# A Complicated Exercise In Addition

Another day, another walk down to the cafe. I was waiting in line for my coffee, wondering what the barista could do this time to make my name look whack. But as I waddled in line, a curious site caught my eye. On the other end of the cafe, in a dark and dusty corner, set an unclaimed chessboard with an assortment of pieces still on it.

After buying my drink, and the usual scowl at my barista, I headed on over to the board. There were plenty of pieces missing from the board. The strange part is that the missing pieces were nowhere to be seen. Nor was there a paper or two to indicate if they had been knocked off the board or not.

This left my brain scrambled as to where they could be. What pieces could be left on the board and what pieces couldn't be? Heck, I didn't know who's turn it was.

The board looked like this.

I've spent the past few days dwelling on it. Even so, I cannot seem to crack it. However the position was reached, it was certainly an ordinary game. Can you tell me how many pieces can be added back to the board, which pieces they are, and where they are located?

Oh, and by the way, I'll give bonus points to whomsoever can find the fastest way to reach the position. It's nothing much, but hey, who doesn't love a good challenge?

This problem was composed by Thierry Le Gleuher and published in an online article called "Economy Records in Add Unit(s) Problems". No looking up the solution whatsoever!

Addendum 2/7: Now that the puzzle has been solved, here is the link to the original publication: http://abrobecker.free.fr/chess/addunits.pdf

There are 5 missing units that need to be re-added for this position to be legal: black pawns on a3,b3,c3,d3,e3. White played last (either Ba8 or Ng1). This is the only solution.

Proof game

Here is a (not fully optimized) solution to reach the diagram after White's 55th move:

First steps

The only missing White unit is

the dark squared bishop. It never could move from its initial position on c1 since Pb2 and Pd2 are still on place but since the white King found its way from e1 to a1, the bishop cannot stand on c1 either. We can deduce that it was taken on place by a black knight (from b3).

The missing Black units are

* 5 pawns from the files a to e (black pawns never changed file because no white unit was ever taken but on c1.)
* the queen, the dark-squared bishop and one knight, which must all have been taken by the white pawns on their way to f3 and f7: gxf3, hxg and gxf

It means that the only possible units that can be added

are a number of black pawns (between zero and five) on the queenside and center.

How many and where

That where a deep retrograd analysis is needed !

There is at least one missing black pawn. Otherwise, what would be Black's last move ? not Kg4-h5, because the king couldn't be twice under check. And even if White played last, that cannot give an ante-move to Black.

Consider the cluster of 11 pieces in the north-east part of the board and ask yourself: which of them played last ?
* Not a pawn because the squares behind them are not free and there is no capture left (so, neither wPe6xf7 nor bPe7xf6 is possible)
* Not a rook nor bishop nor black knight because they have no square to come from.
* Not the white knight because the black king would have been in check by the wRh6 with White to move, which is illegal.

There only remains

the Black king. Its last move necessarily was ...Kg4-h5, escaping from a check by the knight when White played Nf5-h6. This proves in turn that White's pawn wasn't on f3 yet (or it would have doubly attacked the bKg4 in an impossible configuration) and White next move was g2xf3 (taking the queen - neither the bishop which is dark-squared, nor the knight for tempo-related issues - we need the white king on e1 as it will soon become apparent).

But then we can deduce what was White position at this point:

The Bf1 hadn't moved yet, so white king, queen and rook also were still boxed in on a1-e1. White knight was on g1 too. Well, it could be on h3 but that would make us lose a move in the following analysis, and nowhere else because it needs a road back to g1 - and f4 is forbidden, from there the wN would give a check and force Black to react with ...gf4.

So, it means that after ...Kg4-h5 White must have played

at least 19 moves:
g2xf3,Bf1-h3-c8-b7-a8 (for instance, but that's 4 moves by the bishop in any case), Ng1-h3,Ke1-f1-g2,Qh1,Rg1,Kg2-f1-e1-d1-c1-b1-a1,Rg1-b1,Qh1-d1,Nh3-g1

Meanwhile, Black could only move

its 5 queenside pawns that are currently invisible
They can play a theorical maximum of 20 moves (4 each for the 7th rank to the 3rd). However, either the b or the d pawn had already moved beforehand to let the bBc8 go out. And so did the e-pawn, to let the bBf8 go out and get taken on g3 or g5. It remains 18 moves : just enough if it is Black to move in the current position.

You may wonder why

the bBf8 was released when Pe7 moved, and not Pg7 ?

That's because

the black bishop must have been taken before the Pg7 moved at all, in order to allow the white h-pawn a way to f7. The pawn moves on the kingside were h2xg3,g4,g5 (or h2-h4,h4xg5), g6, ...f6, g6xf7, ...g5, ...h5, ...h4, and finally g2xf3.
The alternative way for white h-pawn and black g-pawn to cross ways, with ...g5 (letting bBf8 out) and h2-h4-h5xg6xf7 is impossible because all white captures would be on light squares, and the dark-squared bishop is one of the units that must be taken on the way.

So,

When White played g2xf3, the missing black pawns were on a7,c7,e6, and either d6,b7 or b6,d7. In the 18 next moves White played the aforementioned manoeuvers while Black pushed its pawns one square at a time down the a,b,c,d, and e files. White's Ba8 couldn't take any on its way, so they must still be there in the final position.

A truly magnificient problem by Thierry Le Gleuher !

