2 Fix typos and English
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Short answer: Since after the bishop recpaturerecapture on f8 (and not the rook recapture!) white is tactically and positionally completely busted, with 5 active black pieces against a completely exposed king in the centre and no foreseeable chance of consolidation in order to eventually benefit from the material advantage.

  • 15...Bxf8 is with tempo as it attacks the queen on a3
  • White king is still stuck in the centre and being at least two tempi away from castling, not even sidestepping from the open file with Kf1 is possible.
  • With the bishop recaptuingrecapturing on f8, Bb4 becomes a constant threat, which means Nxc3 is unstoppable for white as Qxc3 falls to Bb4 winning the queen in all ensuing lines.
  • Although white is an exchange up temporarily, one of their rooks and the f1 bishop are completely out of play still and therefore it is in fact black who effectively appears to be material up (with last piece being the a8 rook that comes into play with check in fact Re8+), not in the raw count of pieces but most definitely in terms of the positional compensation.
  • Most important consequence of all these observations: black is well in time to prevent white from ever consolidating, which means even without concrete assessment of the different lines we can safely bet that everything's about to fall apart for white quite shortly after 15...Bxf8 as black's initiative on the white king seemseems to lead to tactically and forcibly winning variations in every scenario.

these lines clearly demonstrate why the f8 rook is in fact not hanging and why white's best chances liedlie in Bc4 creating a Luft on f1 for the king (or even short castles if black allows on a good day) side-stepping (at first glance) most of the tactical shenanigans we just witnessed after 13...Bxf8.

 [title "15. Bc4 only attempt at consolidating"]
 [fen "r4rk1/pp2Bpbp/1qp3p1/8/3Pn1b1/Q1P2N2/P4PPP/3RKB1R w K - 0 1"]

 1. Bc4 Nxc3 2. Bc5 Rfe8+ 3. Kf1 {and the rest is as you see in the real game, with Fischer's devastating attack despite the tucked away king on f1, which is just too little too late.}

Short answer: Since after the bishop recpature on f8 (and not the rook recapture!) white is tactically and positionally completely busted, with 5 active black pieces against a completely exposed king in the centre and no foreseeable chance of consolidation in order to eventually benefit from the material advantage.

  • 15...Bxf8 is with tempo as it attacks the queen on a3
  • White king is still stuck in the centre and being at least two tempi away from castling, not even sidestepping from the open file with Kf1 is possible.
  • With the bishop recaptuing on f8, Bb4 becomes a constant threat, which means Nxc3 is unstoppable for white as Qxc3 falls to Bb4 winning the queen in all ensuing lines.
  • Although white is an exchange up temporarily, one of their rooks and the f1 bishop are completely out of play still and therefore it is in fact black who effectively appears to be material up (with last piece being the a8 rook that comes into play with check in fact Re8+), not in the raw count of pieces but most definitely in terms of the positional compensation.
  • Most important consequence of all these observations: black is well in time to prevent white from ever consolidating, which means even without concrete assessment of the different lines we can safely bet that everything's about to fall apart for white quite shortly after 15...Bxf8 as black's initiative on the white king seem to lead to tactically and forcibly winning variations in every scenario.

these lines clearly demonstrate why the f8 rook is in fact not hanging and why white's best chances lied in Bc4 creating a Luft on f1 for the king (or even short castles if black allows on a good day) side-stepping (at first glance) most of the tactical shenanigans we just witnessed after 13...Bxf8.

 [title "15. Bc4 only attempt at consolidating"]
 [fen "r4rk1/pp2Bpbp/1qp3p1/8/3Pn1b1/Q1P2N2/P4PPP/3RKB1R w K - 0 1"]

 1. Bc4 Nxc3 2. Bc5 Rfe8+ 3. Kf1 {and the rest is as you in the real game, with Fischer's devastating attack despite the tucked away king on f1, which is just too little too late.}

Short answer: Since after the bishop recapture on f8 (and not the rook recapture!) white is tactically and positionally completely busted, with 5 active black pieces against a completely exposed king in the centre and no foreseeable chance of consolidation in order to eventually benefit from the material advantage.

