How Long Can a Game Go?
In trying to answer this, we will assume two parameters. First, of course, full cooperation by the players; second, the 50-move rule must be obeyed - a game will count as automatically drawn if both players make 50 moves without either moving a pawn or making a capture.
A first attempt:
The most obvious approach goes like this:
Both sides move their knights around (without falling into a 3-time repetition), and move 50 Black plays 50...h6. We continue as follows - assume meaningless piece moves in the intervals:
100...h5, 150...h4, 200...h3, and then the same 200 move sequences for all of the pawns: 400...g3, 600...f3, 800...e3, 1000...d3, 1200...c3, 1400...b3, 1600...a3.
For our next stage, White will capture these pawns on his 50th move: 1650.Nxh3, 1700.Nxg3, 1750.Nxf3, 1800.Nxe3, 1850.Nxd3, 1900.Nxc3, 1950.Nxb3 and 2000.Nxa3.
Next, capturing all the Black pieces will use another 350 moves, bringing us to move 2350. At move 2400, let's have White push a pawn - say e3 - and eventually force the Black king to shuttle between a8 and b8 with the White queen on d7 and White king on d8 (just to make visualizing the position convenient). Then we'll have 2450.d3, 2500.c3, 2550.b3, 2600.a3, 2650.f3, 2700.g3, 2750.h3. Repeat in 400 move increments to bring the pawns to the fifth rank (culminating, say, with 3550.h5), and now we'll just take care of the kingside pawns:
3600.h6, 3650.h7, 3700.h8Q, and then 3750.Qa8+ Kxa8 (or 3750.Qb8+ Kxb8, depending on where Black's king is). Repeat with the g, f and e-pawns, and we're up to 4350...Kxa8.
Next, we'll get rid of both rooks, both knights and one of the bishops (if we're assuming Black's 4350th move is ...Kxa8, it will be the light-squared bishop), taking us to 4600...Kxa8. At this point we'll drive the Black king to the h8 corner (clearing the way for the queenside pawns) and there give up the remaining bishop: 4650...Kxh8. Pushing and promoting the pawns and then sacrificing the promoted queens gets us another 800 moves (5450...Kxh8), and now all that remains is to mate the king on the 50th move: 5500.Qg7#.
A second try:
Of course, one normally doesn't want to combine captures with pawn moves, as that would throw away 50 moves. However, there's reason to do so here - 50 moves are lost, but 100 moves are gained by a Black pawn's being able to go two ranks further, for a net gain of 50 moves per eligible pawn.
I believe we can do this with every Black pawn - sometimes by that pawn's capturing a White piece, and sometimes by a White pawn's capturing a Black piece. Thus:
50...h6, 100...h5, 150...h4, 200...h3, 250.g3, 250.Bg2 hxg2, 300.Nf1 g1(Q), 350.Rxg1. One Black pawn down, seven to go.
400.g4, 450.g5, 500.g6, 549...Rh7 550.gxh7, 600.h8(Q), 650...Nxh8, 700...g6, 750...g5, 800...g4, 850...g3, 900...g2, 950.Rh1 g1(Q), 1000.Rxg1. Two Black pawns down, six to go.
1050.h3, 1100.h4, 1150.h5, 1200.h6, 1250.h7, 1299...Ng6 1300.h8(Q), 1350...Nxh8. (Just clearing out the superfluous debris.)
1400.Rg6 fxg6, 1450...g5, 1500...g4, 1550...g3, 1600...g2, 1650...g1(Q) 1700.Nxg1. Three Black pawns down, five to go.
During this next stage, move the Black king out of the way (to a6, say) and the Bf8 to b6, and now: 1750.f3, 1800.f4...2000.f8(Q) 2050...Qxf8, 2100.e3, 2150.e4, 2200.e5, 2250.e6, 2299...Qf7 2300.exf7, 2350.f8(Q), 2400...Nxf8.
Now let's get White's king out of the way - h8 looks convenient, and continue: 2450...e6...2700...e1(Q), 2750.Qxe1. Four down, four to go.
2800.d3...2950.d6, 2999...Ne7 3000.dxe7, 3050.e8(Q), 3100...Nxe8.
3150...d6...3400...d1(Q), 3450.Qxd1. Five down, three to go.
3500.c3, 3650.c6, 3699...Bd7 3700.cxd7, 3750.d8(Q), 3800...Rxd8.
At this point we have more pieces than we need, so let's get rid of some clutter before finishing the pawns off: 3850.Qxd8, 3900...Bxg1, 3950.Ne3 Bxe3, 4000...Bxc1, 4050.Rxc1. We're down now to this: White: queen, rook and two pawns; Black: knight and three pawns.
4100.Rh1 c6, 4150...c5...4350...c1(Q), 4400.Rxc1 (six down), 4450.Rc6+ bxc6...4700...c1(Q), 4750.Qxc1. (Seven down.)
At this point, for convenience' sake, let's remaneuver things so that the Black king is on h8, the White king on e8, the White queen on e7 and the Black knight is hopping along elsewhere. Now:
4800.b3...5050.b8(Q), 5100...Nxb8, 5149...Nb3 5200.axb3...5450.b8(Q)...5750...a1(Q), 5799...Qa8 5800.Qxa8 (mission accomplished!), 5850.Qh8+ (or Qg8+, again depending on where the Black king is) Kxh8, and finally, at long, long last: 5900.Qg7#
Is there some further trick that enables the move count to increase even further? Can the magic 6000 move barrier be broken? Readers who like puzzles of this sort, or with more time than they know what to do with are encouraged to try.