• Can the downvoter please explain how I should improve this answer ? Jan 31, 2022 at 13:46
• I seem to recall the downvote occurring pretty soon after you originally posted the answer, when it was still quite unburdened by any substance whatsoever. The site automatically saves draft versions of unposted answers, there's really no need to hit the "post answer" button when the post doesn't actually contain an answer yet.
– Bass
Feb 2, 2022 at 0:06
• I forgot to say this earlier, but excellent proof game! Is better than 54 possible? (this stops just before the final retraction: 1. h4 h5 2. Rh3 d6 3. Rg3 Kd7 4. Nc3 Ke6 5. Rg6+ Kf5 6. Rh6 Kg4 7. Rh7 Bf5 8. Rh6 e6 9. Rh7 Be7 10. Rh6 Bg5 11. hxg5 h4 12. Rh5 Rh6 13. Nb5 Qf6 14. Nd4 Bh7 15. g6 Qf4 16. Nf5 Nc6 17. Nh3 Na5 18. Ng1 Nb3 19. Nh3 Nxc1 20. Ng1 Nb3 21. Nh3 Na5 22. Ng1 Nc4 23. Nh3 Ne5 24. Ng1 Ne7 25. Nh3 f6 26. Ng1 Nf7 27. gxf7 g5 28. Nh3 Rg8 29. Ng1 Rg7 30. Nh3 Bg8 31. Ng1 Ng6 32. Nh3 Nh8 33. Ng1 Rhg6 34. Rh7 Qf3 35. Nh6+ Kh5 Feb 7, 2022 at 21:23

There's quite a lot of chaff in the question, but as far as I can tell, we're supposed to do retrograde things to the situation to find out which piece(s) have been removed from the board.

## Part 1: Roll call

So, let's start with white: we can see all the pieces except the dark square bishop. Its fate is clear: black has captured it earlier on. (It cannot have moved from its starting square at C1, yet the white king must have moved through that square to get to its current position.)

Since black has only captured at C1, all the black pawns must therefore still be on their starting files. In particular, the g- and h-pawns are still there, so black hasn't promoted any pawns.

Next, let's count the rest of the black (non-pawn) pieces. At least three of them have been taken by the white pawns (1 by the g pawn, 2 by the h pawn), and since we can see two rooks, a bishop and a knight, that actually accounts for all the black pieces.

## Part 2: Previous moves

It clearly cannot be white's turn: no black piece has a square it could have come from.

If it's now black's turn, then white's previous move cannot have been anywhere at the top right: the f7 pawn cannot have come from e6, because while it could have gotten to the e file by taking the e pawn, getting back would require an extra captured black piece. The rook doesn't have any squares it could have come from, and the knight cannot have moved into its current position either: the black king would have been in check during white's turn otherwise. (Because the knight move cannot have captured anything.)

This means black's previous move was not made with any of the pieces still on board. It also cannot be by another piece that got captured since, because black's missing pieces were taken by white pawns, so the previous white move would have needed to be gxf3, which is impossible, because white's light square bishop couldn't have left its starting square with g2 still in place.

Conclusion:

The position as shown is illegal

But since we are given the possibility that there are some pieces missing from the board, there's still hope.

## Part 3: Untangling the clump

The tangle at the top right is super tight. I cannot find any other way to reach it except that the black king must be the very last piece to step in, and that can only have occurred instantly after it was checked by the knight at h5.

But this this is utterly ridiculous. It means that at the time the tangle was finalized, white's g pawn was still on g2. And therefore white's light square bishop was on f1! And this means the white pieces at the home rank cannot yet have changed their order at this time.

FEN: q5bn/5PrR/5pr1/5Np1/6kp/8/PPPPPPP1/R2QKBN1 w - - 0 1

In the picture above, I've returned the black piece to be captured on f3 to the board (the queen at A8). The other white knight is on its home square, only to keep things simple. (We're going to want to dodge with it later, but we'll do it after the bishop is out)

From here, we can now finalize the tangle with 1. Nh6+ Kh5 2. Rb1 Qf3 3.gxf3, but then we are out of a black move. Or about a billion of them, really. Let's count:

bishop to A8: 4 moves
King to f2 and then A1: 8 moves
Knight to dodge the heavy pieces: 2 moves
Queen to h1 and back: 2 moves
Rook to g1 and back: 2 moves

That's quite a few moves. So many, actually, (> 16) that we must return all the black pawns to the board. We cannot return all of them in their starting squares though, because the C8 bishop needs a way out.

Looks like we're not unique yet, though. From here, the given position can be reached with the sequence b6 2.Bh3 c6 3.Bc8 a6 4.Bb7 a5 5.Ba8 b5 6.Nh3 c5 7.Kf1 d5 8.Kg2 e6 9.Qh1 e5 10.Rg1 a4 11.Kf1 b4 12.Ke1 c4 13.Kd1 d4 14.Kc1 e4 15.Kb1 a3 16.Ka1 b3 17.Rb1 c3 18.Qd1 d3 19.Ng1, and there's still ambiguity in the pawn positions:

Maybe there are some hidden complications in the earlier phases of clump creation?

Ah, yes. One of white's h-pawn's captures must be the dark square bishop. That capture must have happened either behind the black g-pawn (impossible, there are no suitable dark squares), on f7 (wrong coloured square), or before the g pawn moved. That means that black's e-pawn must also have moved before the clump was formed, and finally we have a unique solution:

The missing pieces are

The five black pawns, all pushed as far forward as they can go,

and it's black's turn to move from this position: