Proof Games and more Proof Games

I just solved Donati's PG (very entertaining choice, BigDogg!) and can say that there is no flaw in chaotic_iak's reasoning thus far. Everything he wrote is correct.

The bolded part of Frankwho's quote is just part of the crux of the problem. The giant wall of units across the board seem to forbid black's final sacrifice, but there is a chink in the armour. A very tiny one, but it can be seen through with pure logic.

Obviously the task is not impossible - perhaps giving it a few more shots and writing down meticulously every step of reasoning would be illuminating.

Let's continue...

There are two pieces that are problematic: Bc8 and one of Black's rooks. Due to the pawn wall, both of them can only exit the back rank via d7, so we need bPd7xc6 before freeing either of them. This also means wBd5 is present. Either bPe7-e6 is not played yet (so bBc8-e6-out and bR...-d8-d6-e6-out are possible), or bPe7-e6 is played, which means bPd6-d5 must be played to free them (bBc8-d7-c6-out and bR...-d8-d6-c6-out).

However, the latter is impossible. Before bPc6-c5 is played, we need wBc1-a3-f8, so we have done both wPa2xb3 and wPb2xc3. The only remaining pawn capture is wPh2xg3. But now wNa4, wPb3, wPc4, wBd5, bPe6, bPf7-h7 block the upper part from the bottom; neither of those two pieces can get through. So bPe7-e6 is played after bPd7xc6. This also means Black's first three moves are bNb8-c6-e5-f3.

Now, what should Black do? He can free up bPd7xc6 only by move 9 (5. Bf1-c4 6. Bc4-d5 7. c2-c4 8. Qd1-a4 9. Qa4-c6 d7xc6), so Black needs to waste six moves. At the moment, no Black pawn has moved, so the only way to waste moves is by moving Ng8 or either rook along the back rank.

...okay what is this. Why does it seem like Black has plenty of possible moves to waste (read: the game is not unique)?

The game is unique (I can't find a cook, and indeed the solution itself suggests reasons why it should be unique.)

Just try constructing the game. Long PGs usually need several tries once the preliminary analysis is done, which will show up even more difficulties than thought. It's rare that a single construction attempt suffices to yield the solution.

Yes, I was trying a bit and found something more.

After bPd7xc6, the only available moves for White are wNg1-e2-c3, two moves. The others need either a2xb3 or h2xg3 first, but there is no Black piece ready to be captured. Meanwhile, given the positions of the pieces, only one piece can go to b3, and it's Bc8 that needs three moves (Bc8-f5-c2-b3). Thus the next capture is not a2xb3, but h2xg3, which must be done within two moves. The only way to feed a piece to g3 is bQd8-d6-g3. So that is the case; we will have n... d7xc6 n+1. Ng1-e2 Qd8-d6 n+2. Ne2-c3 Qd6-g3 n+3. h2xg3.

After Qd6-g3, White now has three moves (h2xg3 and Rh1-h6-f6). The only piece that can go to b3 is still bBc8, thus we can continue n+3... Bc8-f5 n+4. Rh1-h6 Bf5-c2 n+5. Rh6-f6 Bc2-b3 n+6. a2xb3 and now we have taken care of one problem piece, so bPe7-e6 is now allowed.

Note that n >= 9; d7xc6 can only occur as early as Black's 9th move. Thus after a2xb3, we have at most 8 moves remaining as n+6 >= 15. But this is exactly what we need; there are 8 moves necessary after a2xb3. So we know n = 9, and thus we know precisely what White's moves are.

1. d2-d4
2. Ke1-d2
3. Kd2-e3
4. e2xf3
5. Bf1-c4
6. Bc4-d5
7. c2-c4
8. Qd1-a4
9. Qa4-c6
10. Ng1-e2
11. Ne2-c3
12. h2xg3
13. Rh1-h6
14. Rh6-f6
15. a2xb3
16. Ra1-a6
17. Nc3-a4
18. b2xc3
19. Bc1-a3
20. Ba3-f8
21. Ra6-d6
22. Rd6-d8
23. Rd8-c8

The next capture is b2xc3, coming in three moves after a2xb3 (move 18). Ke8 and Ng8 may not be captured, while the rooks need three moves even to exit rank 6 (R...-d8-d6-e6-out), so only Bf8 can reach c3 in 3 moves: 15... e7-e6 16... Bf8-b4 17... Bb4-c3 18. b2xc3.

