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This is a follow-up to (Almost) all hands on check by @loopy walt.

The task (mate in one with as many essential pieces as possible) stays the same, but you are to go about it more cunningly.

Recap of the original rules:

Construct a legal position where white to move mates in one, subject to:

  1. The mate is unique as defined in the Handbook of Chess Composition, article 13(2) with footnote 16 (thanks @trolley813) where it says: "[...] except that in the final move a promotion into different pieces having partially the same power (for example queen/rook or queen/bishop) may be tolerated.

  2. Removing any piece except a king results in a legal position (white to move, i.e. white king may be in check, even mate, black king mustn't).

  3. An "accessory to regicide" is any piece which when removed leaves a position where white can no longer mate in one.

  4. A "witness" is a non-king piece that is not an accessory.

Let us define the regicide score as number of accessories - 3 x number of witnesses.

Task: Maximise the regicide score.

New requirement:

Our team of overpaid legal mercenaries have identified a legal loophole which you are to exploit to the best of your ablity:

Handbook of Chess Composition, article 16(2): [...] An en-passant capture on the first move is permitted only if it can be proved that the last move was the double step of the pawn which is to be captured.

Accordingly, you are to construct a position subject to

  1. white to move provably has an e.p. capture that also checkmates the black king.

  2. no other white move is a checkmate.

  3. removal of any single non-king piece yields a legal (white to move) position.

In as many of these positions as possible

  1. it cannot be proved that e.p. is available.

  2. no mate in one is available.

Note: I've tried this strategy a bit and it wipes the floor with what was tried so far. A regicide score in the high 20s should not be a problem. If you feel ambitious aim for 30.

Almost example:

enter image description here
In this example we can prove that b7-b5 is the only last move possible: The king couldn't have moved because all the squares he could have come from are threatened doubly. The knights couldn't have moved because the squares they could have come from would have checked the white king. And the other pawns couldnt have moved because the squares they could have come from are occupied. Note how the pawns on e2 and g2 are crucial accomplices because they prove that the only missing white piece, the light squared bishop died peacefully in his home on f1 and not on the battlefield, meaning that none of the black pawns could have arrived at their position via capture.
What makes this a non-example is that the checkmate is not unique.

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  • $\begingroup$ So, you found a better solution to the other puzzle, but decided to post it as a riddle instead? $\endgroup$
    – justhalf
    Commented Mar 12, 2021 at 3:05
  • $\begingroup$ @justhalf Yes, as it relies on a loophole in the rules and feels very different in spirit. Also I thought it was more fun to see what you people can do instead of running away with that little "cheat". $\endgroup$
    – Albert.Lang
    Commented Mar 12, 2021 at 5:03
  • $\begingroup$ Hm, if I understand this correctly, it's only adding the requirement to use en passant in the solution? If that's the case, I feel like it's not a loophole. But perhaps I missed something here. $\endgroup$
    – justhalf
    Commented Mar 12, 2021 at 6:15
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    $\begingroup$ @justhalf Technically, you are, of course, correct. But the whole "remove one piece at a time and check again" business was originally just a way to get a handle on the vague idea "every piece has a role in the checkmate". Now, having an actual role (like blocking a piece from capturing a checking piece) and being a clue in a retrograde analysis to show that the last move couldn't have been anything other than a given pawn's double move are very different and I think it is fair to assume the latter was not on loopy walt's mind at the time. $\endgroup$
    – Albert.Lang
    Commented Mar 12, 2021 at 6:35
  • $\begingroup$ @justhalf I've added an example that I think demonstrates what I mean. To say that the pawns e2 and g2 are actively participating in the checkmate is a bit of a stretch, but they are clearly accessories by the formal definition. $\endgroup$
    – Albert.Lang
    Commented Mar 12, 2021 at 7:22

2 Answers 2

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I think I've found a 30. But I'm not 100% sure since this retro analysis stuff is so error prone. What I am sure of is that some eagle-eyed community member will let me know in due course if I messed up. Anyway, here goes:

enter image description here

Obviously this owes a lot to ideas by @Albert.Lang and @justhalf.

