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Why does Black take the knight on d5 on his 9th move instead of saving the Queen by moving to a4? Is it because then on the next move white plays Nxc7+ followed by NxR? board

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Because that was the move that the computer saw lost the least material. The BEST you can do there is just get two minor pieces for the queen. It gets worse after Qa4. Even just Nxc7 wins so much material that it is easy, but there is even better.

 [FEN ""]

 1. e4 d5 2. exd5 Qxd5 3. Nc3 Qa5 4. Bc4 Nf6 5. d3 Bg4 6. f3 Bf5 7. Bd2 e6 8. Nge2 Nc6 9. Nd5 Qa4 10. Bb5! (10.Nxc7) {And the queen is still trapped, or the Ra8 is lost...at least.} Qh4+ (10... Qxb5 11. Nxc7+ Kd7 12. Nxb5) 11. g3 Qh5 12. Nef4 (12. Nxc7+ Kd7 13. Nf4 Qh6 14. Nxa8) 12... Qh6 13. Nxe6 {And it is just over.}
  • What'd I miss about 10. Nec3? Was I just too focused on the queen? (10. ... Qa5 11. Nb5 Qa4 12. b3) – Arlen Feb 14 at 19:08
  • @Arlen, 10.Nec3 black already has to give the Q for two minors with 10...exc5 11.Nxa4 dxc4, which is just another variant of what I said here" The BEST you can do there is just get two minor pieces for the queen. It gets worse after Qa4." – PhishMaster Feb 14 at 21:06

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