Now, just in one move, f8 will be under attack, and in two moves, f8 will be blocked. So the king must go to kingside quickly. This means the knight is not present on g8 so that 18... 0-0 is possible. (Otherwise Rh8 will be trapped in northeast, since there is no more castling after move 18 with f8 under attack/blocked.)

Now Black only has 5 moves left. One is Kg8-h8, the other is c6-c5, so the remaining three are to return the knight and/or moving Ra8/Rf8 if necessary.

How can Ng8 return home? Or more importantly, where was Ng8 while the whole load of moves occurred? Note that it is not moved from White's 9th move until White's 19th move, so it must not be on any square traversed by the pieces involved in those moves.

The above gives all the squares traversed/occupied during White's 9th-19th moves. Some of the squares are used more than once; I only put the piece that occupies/traverses it the first time.

We can see that there are only nine possible squares. We can rule out d1, f1, g4 as they are checking the king. We can rule out e1 as it's too far from g8 (seven ranks apart while a knight can only step two ranks at a time). We can rule out d2 as the only route to g8 is d2-e4-f6-g8, hitting the rook on f6. We can rule out b5 because it's of the wrong color; with same-colored squares, the knight can only use two moves to return to g8, but they are separated by five files. This leaves g5, b6, and b8 as candidates.

How does the knight return? Not from f6, certainly, as it's occupied. Not from h6 either; it needs to visit f5 or g4 to go to h6, but it checks the king. So it comes from e7. Thus g5 is impossible as it needs four moves to go to e7.

It's time to roll back to the beginning. Clearly the knight moves 4... Ng8-f6 , since Ng8-e7 hits the pawn while Ng8-h6 cannot go any where with f5 and g4 checking the king. But what is the next move? 5... Nf6-d5 is impossible as it checks the king (and that 6. Bc4-d5 hits the knight), so there is an important point: Nf6-d5-b6 is impossible; in other words, the knight still needs four more moves to go to b6. So if it settles on b6, then its last move is to b6. But from where? a8, c8, d7 are still occupied; d5, c4 are occupied by White units, and the important point: a4 is currently occupied by White queen. There is no way the knight goes to b6, so it must go to b8. It also follows that the knight moves b8-c6-e7-g8, the only way to visit e7 in two moves. Also, since wRa6-d6 uses c6, the knight cannot remain there on White's 21st move. Since Black only has one unspecified move (19th), it cannot be Nb8-c6 as it will leave the knight on c6 during White's 21st move. Thus 19... Kg8-h8 20... c6-c5 21... Nb8-c6 22... Nc6-e7 23... Ne7-g8. By the way, because Rf8 is not moved, the bishop captures it: 20. Ba3xRf8.

With this in mind, 5... Nf6-h5 is impossible, as from h5 to b8 the only route is h5-f6-d7-b8, hitting the pawn on d7. Then, d7, e8, h7 are clearly impossible; g8 just rolls back progress; g4 checks the king. So 5... Nf6-e4 is the only option.

To avoid going far from the target, only c5, d6, or f6 can be the destination on Black's 6th move. But f6 rolls back progress; f6-d7-b8 hits the pawn on d7. From d6, the knight cannot reach b8 in two moves. Thus 6... Ne4-c5, and the rest is trivial: 7... Nc5-a6 8... Na6-b8.

And we're done, proving uniqueness too.

Pascal Wassong
Phenix 1992

PG in 12.0

[Solving Wassong]

Black is the one who has the more obvious schedule to keep, so I'll count him first.

• K: 4
• Q: 0
• R: 1
• B: 2
• N: 1
• P: 4

Total = 12

Black has no spare moves; he must stay on schedule at all times.

So what does white do? He doesn't have to capture anything. All he has to do is lose a tempo.

SOLV'D. Good work.

This is a problem that solvers tend to appreciate more than judges. I think the judge only gave it an HM. Robbed!

Next PG:

Composer: P. van den Heuvel

PG in 29.5

^ Bump. No takers? This one's a fun one. Not easy, though.

I was thinking to give this to someone first because I did too many. :P

You could always try it, solve it and sit on the answer while waiting for someone else. Unless you've solved it already too, in which case just keep sitting.

SOLV'D [remellion]

On this one, I'm sure they won't mind if you solve it for them. :)

van den heuvel SOLV'D again [uncia uncia].

Sol:

Next PG:

Composer: R.McCracken

PG in 20.5

McCracken:

SOLV'D

I thought I just sent you a proof game for this. :(

Let the record show that frank sent a message with the solution 53 minutes before caveatcanis posted it.