Highlight reel:

Unique checkmate is g5xf6ep#. E.p. is available here because:
a5 couldn't have moved because there are two black rooks out and no promotions used for black
black king couldn't have moved because white king would have been in check by bishop h8

Removals:

To verify that if any single piece is removed, e.p. is no longer available observe that almost all pieces are on squares a black piece could have come from. Except:
Ra4 if removed there is no longer proof that a5 wasn't moved in the last move.
EDIT: not needed, see @Albert.Lang's comment Rc4 if removed white could have had another rook on e8 and the position could have been reached via 1. Re8-e7+ Kg7-f8+ 2. Re7-g7+ Kf8xg7. End of Edit
Pf5,Pg5 obvious.
Bc8 same as Ra4.
Ng8 does not invalidate e.p., but it is no longer checkmate: 1. g5xf6ep+ Kg7-h6.

Proof game:

[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"] 1.Nf3 Nf6 2.Nc3 g6 3.Nh4 Bg7 4.Nf5 O-O 5.g4 Bh8 6.Rg1 a5 7.Nh6+ Kg7 8.Ng8 Ra6 9.g5 Rb6 10.Rg4 Rb4 11.Nb5 Ra4 12.b3 c5 13.Ba3 c4 14.Bb4 c3 15.Rc4 Nc6 16.Bh3 Qb6 17.Kf1 Qe3 18.Kg2 e5 19.f3 e4 20.Kg3 Ng4 21.Qf1 Qd3 22.Kf4 Re8 23.Qg2 Re6 24.Rh1 Rd6 25.Qf1 Nf2 26.Qd1 Nd4 27.Ke5 Qe3 28.Be6 Nf5 29.Na7 Rd3 30.Nb5 Nd4 31.Re1 Nh3 32.a3 Nf2 33.Rf1 Nh3 34.Rh1 Nf2 35.h3 f5 36.gxf6# *

Online player

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  • 1
    $\begingroup$ Looks good to me. I even like that not all removals disable the mate in one by removing e.p. but that there is one which does it differently. My only gripe: Rc4 simply blocks the square where Pc3 could have come from, so no need for all that parrying-discovered-check-while-giving-ones-own-discovered-check cleverness. But that does not invalidate the solution. $\endgroup$
    – Albert.Lang
    Commented Mar 17, 2021 at 8:51
  • $\begingroup$ Is that me looking silly by trying to be clever too hard? @Albert.Lang $\endgroup$
    – loopy walt
    Commented Mar 17, 2021 at 8:58
  • $\begingroup$ "a5 couldn't have moved because there are two black rooks out and no promotions used for black" this is clever! Pawns are the hardest to cover, and alternative ways to cover pawns will be very useful, like in this case! $\endgroup$
    – justhalf
    Commented Mar 17, 2021 at 9:47
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    $\begingroup$ @justhalf Ah, let me see c8 and d7 in mine, b8 and e8 in yours? I set up the player btw. $\endgroup$
    – loopy walt
    Commented Mar 17, 2021 at 10:18
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    $\begingroup$ @justhalf What do you mean "fortunately"? All part of my grand design =D $\endgroup$
    – loopy walt
    Commented Mar 17, 2021 at 10:32
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While exploring changes to the example given (a 17-score solution can be found below, for historical reference), I found an almost completely different solution with 29 points (no witness):

enter image description here
There is only one capture, which is one black pawn being captured. In the final position, it is clear that the captured black pawn comes from d-file.

The explanation is mostly similar as the example, but for completeness I'll put the explanations here.

Retrograde analysis

Black king couldn't have come from b7, since the checking knight at d6 has nowhere to retract to.
Pawns on c6 and e6 couldn't have come from d7 via capture since white has no missing piece.
No other black pieces have available space to retract to.
Lastly, b5 couldn't have come from b6 since it is a check to white king.
Therefore b7-b5 was black's last move.

Legality and Remove-one-legality

The position is legal.
Removing any piece does not result in a situation where the black king is in check.
A complete proof will require a proof game for each piece removed, which is too tedious. But it seems quite plausible that any piece removed still results in a legal game.

Unique Mate-in-1

Based on the retrograde analysis, black last move was b7-b5, so a5xa6 e.p.# is the unique mate.
Rxc7 is countered by Bxc7.
Nxb5 is countered by axb5 or cxb5.
No other white move can give check to white king.

