if 29... Kg8 30. Qxg6#
if 30... K moves 31. Qg7#
if 30... Rd7/f7 31. QxR+ (leads to the main line)

29. Qg4 also leads to mate, but this line is both quicker and nicer... 29. Qc7 also wins.

However, not bad played for being a Sim.>

Position after 29 ♖g7xh7+!

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Very nice.

29 ♖g7xh7+! both destroys the <PAWN CHAIN BASE> Black h7-pawn which <DEFENDS> the Black g6-knight while vacating (<CLEARANCE>) the <LOOSE> g7-square next to the Black g8-king for the White g3-queen to eventually occupy.

The amazing point of this <DESTRUCTIVE SACRIFICE> 29 ♖g7xh7+! is that after 29 ... ♔h8x♖h7 30 ♕g3-c7+,

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Black has no(!) supported <BLOCK> of the White c7-queen check along the 7th rank.

Notice the importance of <LOOSE SQUARES NEXT TO THE ENEMY KING>: this entire combination is based on the fact that the g7-square is a <LOOSE SQUARE> next to the Black h8-king. The White f5-knight attacks it, only the Black h8-king defends it. Fischer (White) merely needs a way to bring his queen into contact with this <LOOSE> g7-square to create mate threats, which is exactly what 29 ♖g7xh7+! does.

I am not sure if there isn't a misunderstanding here. Fischer played this combination on move 30. with the queen acting on column h and g. <Mate Hunter> indicated that this could be played earlier, with the queen acting on rank 7 instead, with the same motifs but more decisively (forced mate). Did Fischer miss this or regarded the quiet 29.Qg4, preparing the combination, as more elegant?

Personally, I would play straightforwardly 29.Qc7 with mate in a few moves, but without a need for sacrifice. Definitely not as elegant as the combination discussed above.