## Turnary Reasoning

The most difficult puzzle I wrote for the MIT Mystery Hunt 2013 was Turnary Reasoning. I can’t take credit for the difficulty: I designed the checkers positions; they were expectedly the easiest. Timothy Chow created the chess positions, and Alan Deckelbaum created the MTG positions. As the name of the puzzle suggests, you need to find whose turn it is in each position or, as the flavor text suggests, decide that the position is impossible.

I tried to solve the chess positions myself and was charmed by their beauty. The most difficult one was the first chess puzzle presented below. Find whose turn it is or prove that the position is impossible.

Share:

1. #### Earl:

I don’t know how that black rook can get on the baseline like that. But I’m not sure how to prove it.

2. #### Geneal:

It can’t be Black’s turn. If White made the last move, the only move that could possibly have been would be the king moving out of check. But then Black had no way to *place* White’s king in check on the previous move. So either it’s White’s turn or (as I suspect) it’s an impossible position. Each individual piece could get to where it is, I think, but stuffing all those white pieces into the corner like that, with the black rook on the baseline, seems difficult. The only way the black rook could have got there is via the same corner. If there’s a sequence of moves that could accomplish it, I can’t see what it would be.

3. #### Tanya Khovanova:

Yes, it can’t be Black’s turn.

4. #### Jonathan D:

Geneal’s reasoning about White not making the last move also applies to the situation before any possible last moves by Black. So the position is not possible.

Actually…

6. #### Joseph:

I’m thinking that:

1. Jonathan D’s final conclusion (the position is impossible) is correct, but…

2. …his reasoning is incomplete. You need to consider the possibility that Black just made a capture.

7. #### Joseph:

And as soon as I posted that, I realized my mistake. Bleh.

I’m now in the “the position is possible, it’s White’s move” camp.

8. #### Tanya Khovanova:

To prove that the position is possible you need to trace several previous steps from a position that is obviously possible.

9. #### Johannes:

The position is impossible.

Consider the f3 and g4 pawns. The f3 pawn must have captured some piece from g2 and thus the g4 pawn the same from either h2 or h3. The only black pieces missing are two pawns.

Note also that the only white piece missing is the a2 pawn. The only means for pawns to change column is by capture (including en passant). We see that on the black side, neither the “d” pawn nor the “f” pawn were thus able to change column (multiple column changes being needed to capture the white “a” pawn). This also implies that no black pawn was promoted (a black pawn must have reached either the “a” and “h” columns for this). We must conclude that the two missing black pawns are responsible for the positions of the above mentioned two white pawns. But now the g4 pawn is in an impossible position, as we cannot account for the piece captured in its move to g4.

10. #### Johannes:

I should preface this comment by saying I don’t have absolute faith in my answer, given my propensity for silly mistake. In any case, I first encountered these kinds of problems through a German translation of the book “The Chess Mysteries of Sherlock Holmes” by Raymond Smullyan (whose wikipedia page is rather interesting and describes him as “an American mathematician, concert pianist, logician, Taoist philosopher, and magician”). The 50 problems in the book, all in some part involving “retrograde analysis” of chess positions, can be found here:

https://www.janko.at/Schach/Smullyan-SH/index.htm

A little German is needed to read the solutions.

P.S. It seems the whole book (and more) is available online if you simply google the English title, but I’m a little skeptical about its legality.

11. #### Johannes:

So, my presumption was correct, and there was indeed a rather large blunder (the curious absence of the black queen). I still stand by the positions’ impossibility, very tentatively:

Given that the last move could not have been made by any of the white pieces on the board as they stand, we know black must have played before (as has been stated). As per Jonathan D’s conclusion, this applies to white’s penultimate move too, and so on. We must conclude that white’s last move was made by a piece not on the board. The only white piece not on the board is the white “a” pawn. By previous reasoning, it must have been captured on the “a” file. The only black piece capable of having carried out this capture is the black king. So the white pawn was on a6 when captured. Where was the black king before the capture? Certainly not b6 or before a5-a6 was played the black player was in check at the end of his move. So b5. Now where was the black king before a4-a5 (the only possible white move before a5-a6). Certainly not b5, by the same reasoning. For lack of space, it must have been b4. So, continuing, we have established the order of the last moves as:

N. a2-a3, …
N+1. a3-a4, Kc3-b4
N+2. a4-a5, Kb4-b5
N+3. a5-a6, Kb5xa6

The … black move must therefore enable white to move another piece. This can only have been the king moving out of check. So we have N-1. Kf1-g2. … . Which black move preceded this? The white king on f1 must have been out of check before and in check after. So the black rook could not have made this move. The only possibility is discovered check, and the only piece capable of having carried out this move is a knight on e1. In particular, the check move must have been Nd3-e1. We thus have two moves in which black moved a knight from d3 to a position it is currently in. This is impossible.