Accessories check

This is the hardest part, as I claim that all non-king pieces here are accessories.
Removing Ra8 allows Ka8-a7 as the last move, invalidating en passant.
Removing Bb8 allows Ra7-a8 as the last move, invalidating en passant. Rxc7+ is countered by Kb8.
Removing Rc8 allows Nc8-d6+ Kb7-a7 as the last move, invalidating en passant. Nc8+ is countered by Kb7.
Removing Be8 allows Ne8-c7 as the last move, invalidating en passant.
Removing Qg8 allows Ng8-e7 as the last move, invalidating en passant.
Removing Nc7 allows Bc7-b8 as the last move, invalidating en passant.
Removing e7 allows e7-e6 as the last move, invalidating en passant.
Removing any piece around Rf4, Qf5, or Bg6 allows the respective R, Q, or B to be able to retract, invalidating en passant.
Removing a6 allows Ka6-a7 as the last move, invalidating en passant.
Removing a5 removes the possibility of en passant.
Removing b5 removes the possibility of en passant.
Removing c6 allows Nc6-e7 as the last move, invalidating en passant.
Removing Nd6 no longer mates in 1 by Ka7-b7.
Lastly, removing any white pawns in b, c, d, e, f files allows c7xb6 or c7xd6 as the last move, invalidating en passant.
Therefore, all pieces are accessories.

Design thoughts:
Removing black pawn in d-file in my case is necessary to force white pawns on b and f files to be present, as I couldn't find a way to have other pieces force them to be present.
So, my belief is that 29 score is the highest achievable score (the highest possible score is 30, with all pieces on board as accessories).

Old Answer

To get the ball rolling: regicide score of 20 - 3 = 17 points. Witness highlighted in red.

The same idea as the example, just removing the other possibilities for other mates by moving the queen outside.
enter image description here
Retrograde analysis component:
Black king couldn't have come from a8, since Nc7 has no square to retract.
Black king couldn't have come from b8, since Na6 has no square to retract.
Black king couldn't have come from b7, since it is doubly checked by Bc8 and Nd6 with no possibility of discovered check.
Black knight couldn't have come from b7, since it gives check to white king.
Black pawns c6 and e6 couldn't have captured from b7 nor d7, since white has no missing piece.
Black pawn b5 couldn't have come from b6, since it gives check to white king.
Therefore b7-b5 is the last move.

Mate-in-1-uniqueness-check:
Nxc6 is countered by Nxc6
Nxb5 is countered by c6xb5
No other move can give check, and so mates in one with c5xb6 e.p. is unique

Remove-one-legality-check:
Removing any piece results in legal position, since there is no discovered check by white to black.

Remove-one-no-longer-mate-in-1-check:
Bh8 removed makes Nh8-f7 a possible last move, invalidating the only mate in one.
Any white piece (except Bh8) removed makes d7xe6 a possible last move, invalidating the only mate in one.
Black pawn on e6 removed makes black knight last move possible, invalidating the only mate in one.
Any of the black knights removed makes the other black knight last move possible (to each other's location), invalidating the only mate in one.
Lastly, b5 removed makes en passant not possible, and no other moves results in a mate in 1.

So only black pawn at c6 is a witness (removing it allows Ndxb5#), giving a regicide score of 20 - 3 = 17 points.

This can still be extended to higher scores:

by, for example, replacing Bh8 by a chain of black pieces that block each other. The legality will become harder to prove the more pieces added.

Appendix Contender for high scores:

Score: 28 (by removing the h6 pawn, since it is a witness, not an accessories). Still valid, since the missing white h-pawn cannot promote and be captured on d-file by black since there is only one missing black piece.
enter image description here

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  • $\begingroup$ The only missing part for 30 score solution is how to cover the b and f file white pawns with black pieces so the pawn capture argument doesn't need to be used, and so we can add back the black pawn on d-file. $\endgroup$
    – justhalf
    Commented Mar 13, 2021 at 8:16
  • $\begingroup$ I think that you misread the definition of accessories (or maybe I did): I think that removing the piece should invalidate the existence of a mate in one, that's all. Not invalidate the retrograde analysis. $\endgroup$
    – Arnaud Mortier
    Commented Mar 19, 2021 at 18:44
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    $\begingroup$ Yep, in this case invalidating the retrograde analysis invalidates the mate in one since the mate in one relies on it, and doesn't create avenue for another mate in one, which I why I also explored some alternatives mates and why they don't work. $\endgroup$
    – justhalf
    Commented Mar 20, 2021 at 9:39
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    $\begingroup$ That's right, my bad! Good job! $\endgroup$
    – Arnaud Mortier
    Commented Mar 20, 2021 at 9:42
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    $\begingroup$ Ah, sure then. Thanks Albert and loopy! =) You did a great job too with your 30 points answer :) $\endgroup$
    – justhalf
    Commented Mar 22, 2021 at 11:37

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