User:Chuck Carroll/Sandbox/Chess problem

The problem
Chess.com posted this problem to the "Chess" Facebook page on February 10, 2011. This page shows the detailed solution, as well as some other frequent attempts made in the comments which don't work, and explains why they don't work.

A few of the people commenting on Facebook seem unclear on the conventions of chess problems such as this one. As a reminder:
 * "White to move and mate in 3" means that White moves first, and White must checkmate Black within 3 (White) moves.
 * Black still gets to make their regular moves. "3 moves" doesn't mean White gets three moves without any Black moves in between!
 * The solution must account for all possible Black moves. You can't just pick the worst moves for Black to make.
 * "Mate in 3" means it will take White three moves against the best possible Black defense. It may take fewer than three moves against inferior Black moves, and that's OK. It doesn't have to be exactly three moves against every possible Black move, just three or fewer.
 * Since the requirement is "mate in 3," then it's not a solution if Black has a defense which makes White use four or more moves. That might still be a fine strategy for White in an actual chess game (where checkmate is checkmate regardless of whether it's delivered in 3 moves or 10), but it's not a solution to the puzzle.
 * White is at the bottom of the board, Black is at the top. The White pieces started on rows 1 and 2, and the Black pieces started on rows 7 and 8.  That is, the Black pawns are moving down the board.

This page uses algebraic notation to describe the moves. If you're unfamiliar with that, you may want to review that article before proceeding.

The solution: 1. Qb8
The solution begins with White playing 1. Qb8. This may at first seem counterintuitive—sacrificing the queen—but the key is that it frees up the critical square e5 for a knight to move to.

White's second move will depend on Black's response. The various possibilites are described below.

Black takes the queen
The most obvious Black response is to take the White queen, after which White playes Ne5. Now white threatens mate in two different ways, with 3. Nef7#, or with 3. Nxg4#. Black can guard one square or the other, but not both.

For example, if Black tries to defend the f7 square with 2... Rf8, White wins with 3. Nxg4#.

Or if Black tries to defend the g4 square with, say, Bf5, White wins with 3. Nf7#.

There is no Black move which defends both f7 and g4. White can mate with a knight on at least one of those two squares regardless of Black's second move.

So, if black takes the queen, he loses. Maybe black should try something else. (Remember, to solve the problem, we must show that our solution wins against any possible Black moves.)

Black moves the rook along the a file
Moving the rook to any other square on the a file loses immediately to 2. Qh8#

Black plays 1... Ba2
Moving the bishop to a2 might look good at first. If White takes the rook with his queen, black can then move the bishop to g8, which blocks the queen from moving directly to h8. White would still mate, but it would take four moves.

If White plays 2. Ne5, trying to set up the double mate threat, Black can play 2... Be6, defending both of the key squares g4 and f7.

However, after 1... Ba2, white can move 2. Qh2 instead. This threatens discovered mate on the next move with either 3. Kg3# or 3. Kxg4#, and black cannot stop both threats.

(It might seem that White could just play 1. Qh2 right away. This is addressed in the "other proposed solutions" section below.)

If Black defends the g4 pawn with, for example, 2... Ra4, White wins with Kg3#.

If Black stops the Kg3 mate by playing 2... gxf3 (White could still play 3. Kg3+, but it wouldn't be mate since the knight on g5 would be undefended) White simply wins with Kg4# instead.

Black plays 1... Bf5 or 1... Be4
1... Be4 defends the rook, and 1... Bf5 allows the bishop to move to c8 if the queen takes the rook (delaying mate by one move which White cannot afford), so 2. Qxa8 does not work against these moves. Instead, checkmate can be delivered by the knights.

2. Nf7+ forces 2... Kh7, which is followed by 3. N3g5#. Note that the White queen attacks the g8 square, which would have been an escape square for the Black king if White had tried the mate with the knights without moving the queen first.

1... Bf5 is shown in the diagram here, but the mate plays out the same if the bishop is on e4. (2. Qh2 also works against 1... Be4, but not against 1... Bf5.)

Any other move
Against any other Black moves that haven't already been covered, White wins with 2. Qxa8 and 3. Qh8#.

