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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.

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    $\begingroup$ Are we assuming that no pawns were promoted to queens? I suppose not if Alice is able to determine a unique board position? $\endgroup$ – Ian MacDonald Apr 10 '16 at 12:56
  • $\begingroup$ @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). $\endgroup$ – klm123 Apr 10 '16 at 12:57
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The pieces and number of mates were:

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

The position was:

enter image description here

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.

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    $\begingroup$ @IanMacDonald Wouldn't there only be 2 mating moves in those positions? $\endgroup$ – Zandar Apr 10 '16 at 13:05
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    $\begingroup$ @Carl Rd2 blocks Kd2# $\endgroup$ – ffao Apr 10 '16 at 17:11
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    $\begingroup$ @DarrelHoffman, queen + rook won't work either. $\endgroup$ – klm123 Apr 10 '16 at 20:25
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    $\begingroup$ @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. $\endgroup$ – f'' Apr 11 '16 at 1:24
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    $\begingroup$ You can't replace Rc2 by Qb3 because the queen could also be on c2 for four mates. $\endgroup$ – f'' Apr 11 '16 at 19:17
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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 !

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  • $\begingroup$ 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. $\endgroup$ – klm123 Oct 23 '18 at 10:47

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