  • 15...Bxf8 is with tempo as it attacks the queen on a3
  • White king is still stuck in the centre and being at least two tempi away from castling, not even sidestepping from the open file with Kf1 is possible.
  • With the bishop recapturing on f8, Bb4 becomes a constant threat, which means Nxc3 is unstoppable for white as Qxc3 falls to Bb4 winning the queen in all ensuing lines.
  • Although white is an exchange up temporarily, one of their rooks and the f1 bishop are completely out of play still and therefore it is in fact black who effectively appears to be material up (with last piece being the a8 rook that comes into play with check in fact Re8+), not in the raw count of pieces but most definitely in terms of the positional compensation.
  • Most important consequence of all these observations: black is well in time to prevent white from ever consolidating, which means even without concrete assessment of the different lines we can safely bet that everything's about to fall apart for white quite shortly after 15...Bxf8 as black's initiative on the white king seems to lead to tactically and forcibly winning variations in every scenario.

these lines clearly demonstrate why the f8 rook is in fact not hanging and why white's best chances lie in Bc4 creating Luft on f1 for the king (or even short castles if black allows on a good day) side-stepping (at first glance) most of the tactical shenanigans we just witnessed after 13...Bxf8.

 [title "15. Bc4 only attempt at consolidating"]
 [fen "r4rk1/pp2Bpbp/1qp3p1/8/3Pn1b1/Q1P2N2/P4PPP/3RKB1R w K - 0 1"]

 1. Bc4 Nxc3 2. Bc5 Rfe8+ 3. Kf1 {and the rest is as you see in the real game, with Fischer's devastating attack despite the tucked away king on f1, which is just too little too late.}
1
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Short answer: Since after the bishop recpature on f8 (and not the rook recapture!) white is tactically and positionally completely busted, with 5 active black pieces against a completely exposed king in the centre and no foreseeable chance of consolidation in order to eventually benefit from the material advantage.

First observations:

  • 15...Bxf8 is with tempo as it attacks the queen on a3
  • White king is still stuck in the centre and being at least two tempi away from castling, not even sidestepping from the open file with Kf1 is possible.
  • With the bishop recaptuing on f8, Bb4 becomes a constant threat, which means Nxc3 is unstoppable for white as Qxc3 falls to Bb4 winning the queen in all ensuing lines.
  • Although white is an exchange up temporarily, one of their rooks and the f1 bishop are completely out of play still and therefore it is in fact black who effectively appears to be material up (with last piece being the a8 rook that comes into play with check in fact Re8+), not in the raw count of pieces but most definitely in terms of the positional compensation.
  • Most important consequence of all these observations: black is well in time to prevent white from ever consolidating, which means even without concrete assessment of the different lines we can safely bet that everything's about to fall apart for white quite shortly after 15...Bxf8 as black's initiative on the white king seem to lead to tactically and forcibly winning variations in every scenario.

Brief concrete assessment:

Now let's look at a few candidate lines concretely (by no means exhaustive, please make sure to check these and the connected sidelines with an engine on your own)

 [title "after 15. Bxf8 Bxf8"]
 [fen "r4bk1/pp3p1p/1qp3p1/8/3Pn1b1/Q1P2N2/P4PPP/3RKB1R w K - 0 2"]

 1. Qb3 Nxc3 2. Bc4 (2. Ra1 Re8+ 3. Kd2 (3. Ne5 Rxe5+ 4. dxe5 Bb4 {and the mating net is set, white's completely defenseless.}) Ne4+ {and black's at least winning the f3 knight after for instance Kc2 and Qa5}) (2. Qxb6 axb6 3. Ra1 Bb4 4. Bc4 Ne4+ 5. Ke2 Re8 {only way to avoid the discovery is to play Kd3 but after Nxf2 Nxh1 we're back to a quite similarly lost endgame compared to the 2...Nxd1 Qxb6 line}) Nxd1 {and black is up a clear piece as white cannot in any way recapture the knight} 3. Qxb6 (3. Qxd1  Qb4+) (3. Kxd1 Qxd4+ 4. Qd3 Qa1+ {and the h1 rook is lost, notice the Bxf7 trick doesn't lead to anything for white.}) axb6 4. Kxd1 Ra4 5. Bb3 Rxd4+ 6. Ke2 Re4+ 7. Kd1 Bb4 {and black's clearly winning considering the bishop pair, the 2 pawns up and a passed one already, the exposed white king under constant threats and the out of play white rook.}

these lines clearly demonstrate why the f8 rook is in fact not hanging and why white's best chances lied in Bc4 creating a Luft on f1 for the king (or even short castles if black allows on a good day) side-stepping (at first glance) most of the tactical shenanigans we just witnessed after 13...Bxf8.

 [title "15. Bc4 only attempt at consolidating"]
 [fen "r4rk1/pp2Bpbp/1qp3p1/8/3Pn1b1/Q1P2N2/P4PPP/3RKB1R w K - 0 1"]

 1. Bc4 Nxc3 2. Bc5 Rfe8+ 3. Kf1 {and the rest is as you in the real game, with Fischer's devastating attack despite the tucked away king on f1, which is just too little too late.}