by Thomas Volet
Die Schwalbe 1980 First Prize
Dedicated to J.G. Mauldon
8+16. On what squares did captures occur? In what order?
There are two ways to approach a solution, which ultimately converge. We can first inquire as to how the arrangement of units in the North of the board (the "North Matrix") came about, or we may ask "What was the last move?" For a solver not familiar with retroanalysis, the second approach is perhaps more effective.
White could not have made the very last move, as no White unit could have made a move that results in the above position (the "diagram position"). None of the White units had the physical freedom to make the last move other than the White King. The only move possible for the White King was to have moved from e8 to f8. However, if the White King had moved from e8 to f8 it would have been escaping simultaneous checks from both the Black Queen and the Rook at e7. Unless both those checks could have been delivered by Black on its immediately preceding move, the position would have been impossible. (Black could not make a move that delivers a check while another check was pending.) In fact, neither check, alone, could have been delivered on the last move. Therefore the White King could not have come from e8 on it last move. Thus White did not make the last move.
Black must have made the last move.
There are seven Black units that had the physical freedom to have made the last move. Three units, the Rooks and the Queen, could only have come from squares on the eighth rank from which they would have been delivering checks. As one side cannot be moving if the other side is in check, none of those three units could have made the last move. That leaves the Black King, the Knights, and the Bishop at g2 as possible last moving units.
Consider any position one move ago, in which one of those four units occupies a different square from which we suppose that it has moved to reach the diagram position. It will be seen by inspection that simply changing the square occupied by any of those units to a square that it could immediately have moved from, alone, will not result in a position in which White could have had a preceding move. Our conclusions in the second paragraph as to White's inability to have moved will continue to hold for any position that differs only in that the Black King, one of the Black Knights, or the Black Bishop, is on a square that it could have occupied before moving to its diagram position. The only possible way that Black's last move can have been made from a position in which White had a preceding move is if Black's last move was the capture of a White unit and it was that captured White unit that made White's last move.
Thus the Black King, a Black Knight, or the Black Bishop at g2 captured a White unit on its last move. In order to continue the analysis, we must distinguish between the capture of a White Pawn and the capture of a White unit other than a Pawn. (We will refer to a unit other than a Pawn as a "Piece".)
Could Black have captured a White Piece (i.e., a White unit other than a Pawn) on the last move? The answer to that question depends upon some unusual positional features of the diagram.
All Black's units are present. Thus White made no captures. Eight White units are present. Thus Black made eight captures.
The only units missing from the diagram are the eight White Pawns. The diagram presents a White Queen, two White Bishops, two White Knights, and two White Rooks, along with the King: a full complement of eight White Pieces.
As White has made no captures, none of its Pawns has swerved off its original file. Each White Pawn was either captured on its original file or reached the eighth rank on that file and promoted.
The 8 Black Pawns are arrayed one per file. If any Black Pawn swerved off its original file, either it returned to that file or another Black Pawn captured over to that file, thus restoring the Black Pawn array. For example, if the Black Pawn at h5 originated on f7, then either the Black Pawn originating at h7 moved to f5 or the Black Pawn originating at h7 moved to g5 and the Black Pawn originating at g7 moved to f7. The point is that given the Black Pawn array in the diagram, for each file shift of a Black Pawn there must have been one other file shift of a Black Pawn. Each capture by a Black Pawn implies another capture by a Black Pawn.
For example, if the Black Pawn that originated on the h-file (the "BhP") swerved to the g-file to capture the White Pawn that originated on the g-file, the diagram position could be reached if the BgP swerved to the h-file by capturing the WhP. Similarly, the following could have occurred: the WgP could have advanced to g6, the BhP then captured a White Piece (say a Knight) at g5, the WhP then promoted at h8 (say to a Rook), the promoted Rook moved to h6, the BgP then swerved to the h-file to capture the promoted Rook, the WgP then promoted at g8 to a Knight, and the diagram position could have been reached. However those examples all involve captures by Pawns. None of the examples addresses the case in which a Black Piece has captured a White Piece.
Can the position be reached if a Black Piece has captured a White Piece?
If Black at any point had captured a White Piece, the diagram could present a full complement of White Pieces only if a White Pawn had promoted and the promoted Piece either (i) appears in the diagram in place of the captured piece or (ii) was itself the captured Piece. As no White Pawn swerved off its file, a White Pawn could have promoted only if a Black Pawn swerved off its file to permit the White Pawn direct passage to the eighth rank. Thus the capture of a White Piece implies a capture by a Black Pawn.