12. #### ama:

I think the position is possible:
Last move is a black one, king takes pawn on a6 from b6, eureka !
pawn took queen in a5 from b5
backward, put the black tower away from c4 (f4?)
white pawn came from b4
put the bishop b3 away (e6?)
white pawn came from a3, removing a black pawn
knight a1 came from b3
white pawn came from a2
and now, white can move the tower b1-a1-b1-a1-b1 as long as needed.
So we can put, after x moves, the black knight on e1, this allows the white king to go to f1.
The right edge can now be explained (move back bishop, queen, knight g1, tower g3, other knight, ….), while black plays some ridiculous moves. We can in the end move white king toward h1…h5, g5, remove black knight from e1 and move the black tower the same way as the king.
Am I right ? What subtlety have i nevertheless missed in my explanation ?
Thx for the three nice hours i spend on this riddle !
Marc, french guy.

13. #### Tanya Khovanova:

The position is possible.

14. #### Olaf Hansen:

Of course the position is possible.
I know it, because I immediately recognized the position from one of my games. I’ll give you the chance enjoy my fabulous play. I’m playing the white pieces:

1. Knight b1-c3 , d7-d5. 2. Knight c3xd5, f7-f5. 3. Knight d5-e3, f5-f4. 4. Knight e3-g4, f4-f3.
5. g2xf3, Queen d8-d7. 6. h2-h3, e7-e5. 7. Knight g4-h2, Queen d7-g4. 8. h3xg4, Knight b8-c6.
9. Bishop f1-h3, Knight c6-d4. 10. King e1-f1, Knight d4-b3. 11. King f1-g2, Knight g8-f6.
12. Knight h2-f1, Knight f6-d7. 13. Rook h1-h2, Knight d7-c5. 14. King g2-h1, Knight c5-d3.
15 Knight f1-g3, Bishop f8-c5. 16. Bishop h3-g2, Bishop c8-d7. 17. Knight g1-h3, Bishop d7-a4.
18. Knight g3-e4, Rook h8-f8. 19. Knight h3-f4, Rook f8-f6. 20. Rook h2-h5, Rook a8-d8.
21. Queen d1-g1, Rook d8-d4. 22. Queen g1-h2, Rook f6-h6. 23. Rook h5-f5, Rook h6-h3.
24. Rook f5-g5, Rook h3-g3. 25. Bishop g2-h3, c7-c6. 26. Queen h2-g2, King e8-d8.
27. King h1-h2, King d8-c7. 28. Queen g2-h1, Rook g3-g1. 29. Knight e4-g3, Rook g1-d1.
30. Rook g5-h5, Knight d3-e1. 31. King h2-g1, King c7-b6. 32. Queen h1-h2, King b6-a6.
33. Knight g3-h1, King a6-b6. 34. Bishop h3-g2, King b6-a6. 35. Rook h5-h3, King a6-b6.
36. Rook h3-g3, King b6-a6. 37. King g1-f1, King a6-b6. 38. Knight f4-h3, Rook d4-c4.
39. Knight h3-g1, King b6-b5. 40. Bishop g2-h3, Knight e1-d3 (check) 41. King f1-g2, Knight d3-b4.
42. Rook a1-b1, Knight b4-a6. 43. Rook b1-a1, Knight a6-b8. 44. Rook a1-b1, Knight b3-a1.
45. a2-a3, Bishop a4-b3. 46. a3-a4 (check), King b5-b6. 47. a4-a5 (check), King b6-b5.
48. a5-a6, King b5xa6……… and I went on to lose the game!

15. #### Tanya Khovanova:

Thank you Olaf.

16. #### Timothy Chow:

I’m eight years late to this conversation, but I wanted to point out that Geneal’s reasoning that it can’t be Black’s turn has a loophole. Conceivably, Black’s last move could have been to play a knight from e1 to g2, discovering check on the White king on f1. White could then have responded by capturing the Black knight with his king, Kf1xg2. Well, actually this hypothetical scenario turns out to be impossible, but for reasons that are more subtle than what Geneal proposed.