 b4 4.e6 b3 5.axb3 a6 6.b4 a5 7.b5 a4 8.b6 a3
  9.b7 c6 10.b3 c5 11.b4 c4 12.b5 c3 13.b6 a2 14.g3 axb1N 15.g4 dxe6 16.g5
e5
 17.g6 fxg6 18.dxc3 g5 19.c4 g4 20.c5 g3 21.c6 g2 22.c7 gxh1N 23.bxa8N g6
24.c3 g5
 25.c4 g4 26.c5 g3 27.hxg3 h6 28.c6 h5 29.b7 e4 30.bxc8N e3 31.g4 e2 32.g5
exd1N
 33.g6 h4 34.g7 h3 35.cxb8N h2 36.gxf8N hxg1N 37.c7 e6 38.cxd8N e5 39.f3 e4
 40.f4 e3 41.f5 e2 42.f6 exf1N 43.f7+

(The final position is:  NNNNkNnr/5P2/8/8/8/8/8/RnBnKnnn b Qk - 0 0)

 So a lower bound is 85 half-moves, it remains to be proven that
 this is also an upper bound.

A complete proof would be to long to provide it here, but the main
points are:

 (let  "wpc"  be the number of white pawns that were captured
       "bpc"  be the number of black pawns that were captured)

- to allow the pawns to reach the 8th (resp. 1st) rank,
  some pawns have to be captured.

- if 3 white pawns have been captured, then each of them could have moved
  at most 4 times and even if the remaining 5 white pawns all would have
  reached the 8th rank, the maximum number of white plies is (3*4)+(5*6)=42.

  Therefore, if a game with more than 85 plies exists, at most 2 white
  pawns have been captured (i.e. wpc<=2)

- a similiar reasoning shows that at most 2 black pawns have been captured.

  Rem.: The above mentioned game was found after I observed, that
        with wpc=2 and bpc=3 theoretically a game of 85 plies
        could exist.

- if the final position contains a blocked file (i.e. a black pawn
immediately
  above a white pawn on the same file), the game leading to that position
  could not have taken more than 85 plies.

- to reach a position where no file is blocked requires at least
  4 pawn captures

  we therefore have: wpc<=2; bpc<=2;  wpc+bpc>=4

  and therefore (wpc=2, bpc=2)

      The captures reduce the achievable ply number to 44
      for each side.

      Each file contains at least one pawn, especially the
      files d and f. If they are of the same colour, then
      at most 41 plies for each side are achievable.
      Therefore they are of different colour and a game cannot
      be longer than 85 plies, because not later than on the 43th
      move white must play 43. d/f7+.
      (Rem.: It can even be shown that with wpc=bpc=2 only
             a maximum of 84 plies is possible.)


But this already completes the proof, that 85 plies is also
an upper bound, and therefore the longest possible game in which only
pawns move, has a length of 85 plies.


greetings

   mario

P.S.: Thanks to "nicolasdupont" for forwarding the "pawn-only question"
      to the France Echecs discussion forum!