But we have established that each Black Pawn capture implies a "matching" Black Pawn capture (to restore the Black Pawn array). Thus the capture of a White Piece implies at least two captures by Black Pawns. As shown in the example above, those captures could have been of White Pieces (including promoted Pieces).
The White Piece hypothetically captured by a Black Piece was either an original White Piece or a promoted White Piece. But if a Black Piece has captured an original White Piece, at least one promoted White Piece must have survived to replace the captured one in the diagram array of White Pieces. Consider the file on which the White Pawn promoted to become that replacement White Piece The capture onto the file of promotion by the Black Pawn that now occupies that file (in the diagram) could not have been of a White Pawn, because that White Pawn is the one that promoted. The Black Pawn capture to that file must have been of a White Piece.
If the Black Piece has captured a promoted White Piece, the promotion could not have taken place without captures by two Black Pawns. As above, the capture onto the file of promotion by the Black Pawn that now occupies that file (in the diagram) could not have been of a White Pawn, because that White Pawn has by hypothesis promoted. The Black Pawn capture to that file must have been of a White Piece.
Thus, whether the White Piece captured by a Black Piece was an original White Piece or a promoted White Piece, the capture of a White Piece by a Black Piece entails an additional capture of a White Piece, by a Black Pawn. How is that last White Piece to be found in the diagram array of White Pieces unless it, too, was either a promoted Piece or was replaced by a promoted Piece? The hypothesis results in an infinite regress.
Concretely, imagine that a Black Bishop had captured a White Knight early in the game. A White Pawn must have promoted to provide for the two White Knights in the diagram. In order for the White Pawn to have promoted there must have been two captures by Black Pawns (one to clear the file of promotion, one to restore the Black Pawn array). The capture onto the file of promotion must have been a capture of a White Piece, because the White Pawn on that file has promoted or will promote. If this last captured White Piece was a Knight, we are back where we started, needing to replace a White Knight. But whatever other White Piece it might have been will leave us with the same difficulty, as we must find a way to restore a counterpart of that White Piece to the diagram array.
A Black Piece cannot have captured a White Piece (i.e., a unit other than a Pawn) in any game of chess that reached the diagram position.
Therefore the last move must have been the capture of a White Pawn.
The Knight at a7 could not have captured a White Pawn on its last move as a White Pawn could not have reached a7 in the diagram unless it had made a capture. If the other Knight had captured a White Pawn on e2, or if the Bishop had captured a White Pawn on g2, neither of those Pawns could have made the last move as both e2 and g2 are squares on which Pawns originate.
Therefore the last move was the capture of a White Pawn at h3 by the Black King.
We need to determine which square the Black King occupied before it captured on h3. Consider the possible preceding move by White, which could only have been Ph2-h3. As the White Pawn was on h2 before its move to h3, the Black King certainly did not come from h2. Furthermore, the Black King could not have been on g3 when the White Pawn was on h2, or Black would have been in check with White moving. That leaves g4 and h4 as possibilities. Figure 1 shows the position assuming that the Black King was on h4 and the White Pawn on h2 (Figure 1 being reached, hypothetically, in forward play, before White played Ph2-h3 and Black replied Kh4xPh3):
Figure 1. Hypothetical position. What was the preceding move? (Position is impossible.)
Black has made the preceding move. But, as in the diagram position, we must be able to account for the full complement of White Pieces and the Black Pawn array (one per file), so that only the capture of a White unit will give White the possibility of a move preceding Black's last move. We also know that the captured unit must be a Pawn. As above, the Knight at a7 cannot have captured a White Pawn on a7 because no White Pawn could have reached a7. A White Pawn captured at e2 or g2 could not have moved at all. Once again, the captured White Pawn must have been captured by the Black King. But no White Pawn swerved off its file. In Figure 1, the Black King could not have captured a White Pawn on the immediately preceding move. Figure 1 is impossible.
The only last move that could have resulted in the original diagram position was 1.Kg4xPh3. Immediately before that, White played 1….Ph2-h3. (A King may move into check in retroplay, but only if the check can have been delivered by the other side on the immediately preceding move.) See Figure 2.
Figure 2. Actual position before each side's last forward move.
(Forward play of 1.Ph2-h3+ Kg4xPh3 results in the diagram position.)
