Solutions for "Adding Units"

{A}
+wPg7 allows 1 ... Kb4 2 Kg6 Kc5 3 Kf7 Kd6 4 Ke8 g8Q#.

{B}
Completely different in approach, and with an extra degree of freedom in the king moves. +wQa7 allows 1 ... Kc5 2 Kg3 Kd4 3 Kf2 Kd3 4 Ke1 Qg1#.

{C}
(a) Black has made six pawn captures, accounting for all the missing White units, while White has made two obvious ones. However, if Black's castling rights remain, then the White a pawn must have captured once, to be captured or to promote on the b file. Similarly, the White e pawn must have captured once to promote on the d or f file. This would account for all the missing Black units. So the solution is 1 0-0-0 b6 2 cxb6 Nxb6#.

(b) If a unit is added, it must be Black, and now Black's queenside castling is disabled, since one of the castling pieces must have moved before. However, +bRh8 leads to 1 0-0 h6 2 gxh6 Nxh6#. This is viable, since the White a pawn might have promoted on a1, while the White e pawn captured on d/f8. One try is +Ne7, which allows 1 Nf7 Bxg7 2 Rd8 Nf6# (or a transposition of the Black moves). However, Black already has two knights and eight pawns.

{D}
(a) There is no immediate mate in 1 for White. But then we realize that with White to move, Black has no legal last move. So it must be Black to move, and the direct mate solution is 0 ... a~ 1 Nb6#.

(b) The unit added cannot be wBf1, because wRa1 exited over f1 which must have been emptied. So the missing unit is Black. Most possible locations for a Black piece allow Black a last move, without allowing White an immediate mate in 1, so do not permit a solution. A Black pawn on its starting square prevents wRh7 from covering a7. The only possibility is some unit X on h3. X cannot be P, because then the pawns are doubled but no White piece may have been captured. If X is B/Q then White is in impossible check. If X is R, then whilst it is true that Black has no last move, White has no answer to 0 ... R~!. So X is N, White is to play, and +Nh3 allows 1 gxh3#.

{E}
(a) 1 0-0 Nc6 2 Nh8 Sxe7#. The Black queen & bishop were captured in the home cage, and the White h pawn captured two units. Therefore, the Black a pawn must have promoted, capturing 4 times. This, together with the White queen bishop, accounts for all missing units.

(b) Given the pawn capture constraints, the only units which can be added are those captured at home. But if wBc1 is added, then wRa1 could never have escaped. So the only two pieces which can be added are bQd8 & bBf8. The former allows no mate, but the latter +bBf8 gives 1 Nc6 Na6 2 Nd8 Nxc7#.

(c) The only legal addition is +bQd8, which permits 1 h5 fxg6 2 fxg6 Bxg6#.

{F}
The Black kingside pawns made at least 3 captures, so no White unit can be added. waP captured once, but note that wKR was captured in the cage, and so one of the wRs is promoted. This must be wdP. Black is missing 2 Bs, N & 2 Ps which might have promoted without capturing: 4 are available to wdP.

If wdP reached h7, then that took already 4 captures. It could only promote on h8, and then Black cannot castle kingside (although queenside is still on).

If wdP reached a7, then that took already 3 captures, and promotion might happen on a8 or b8 (in the latter case taking another capture). The rook might extricate itself only disturbing the bQR, and so Black cannot castle queenside (although kingside is still on).

Otherwise, wdP promoted via d7-f7, disturbing bK, so Black cannot castle at all.

(a) 1 Na6 Rxa6 2 0-0-0 Ra8#

(b) Adding a piece means that queenside castling will not work. The only way to add a piece to allow a helpmate is +bRh7. Now 1 0-0 Rxf6 2 Kh8 Rxf8#

(c) Adding another piece means that kingside castling will not work, but +bBf5 permits a different mate,
by 1 Kf7 Nf3 2 Kg6 Ne5#

One quick way to converge upon a solution for c) is to note that this third mate must involve bRh7: if it didn't, then the solution to c) would also cook part b).

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