Solutions for "Adding Units"
+wPg7 allows 1 ... Kb4 2 Kg6 Kc5 3 Kf7 Kd6 4 Ke8 g8Q#.
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#.
(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.
(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#.
(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
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
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
(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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