The inquiry continues with the question as to what move Black could have made such that White had a preceding move with the result being the position of Figure 2.
Applying the same logic employed in determining Black's last move in the original diagram position, Black must continue by retracting 2.Kf3xPg4, preceded by 2….Pg3-g4. (Henceforth we will refer to a move backward in time as "retraction" or a "retromove". Play in that direction is referred to as "retroplay".) The retraction 2. Kg3xPg4 obviously does not work because the Black King blocks the retraction of the White Pawn at g4 (i.e., in forward play, it blocks the move Pg3-g4). 2.Kf4xPg4 doesn't work for the same reason that the first retraction by Black could not have been 1.Kg3xPh3, i.e., that with the Black King at f4, Black would have been in check (with White moving) given the WKNP on g3. (Obviously Kh4xPg4 leaves White no preceding move, and Kh3xPg4 will leave the White King stranded on the h-file, with White out of retromoves after the retraction of the WgP.)
Having established a possible immediate past for the diagram position, it is appropriate to consider the mechanism by which the retroplay can unwind the blocked position in the North Matrix (i.e., the means by which the position was reached in forward play). Inspection will show that in order for the diagram position to be have been reached, at some point earlier in the game the Black King must have occupied the square d8. With the Black King on d8 the Black Queen was able to move from the eighth rank to its diagram position without delivering a check to which White could not have responded (and which therefore would have been impossible). See Figure 3.
Figure 3. Unwound position in North Matrix. (Only North half of board shown.)
In Figure 3 the last move was Rd7-e7 by Black, preceded by Re7-f7 by White. From Figure 3, in forward play ultimately resulting in the original diagram position, White made a Pawn move and Black responded Qc8-d7. White made Pawn moves while the Black King went to c8, b8, a7, and b6, and the Black Kn
As the Black units were making their final moves to reach their respective positions in the diagram, and during the Black King's long march to h3, White's moves must have been supplied by Pawns not present in the diagram that were captured after the North Matrix was fixed. Thus, in the retroplay, it is necessary for the Black King to uncapture White Pawns whose retractions (advances, in forward play) will constitute White's retromoves as the Black King travels to d8.
Returning to the retroplay, Black's third retraction must be 3.Ke4xPf3, followed by 3…Pf2-f3. It appears that Black could remove the BQB from g2, but after the retraction by White of the KNP to g2, the pressure to find a W retromove remains. The removal of the BQB from g2 seems to be optional at this point, adding a White retromove but at the expense of a Black retromove that does not advance the progress of the Black King to d8. If optional, we may not conclude, as with the retromoves that we have deduced, that it must have taken place as a condition to reaching the diagram position. However, as explained below, it is in fact necessary that the BQB not retract at this point.
Even if the retraction of the BQB were available, White would soon need another uncaptured Pawn to retract. 3.Kf4xPf3 would leave the Black King on the f-file after the White retraction, with no move by Black allowing for a preceding move by White (except for the retraction of the BQB away from g2, which, after the retraction of the WKNP to g2, leaves the same problem of a preceding White move).
The uncapture of a White Pawn on the e-file is now in order. Black must continue to retract with the objective of maximizing the retromoves available to White, as White must have retromoves while the Black King retracts along a path to d8. Black retracts 4.Kd5xPe4. White responds 4…Pe3-e4. (At this point Black seemingly could retract the Knight from e3, using a tempo to create a White tempo, but not only is it pointless at this stage of the retroplay, but, for the same reasons applicable to the equally pointless (at this time) retraction of the BQB from g2, is not a valid possibility. The reason will become clear in connection with Black's tenth retromove.) Black continues 5.Kc4xPd5, followed by the retraction (i.e., preceded, in forward play by) 5… Pd4-d5.
Black continues to retract with the principle that White is to be provided with the most possible Pawn retractions, and that therefore any Pawn uncaptures take place as high on the files as possible. Black continues 6.Kc5-c4 to gain the highest position for its next uncapture (a maneuver the necessity of which ultimately can be demonstrated only by trial and error). The check into which Black has thus placed itself must have been delivered on the immediately preceding move by White, which has available the retraction 6…Pd3-d4. This allows the Black King to reach b6 on its next uncapture, 7.Kb6xPc5. The retraction into check, once again, is immediately preceded by the checking move, here 7…Pc4-c5. The position resulting from the retroplay thus far is shown in Figure 4, below.