Other proposed solutions (and why they don't work)
This addresses some other proposed solutions frequently seen in the comments on the Facebook page, and explains why they don't work.

1. Nf7+ Kh7 2. N3g5+
Many people in the Facebook comments have suggested 1. Nf7+ Kh7 2. N3g5#. Only problem is, that's not checkmate. The black king can move to g8, and after that there's no third move that gives checkmate. 3. Qxg7+ doesn't work; the queen is unguarded and black can play 3... Kxg7. The White queen can't give check along the diagonal, as that's blocked by the White knight. If the queen gives check along the back rank, it's captured by the Black rook.

1. Nf7+ Kh7 2. Qh5+
Some people have suggested 1. Nf7+ Kh7 2. Qh5#, but that's not mate either. h5 is attacked by the Black pawn on g6, so Black simply takes the queen with his pawn, 2... gxh5.

1. Ne6
1. Ne6 has been suggested, threatening 2. Qxg7#. Black can meet this threat with 1... Rg8. Now the g7 square is defended, so 2. Qxg7+ is followed by 2... Rxg7. 2. Qg5+ also fails (2... Kh7) as does 2. Qh5+ (2... gxh5).

One other interesting try from this point is 2. Qh2, but this is met by 2... g5+. Then after 3. Kxg4+ (or Kg3+) the black king can move to g6 which was just vacated by the pawn.

1. Qh2
In the correct solution, there's a line where White plays Qh2. (1. Qb8 Ba2 2. Qh2) So why not just play 1. Qh2 from the start, threatening Kxg4# or Kg3#? It's met by 1... Ra2 (not available in the main solution because the bishop has already moved to a2). If 2. Kxg4+ or 2. Kg3+, Black responds with 2... Rxh2. Another try for White is 2. Qh1, which is then met by 2... Rh2. In that case White will still mate after 3. Qxh2 and 4. Kxg4# or 4. Kg3#, but that's four moves and the problem is mate in three. White might also try 2. Qb8, but then 2... Rh2+ means White will still need two more moves to mate.

1. Qf4 or 1. Qe3
1. Qf4 or 1. Qe3 have been suggested, with the idea of giving a double check with 2. Nf7+. Problem is, this isn't really much of a threat. Black can play almost anything after White's first move. After 2. Nf7+, Black plays 2... Kh7 and White does not have a mate for his third move. 3. Qh6+ doesn't work because h6 is defended by the g7 pawn, 3... gxh6. 3. N3g5+ doesn't work because Black can play 3... Kg8.

1. Qe4
If sacrificing the queen to the rook succeeds, maybe sacrificing the queen to the bishop does too? Well, it does if Black accepts the sacrifice: 1. Qe4 Bxe4 2. Ne5 then it follows the same idea as the 1. Qb1 Rxb1 2. Ne5 solution — White threatens both 3. Nef7# and 3. Nxg4#, and Black cannot stop both threats.

However, 1. Qe4 fails if Black does not accept the sacrifice and responds with 1... Ra5 (to stop the threat of 2. Ne5 which would otherwise generate the familiar double threat of 3. Nef7# and 3. Nxg4#). 2. Qe8 then wins eventually, but 2... Ra8 delays it for a move, and White takes a total of four moves.

(Another Black defense is 1. Qe4 Rf8, then if 2. Ne5 Rf4 and the rook now guards both of the critical mating squares, f7 and g4. Meanwhile 3. Qxg6+ is not mate either because of the Black bishop on b1.)

1. Kxg4
1. Kxg4 has been suggested, with the idea that White can follow up with 2. Qh2#. However, this is stopped by 1... Bf5+. Now the White king must move out of check...but where? If it moves 2. Kg3 or 2. Kf4 it blocks the queen's path to h2. If it moves 2. Kh4, then 3. Qh2 will not give check, as it's blocked by the White king. If White doesn't move the king but instead plays 2. Qxf5, Black responds with 2... gxf5+ and White must use his third move to get out of check.

1. Nh2
1. Nh2 Bf5 (details and diagram to be added)