Finding the Invalid Move

d3+ d4+ d5+ d6+ b3+ b4+ b5+ b6+ R3h2 R1h2 Rgh2 R1a2 R3a2 Be4#

There should be 8 valid moves. Find them.

a8=Q# a8=B# a8=N# a8=R#

2 invalid moves

Qxf7+ Qxe5+ Bxf7+ Qxh7 Rxd7 for white

Nxh5 Bc5 d5 a3 Bg4 for black

all the pieces on the board and black can still castle

@chessniu: I'd let someone else have a shot, but your "d3+ d4+ d5+..." one, I can get up to 10 valid moves.

I offer one too:
(1 invalid move) axb6+ exd6+ f7 Nc7 0-0-0 Kxd2 Kxe2

@chessniu -- for your four-checkmate question, a8=N# and a8=B# is invalid. The position would be White queen at b1, black king at a5 and White pawn at a7.

0-0-0 and Kxe2 cannot co-exist, regardless of other possible moves. There is no black unit I can place on e2 that allows white to 0-0-0. So it is a question of proving which one of those moves must go.

Nc7 is not a check or a capture, so bK is not on c7 or any square a Knight's move away from c7. This means at least one of the two moves axb6+ and exd6+ must be a discovered check.

f7 is not check, and is not blocked, and must come from f6, so bK isn't on f7, e7, e8 or g8. It also proves exd6+ is a discovered check [in conjunction with above, which ruled out bK@c7].

Is axb6+ direct or discovered? bK@a7 leaves no way for exd6 to be check. There is no line for a discovery.

So both P captures are discoveries. The only way I could find to make this work is:

After Black's move 1...d5, all of the listed moves except for 0-0-0 are possible.

True

Yup. In that set 0-0-0 was impossible. The arrangement BigDogg found is the only possible checking scheme (at least, it was the intent, and there shouldn't be other schemes.) The bK must be on a2, and therefore wRa1 cannot exist for castling.

Sooo, two more sets. I'm slowly discovering some curious geometries just thinking about this.

(1 invalid) fxg5+, gxh5+, Nh2+, 0-0+
(1 invalid) fxg6+, gxh6+, Nh3+, 0-0-0+

First one:

fxg5+ and gxh5+ are not both valid. If fxg5+ is a contact check, then the king is on f6 or h6, and gxh5+ is a discovered check that passes via g4, but only a nightrider can do that. Similar with gxh5+ being contact check. If both are discovered checks, then clearly neither can discover a rank check (because the other pawn is present), and clearly not both file checks (because the king needs to be present on both f- and g-files otherwise), and clearly not both diagonal checks (because the king needs to be present on both white squares and light squares otherwise). So one is file check and another is diagonal check. gxh5+ is not the file check, since there's still a Black unit on g5. So fxg5+ is the file check and gxh5+ is the diagonal check, but then the king is on f5 and the position is illegal (g4 is checking f5 already). So one of them is illegal, and thus the other two are legal.

0-0+ means the king is along the f-file or the first rank. Nh2+ can only come from f3 or g4 (from f1 it blocks the castle). If Nh2+ comes from f3, the Black king cannot be on f-file (f1/f2 impossible due to White king, f3 impossible due to White knight, f4-f8 ruled out because now the knight blocks the check by the rook), so it's on the first rank. Also, Nh2+ cannot be a knight check, since it means Black king is on f1, so it's a discovered check. Clearly not file or rank check, so it's a diagonal check, but that means Black king is on d1 or h1, both impossible. Thus Nh2+ is not from f3 and is from g4 instead, ruling out gxh5+. Indeed, the rest is possible:

Seemingly cooked; it is possible to rule out either fxg5+ OR gxh5+ [I don't like the idea that black-to-move MUST have "..."]

Actually, I like the idea of twinning for these.

Give this one as:

• [1 invalid] fxg6+, gxh6+, Nh3+, 0-0-0+ a) white to move b) black to move

A)

B)

Only problem is that B) is cooked by putting wK on d4 and using a Q on h8 for gxh6+ discovered check.

Second one is similar. fxg6+ and gxh6+ cannot both be legal, so Nh3+ and 0-0-0+ are both legal. Also, the king is along d-file or first rank. So fxg6+ or gxh6+, whichever is legal, is a discovered check. Clearly not a discovered check along the rising diagonal (bottom-left to top-right) because the pawn moves along the diagonal. If it's along the falling diagonal, then Black king is on d7 or d8 (on the same diagonal as f5 and g5), so Nh3+ is a discovered check...but from where? Nh3+ must also be a discovered check along the falling diagonal (since g1, f2, f4, g5 don't share any rank/file with d7/d8), which means the falling diagonal containing the king has both the pawn and the knight, impossible to cause either to be a discovered check.

If fxg6+ or gxh6+ is a discovered rank check, then Black king is on d5, and Nh3+ is a discovered check too. But checking each possibility of where the knight comes from, either it comes from g5 in which both it and the pawn check cannot be discovered (the other piece is blocking the rank), or f4 in which it's illegal (already checking), or f2 or g1 in which there is no piece that can be discovered. So fxg6+ or gxh6+ is a discovered check along the file.

If fxg6+ is discovered, then Black king is on f1, impossible (due to White king on e1). So gxh6+ is discovered, and thus fxg6+ is illegal, which is indeed possible:

@BigDoggProblem: Oh, whoops, I assumed it's always White to move, although "according to conventions"... Blah what conventions are there for this thing.

Right, when it's Black to move, the other solution is possible.

Congratulations, I have successfully confused myself. I hope this work of a devil is not cooked.

White to move, 1 invalid move. 0-0, a8B, B5e3+, Bba3, Bba5, Bf6e5, Bh6g7+, Bhf8+, Kf2, Nxg3+, Rc8, Rhe7.

Something like this attempts to grant all possibilities:

The only problem is that we have 8 promoted wB + 1 promoted wR + 1 wPa7 = 2 too many.

So it's a matter of scratching off a move that saves two pawns and/or promoted pieces.

The only one I can see at the moment is Bf6e5. This saves 2 promoted units and makes the position legal.

All the other moves are still possible. I'll go with this as the intent and let others try to cook it.

Unless I missed something, this is cooked - all moves can be played.

Creating a proof position requires a bit of care, though.

First of all, Kf2, Nxg3 and 0-0 are incompatible unless the black piece on g3 is a rook and there is something to intercept it on g2. (a Ng3 prevents the castle, other options prevent Kf2).

Bhf8 is necessarily Bh6f8, hence Bhf8+ and Bh6g7+, if compatible, both make check by discovery. This means that there is a rooklike white piece on h7 or h8, and the black king is on the h file, somewhere between h1 and h5, and nothing between h6 and the king.

However then, B5e3+ must be a discovery, because no king on h1-h5 is a bishop's move away from e3. Hence this is another discovery check; as a consequence, there is a rooklike white piece on the 5th rank, and the black king is on h5.

Fiddle, twiddle, and you get this:

EDIT: well, unless B5e3+ requires that there is a bishop not on the 5th that could go to e3 with check. If that is the kind of thing involved, I bow to our devil master - my quick solution will need more promotions than pawns to fix.

3 problems in this position [perhaps even more, but these are the first ones I noticed].

1. No Rhe7.
2. Bh6g7 requires a B on h8 to make it "Bh6g7" instead of simply "Bhg7". I am assuming minimal disambiguation will be used whenever possible. Or, to put it plainly, "Bh6g7" cannot possibly be expressed with fewer than 5 characters.
3. Same goes for "Bf6e5" - it can be expressed as simply "Be5".

Qgh1 Qgh2 Qgh3 Qgh4 Qgh5 Qgh6 Qgh7 Qgh8 for white

Qba1 Qba2 Qba3 Qba4 Qba5 Qba6 Qba7 Qba8 for black

how many invalid moves and find them