What's the Maximum Number of Simultaneous Queens from This Position?

[@y_prime]

The maximum number of queens that can be created during the game is 15, I think; after the first capture, future confrontations can be handled by sacrificing a queen to break the stalemate. For example, this would work: 1. b4 a5 2. bxa5 b6 3. a6 b5 4. a7 d6 5. a8=Q+ Kd7 6. a4 Ke6 7. a5 Kd7 8. Qd5 Ke8 9. a6 Kd7 10. Qd4 Kd8 11. Qa4 bxa4 12. a7 a3 13. a8=Q+ Kd7 14. c4 a2 15. f3 a1=Q+ 16. Kf2 Qb2 17. Qd5 Qb5 18. cxb5 c5 19. b6 c4 20. b7 c3 21. b8=Q c2 22. Qc5 dxc5 23. d4 Ke6 24. d5+ Kf6 25. d6 c1=Q 26. d7 c4 27. d8=Q Qf4 28. Qdd6+ exd6 29. e4 d5 30. Qb6+ Ke7 31. e5 c3 32. e6 c2 33. g4 Kf8 34. e7+ Ke8 35. Qe6 fxe6 36. h3 Kd7 37. e8=Q+ Kd6 38. h4 c1=Q 39. Qd7+ Kxd7 40. g5 e5 41. Ke2 e4 42. Kf2 e3+ 43. Ke2 Qe4 44. fxe4 d4 45. Kf3 d3 46. e5 d2 47. e6+ Kd6 48. e7 d1=Q+ 49. Kg3 Qg4+ 50. Kxg4 Qf1 51. e8=Q Qf6 52. gxf6 e2 53. Qg6 hxg6 54. f7 Kd5 55. f8=Q g5 56. h5 g6 57. h6 Kd4 58. Kh3 g4+ 59. Kh4 g5+ 60. Kh5 g3 61. Qf1 exf1=Q 62. h7 g2 63. h8=Q+ Ke4 64. Qa1 g1=Q 65. Qxf1 g4 66. Qxg1 g3 67. Qf1 g2 68. Qd1 g1=Q

[@LetsSolveMathProblems]

Yes, that is the answer to the bonus question. =)

[@ayushdeep7900]

I really appreciate your effort

[@ayushdeep7900]

@@flypelican1555 thank you so much

[@zubin8010]

I love this kind of discussion and problem solving :)

[@particleonazock2246]

Do you play chess, my friend?

[@LetsSolveMathProblems]

As a quick challenge, feel free to comment the solution for the maximum possible number of queens that can be CREATED DURING THE COURSE OF THE GAME starting from this position. I will pin the first correct answer.

[@lucas29476]

LetsSolveMathProblems question is ambiguous; you didn't mention it's **this** game; thought you were asking about normal chess game

[@LetsSolveMathProblems]

@@lucas29476 That is a fair point, although I hoped that it was implied we are playing from the given position. I edited the comment accordingly.

[@haydenkarkainen1167]

Wow this is really cool to see a channel I’ve followed for so long do something chess related that I didn’t expect (Aside: Do you play on lichess and can we follow you? That’d be really cool). I remember an interesting theoretical question I posed on Quora was (paraphrasing): “What material is sufficient to mate a lone king on an unbounded (infinite) chess board where all pieces are within finite distance from each other”, and there was an interesting response showing how 2 opposite colored bishops and 2 connected pawns were sufficient in finite moves to mate the king from any non-trivial (no pieces hanging) starting position, and the process of finding these piece combinations that worked and their informal proofs was quite fun. These kinds of chess problems that require deep thinking about seemingly small ideas are quite beautiful, and I think it would be cool to see you tackle other chess/ board game questions (like the one I mentioned above maybe even) that have very deep and interesting solutions even when those solutions aren’t so immediately tied to concrete math/math theorems like the typical focus of the channel. Thanks for video! Good stuff ( :

[@erickherrerapena8981]

Buen video!

[@typha]

The argument doesn't quite work, I don't think (though I don't doubt the result). You can get situations where you have say, two white pawns and a black pawn in a file (you'd probably have to sacrifice a queen to get there, but the point is it is possible). Then in that file alone we have 2 confrontations, and we could still have another confrontation in a neighboring file. Then if a white pawn in that neighboring file takes the black pawn in the first file, the number of confrontations goes down by 3. Thus it's not entirely accurate to say that "the number of confrontations goes down by at most 2" per pawn taken.

[@LetsSolveMathProblems]

I believe the argument in the video still works. Although I arguably should have put more emphasis, at 2:49 I clarified that the number of confrontation PER FILE by definition is at most 1, even if we have double pawns in that file. Using this definition in the video, the number of confrontation can only decrease by 2 at the most. (Certainly, it can also increase, which we do not need to worry about.) Since each decrease in confrontation can only result from taking a pawn (or a piece), the argument in the video goes through as explained.

[@typha]

​@@LetsSolveMathProblems Oh, I see, that was the definition. Okay, I guess that works then. If I were going to explain it to someone else though, I would probably refer to your "confrontations" as "conflicted files" then, or something like that. then the number of conflicted files starts at 8, goes down by at most 2 every time a pawn is taken - one for each file effected by the capture move - and if the number of conflicted files is non-zero we can always squeak at least one more queen out of the situation. Thus at least 4 pawns must be taken, qed. Yes, that makes sense to me now, thanks for the clarification :)

[@alexting827]

Also I believe this position is 100% impossible since without moving pawns the rooks could not get out. Am I thinking too much about a hypothetical problem?

[@LetsSolveMathProblems]

Technically, this position is possible to achieve from the usual starting position by using the knights to take all minor/major pieces, although that is certainly not the main focus of the video.

[@chunkynorris100]

no its healthy to think

[@alexting827]

@@LetsSolveMathProblems LOL I didn't think about that XD. wait no, wait yes. TOO MUCH THINKING ABOUT KNIGHTS MOVES brain overload

[@cycklist]

Very cool indeed! Love chess.

[@hassanalihusseini1717]

How many queens are possible when starting with a full set?

[@tomyao7884]

This man is too smart for me

[@lucas29476]

Is it not 12? For each two files (let's say G and H), you can have black G pawn take white H pawn, then every pawn (other than the taken pawn) on those two files can promote. I don't think you can do better since there's no other piece for the pawns to take (well, you can have a pawn take a queen, but that defeats the whole purpose)

[@lucas29476]

oh whoops this is stated in the first half of video

[@lucas29476]

Oh, this is not a weekly maths puzzle?? HAHA

[@leonthethird7494]

Can you sacrifice queens to move the pawns ? Or is it maximum number of queens at once?

[@sohamchatterjee4288]

Are you same as BpRp?

[@michaelempeigne3519]

if picard's iteration method is applicable to a function, then is the existence and uniqueness theorem also satisfied automatically ?

[@pankajbhardwaj8807]

We missed your questions.

[@particleonazock2246]

Annyeonghasaeyo, @LetsSolveMathProblems 너는 어때, 너는 체스를하니?

[@hariomrockzi4399]

Rokzi

[@alexting827]

Hmm, I'm first, but I don't care, the problem is much more interesting

[@chunkynorris100]

oh ok