Figure 4. After White's retraction 7…Pc4-c5. Black to retract.
Figure 4 is an important position. Inspection shows that Black needs a minimum of 6 retromoves, each matched by its White counterpart, until a position is reached in which Black can retract Rd7-e7 allowing White to retract Re7-f7 and unwind the North Matrix: the Black Knight at a7 must remove from the path of the Black King (one retromove), the Black King needs to retract to a7, to b8, to c8, and to d8 (4 retromoves), and the Black Queen needs to retract to c8 (one retromove). We must focus on the number of available White Pawn retractions as the Black retractions take place.
In Figure 4 White has 3 Pawn retractions available using the Pawns on d3 and c4. The retractions Pe3-e3 and Pg2-g3 (only one of which is available as one of these Pawns must remain on the third rank to account for the WKB's earlier exit from its home square) have not been included because each requires a Black retromove that seems not to advance the objective of retracting the Black King to d8, and thus would appear to gain a White retraction at the cost of a Black retraction that seems pointless. Thus White needs to find 3 more Pawn retractions, which can only derive from the uncapture of one or both of the White Pawns that originated on a2 and b2.
The uncapture of the WQRP would involve a departure of the BK from the shortest path to d8, so let us first examine the efficacy of uncapturing only the WQNP. There are a number of possibilities. Black could retract Nb5-a7 (White using a retraction of the QBP) and then uncapture the WQNP at b5 by Na3xPb5. That would use up the last retraction of the WQBP, leaving 3 retractions for the uncaptured QNP (or 2 by the QNP and one by the QP, as an exit square must be present for the WQB), which will fall 2 short of matching the 5 retromoves still needed by Black to reach the unwound position (Figure 3). Alternatively the Knight at a7 could retract to c8, allowing the Black King to uncapture the WQNP a square higher, at b6. But that possibility gains 1 retromove for White, still leaving it short by 1.
In order to fully analyze the issues presented by the position of Figure 4, it is helpful to consider the constraints on the array of the White Pawns if all 8 are retracted as far as possible prior to the return of the White Pieces to their respective home squares. At least one White Pawn must remain on the fourth rank so that the White Rooks have a pathway to retract to their home squares (i.e., in forward play, to provide them a means of egress). Also, squares permitting the retraction of the White Bishops must be open, including at least one of e2 or g2, and one of b2 or d2. See Figure 5A.
Figure 5A. Sample array of 8 Pawns allowing exit of all White Pieces with no captures by White.
However, if only 7 White Pawns are uncaptured and retract, it is not necessary for one to remain on the fourth rank because the White Rooks can return through the unoccupied file. See Figure 5B.
Figure 5B. Sample array of 7 Pawns allowing exit of all White Pieces with no captures by White.
Therefore the uncapture of the eighth White Pawn requires that one Pawn remain on the fourth rank, and thus entails the loss of a White retromove (the retraction of the WQNP to b3) in addition to the two retromoves consumed by the diversion of the Black King to the a-file and its return to b6 (which is a necessary stop on the route to d8). But a White Pawn uncaptured at a5 can only retract three squares, to a2. Thus it would appear that the uncapture of the WQRP at a5 will merely recoup the three retromoves it consumes, leaving the Black King no closer to d8 than it stands before the maneuver. Like the removal of the Bishop from g2 or the Knight from e2, each of which creates a White Pawn retraction but requires a Black retromove that does not advance the progress of the Black King to d8, it appears pointless. How can we proceed from Figure 4 to find enough retromoves for Wite?
It is clear that the WbP must be uncaptured. But how can the retroplay continue with only one retromove by the Black Knight at a7, given that when the Knight occupies the only square to which it can possibly make a single retraction, b5, it blocks the same WP, at b6, that was uncaptured to provide the needed White retromoves?
The answer lies in the execution of two consecutive retro sequences, each of which taken alone uses as many Black retromoves as the number of White retromoves it adds, but the two of which, taken together, permit the needed interplay of the Black Knight and White bP by allowing the White bP to retract out of the way of the Knight at a7.
Starting at Figure 4, the first maneuver is to uncapture the WaP. Proceed 8.Ka5xPb6, Pb5-b6. See Figure 6A.
Figure 6A. Position after retraction of 8....Pb6-b6. Black to retract.
At this point, the WbP has been uncaptured, which adds 3 White Retromoves as discussed above.
The WbP cannot retract to b4 at this point (which would mean, in forward play , that it was moving with the Black Kicheck). We must remove the Black King from a5. Proceed 9.Kb6xPa5, Pa4-a5. The WaP must retract immediately (as in forward play it delivers a check immediately before being captured.) See Figure 6B.
Figure 6B. Position after 9….Pa4-a5
This completes the uncapture of the WaP, a maneuver that in its gains no additional net retromoves above those gained from uncapture of the WbP, yet does not cost net retromoves. Furthermore, this maneuver leaves us with the Black units in the identical positions they occupied in Figure 4 (from which we know that at least 6 Black retromoves and 6 White retromoves are needed before Rd7-e7), but, very significantly, with the WbP ab5, only one square away from a position in which that WP no longer interferes with the retraction of the Black Knight to b5 (a square from which the Knight need not retract again during the Black King's trip to d8).
But it is now Black's turn to retract, and the only retromoves available for either the Black King the Black Knight at a7 would result in the squandering of precious White retromoves - - the King can retract only in a wrong direction from d8, lengthening its journey, and the Knight can retract only to c8, which will put it in the King's eventual path and require it to retract yet again.
It seems that we need it to be White's turn to retract rather than Black's, but even if it were, we would be lacking one White's retromove.
At this point, it is necessary to execute the second maneuver, which also by itself gains no additional retromoves, but which in effect allows White to make the needed retraction of the Pawn at b5 without disturbing the otherwise optimal positions of the Knight at a7 and the Black King. Retract 10. Bh3-g2! This surprising retraction by Black (which remained available because of our earlier forbearance) uses a Black retromove in addition to the 6 needed before Rd7-e7, but it also gains a White counterpart in the now-available Pg2-g3. (Any other retraction of the BQB except Bh1-g2 would work just as well, as would any retraction of the WN at e2 except Nc4-e2, the Knight retraction making available Pd2-d3 instead of Pg2-g3.)
The key to the remaining retroplay, however, is that we may defer the newly-available White Pawn retraction to g2, and instead immediately take advantage of the opportunity to get the White Pawn at b5 out of the way of the Knight at a7. Thus, the response is 10....Pb4-b5. See Figure 6C.
Figure 6C. Position after 10....Pb4-b5
As noted above, we chose not to earlier retract the Bg2 or the Ne2 not merely because it seemed inefficacious. Had we done so, we would not have had available, when needed, a harmless Black retromove (the seemingly pointless Bh3-g2, without which the only retractable Black units, the K and the Na7, would have had to move away from their most efficient destinations), which effectively makes it White's rather than Black's turn to retract in the Northwest corner precisely when necessary to allow the WbP poised at b5 to retract out of the path of the Black Knight. Continue 11.Nb5-a7, and we have removed the Black Knight from the Black King's path with a single move. In other words, at the point reached in Figure 6B, when it is Black's turn to retract but the solution requires the White pawn at b5 to vacate its square in favor of the Black Knight, we were able to use the Pawn position in the Southeast as a time "battery" holding one stored retromove. The requirement that Black take its turn was fulfilled by Bh3-g2, which, by creating the White retromove Pg2-g3, had no net cost. Thus as to the position in the West side of the board, effectively the move was shifted from Black (which had only costly retractions until Pb4-b5) to White (which could then retract the bP).
The seemingly pointless uncapture of the WaP allowed us to get the WbP uncaptured and retracted to b5 without cost, and the seemingly pointless Black Bishop move satisfies the requirement thar Black retract on its required turn, but without involving an inefficient retraction by the Black King or the Knighat a7.
The retroplay now continues straightforwardly: 11...Pd2-d3 12.Ka7-b6, Pc3-c4 13.Kb8-a7, Pc2-c3 14.Kc8-b8, Pa3-a4 15.Kd8-c8, Pa2-a3 16.Qc8-d7, and the "stored" retromove, 16...Pg2-g3! Black then retracts 17.Re7-d7, and, finally, we have the first retraction by a White unit other than a Pawn, 17...Re7-f7. From that position (Figure 7) the remaining unwind (however tedious) is clear.
Figure 7. Position after White's retraction 17...Re7-f7.
The White Pawns must have been captured by the Back King in the following order on the following squares: a5, b6, c5, d5, e4, 13, g4, and h3. In each case the capturing move is unique.
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