Are you more clever than Alice?

Question

Once upon a time, Bob was telling Alice about a chess match: "One turn before the end only 4 pieces were on the board, and White could guarantee checkmate for themselves in several ways". Once Alice has learned the number of ways, and the names (and colors) of these 4 pieces, she was able to uniquely determine position of each figure on the board. I bet you are even more clever than Alice and don't need to know all these little things. What are these pieces and their positions Bob was talking about?

Explanations:
1. That was a usual match with all chess rules applied. Thereby 2 of those pieces must be Kings.
2. All the information Bob gave to Alice is mentioned in the puzzle formulation.
3. Alice knew only name and color of each piece, no other information about them; like starting position of a pawn or a cell color of a bishop - nothing of it.

Are we assuming that no pawns were promoted to queens? I suppose not if Alice is able to determine a unique board position? — Ian MacDonald, Apr 10 '16 at 12:56
@IanMacDonald, no. we do not know this. But a queen is a queen independently of where it come from (at least Alice wouldn't know it's history). — klm123, Apr 10 '16 at 12:57

score 18 · Answer 1 · edited Apr 10 '16 at 12:57

18

The pieces and number of mates were:

black king, white king, and two white rooks; four mates.

The position was:

BKa1, WKe1, WRh1, WRc2

and the mates were:

Kd2#, Ke2#, Kf2#, 0-0#

Reasoning behind this answer:

If Alice was able to determine the position uniquely, it must not be possible to flip the board vertically. Only one type of move is affected by this: castling. So one of the mates must be from castling. Now there has to be some way to exclude the position with castling on the other side. For 4 mates, the position with queenside castling is not possible because the second rook would either block one of the king's moves, be blocked by the king's move, or be capturable by the black king. With kingside castling, only one location for the rook and black king prevents all of these: c2 and a1, respectively.

edited Apr 10 '16 at 12:57

Sleafar

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answered Apr 10 '16 at 12:42

f''

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BKa1, WKh8, WRb(3-7), WRb8. Board position isn't unique to guarantee mate on white's next move. – Ian MacDonald Apr 10 '16 at 13:00
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@IanMacDonald Wouldn't there only be 2 mating moves in those positions? – Zandar Apr 10 '16 at 13:05
This would also work with 2 Queens in the same position as the 2 rooks. The only different mate would be Qha8# – stackErr Apr 10 '16 at 13:41
Why can't we have Rd2? – Carl Apr 10 '16 at 14:15
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@Carl Rd2 blocks Kd2# – ffao Apr 10 '16 at 17:11
@stackErr - That wouldn't be mate (yet). The black king could move to b8. – Darrel Hoffman Apr 10 '16 at 17:22
@DarrelHoffman No he wouldn't. There would be a queen at C2. And the king can't move to b8, he would have to move to b1...I thnk thats what you meant. – stackErr Apr 10 '16 at 17:22
@stackErr - Oh wait, I'm dumb, I was changing only one of them to a queen. However, I think with 2 queens, there would be more than just the one solution. For example, mirroring the board on either axis would have the same result. – Darrel Hoffman Apr 10 '16 at 17:26
@DarrelHoffman Ahh yes! Missed the part about uniquely identifying the position – stackErr Apr 10 '16 at 17:32
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@DarrelHoffman, queen + rook won't work either. – klm123 Apr 10 '16 at 20:25
@ffao I guess I'm missing something then, why do we need four methods of mate? The only word I see is "several". – Carl Apr 11 '16 at 01:22
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@Carl There need to be 4 so that the position is unique. There are multiple possible positions with 3 mates, so Alice wouldn't be able to determine which of those positions was the right one. – f'' Apr 11 '16 at 01:24
@ffao I'm replying to you so that you stay notified. I'm still lost and maybe beyond help. Qb3, instead of Rc2, would have four moves for mate as well. – Carl Apr 11 '16 at 01:32
@Carl All you're asking is answered by this sentence in the question "Once Alice has learned the number of ways, and the names (and colors) of these 4 pieces, she was able to uniquely determine" So you can't change a rook for a queen, as the pieces are fixed. If you had fixed a queen, then there would be more than one possibility for 4 mates. – ffao Apr 11 '16 at 01:34
@ffao okay, I think I get it. If there were three moves, then a castle on the other side would be possible, and if there was a queen, then perhaps there is another four move solution. Thanks. – Carl Apr 11 '16 at 01:40
@ffao, Carl is asking a good and right question. The OP's question asks for what was the pieces, you can't fix an answer to a question to anything you want then tell this is the answer and queen is not possible) But still, there is a reason why you can't replace Rc2 by Qb3. I'm still hoping someone will make a complete answer with thorough proof, though it is definitely a hard job and not so much fun as to find the position itself, which f'' already did correctly. – klm123 Apr 11 '16 at 19:04
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You can't replace Rc2 by Qb3 because the queen could also be on c2 for four mates. – f'' Apr 11 '16 at 19:17
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One of the mates must be by castling for the position to be unique against reflection. This is only possible if the two pieces are R/R or R/Q. R/R with 1, 2, or 3 mates is not unique, and neither is R/Q with 1, 2, 3, or 4 mates. This leaves only R/R with 4 mates. – f'' Apr 11 '16 at 19:23
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@f'' I think you need to include your comment on why R/R with 4 mates is the only one yielding unique position. You only mention "For 4 mates, ..." but does not describe why 3 mates won't make it, until I read your comments. So I think your answer can be improved. Other than that, great answer =) – justhalf Apr 14 '16 at 04:17

score 0 · Answer 2 · answered Oct 23 '18 at 07:50

f' gave the great and correct answer (why hasn't it been accepted yet ?), but as an example of the difficulty of this smart problem here is another attempt that fails short:

If Bob had mentioned

Two kings, one white knight, one black pawn, two mating moves

Then the only possible matrix would have been:

White: Kc1, Nd4 / Black: Ka1, Pa2 / Mates by Nc2# or Nb3#

With just one caveat...

The board can be flipped left-right to reach this other position White: Kf1, Ne4 / Black: Kh1, Ph2

...that only f''s clever trick can avoid !

f'' just given an example of what the set of pieces could be so Alice could have figure position out. But he didn't prove that that was The set of pieces Bob was talking about. — klm123, Oct 23 '18 at 10:47

Are you more clever than Alice?

2 Answers2