Bishops standing on their toes

Question

Another silly idea:

[FEN "1B1B1BkB/B1B1B1B1/1B1B1B1B/B1B1B1B1/1B1B1B1B/B1B1B1B1/KB1B1B1B/BNB1B1B1 w - - 0 0"]

How many bishops must be removed such that the position is won? How long does the mate take then?

Obviously, this would best be solved by creating a tablebase...

(Note: I once gave an algorithm that 2 suffice, and 1 most probably won't, but I trust my algorithms that >< far. Also, I never computed how long it takes, but "of the order 100" is my best guess.)

Answer 1

At least three... for example

[FEN "1B1B1Bk1/B1B1B1B1/1B1B1B1B/B1B1B1B1/1B1B1B1B/B3B1B1/KB1B3B/BNB1B1B1 w - - 6 46"]

1. Nc3 Kf7 2. Nd1 Ke6 3. Nf2 Kd5 4. Kb3 Ke6?? 5. Kc4 Kd7 6. Kb5 Kc8 7. Ka6 Kd7 8. Kb7 Ke6 9. Kc6 Kf5 10. Bd4c3 Ke6 11. Be5d4 Kf5 12. Bh8 Ke6 13. Bf6g7 Kf5 14. Bge5 Ke6 15. Bgf6 Kf5 16. Kd5 Kg6 17. Ne4 Kf5 18. Ng5 Kg4 19. Ke4 Kh5 20. Kf3 Kg6 21. Ne6 Kf5 22. Ng7+ Kg6 23. Kg4 Kf7 24. Kf5 Kg8 25. Kg6 Kxh8 26. Bh6g5 Kg8 27. Bg1f2 Kh8 28. Nf5+ Kg8 29. Nh6#

But in this game black makes a fatal error on move 4... Ke6?? after which it is forced mate in 25. If black runs to a white corner, I am not sure :/. It will obviously be much harder to force black to take a bishop there (= make use of a black square so stale mate is avoided).

For this reason, I think that more than four need to be removed - but I didn't figure out yet how many.

Here is another version that is slightly more human and, I think, easier to follow:

[FEN "1B1B1Bk1/B1B1B1B1/1B1B1B1B/B1B1B1B1/1B1B1B1B/B3B1B1/KB1B3B/BNB1B1B1 w - - 6 46"]

1. Nc3 Kf7 2. Nd1 Ke6 3. Nf2 Kd5 4. Kb3 Ke6?? 5. Kc4 Kd7 6. Kb5 Kc8 7. Ka6 Kd7 8. Kb7 Ke6 9. Kc6 Kf5 10. Kd5 Kg6 11. Ke6 Kh5 12. Bd4c3 Kg6 13. Bh8 Kh5 14. Bh6g7 Kg6 15. Ng4 Kh5 16. Nh6 Kg6 17. Be5d4 Kh5 18. Kd5 Kg6 19. Ke4 Kh5 20. Kf3 Kg6 21. Kg4 Kh7 22. Bf6e5 Kg6 23. Bg7f6 Kh7 24. Nf5 Kg6 25. Ng7 Kf7 26. Kf5 Kg8 27. Kg6 Kxh8 28. Nf5+ Kg8 29. Nh6#

EDIT

But then again - here is a version where black runs to a white corner...

[Variant "From Position"]
[FEN "1B1B1Bk1/B1B1B1B1/1B1B1B1B/B1B1B1B1/1B1B1B1B/B3B1B1/KB1B3B/BNB1B1B1 w - - 0 1"]

1. Kb3 Kf7 2. Kc4 Kg6 3. Nc3 Kf5 4. Nd1 Ke4 5. Nf2+ Kf3 6. Kd3 Kg2 7. Bd4c3 Kf3 8. Be5d4 Kg2 9. Bf4e5 Kf1 10. Bg3f4 Kg2 11. B2g3 Kf1 12. Bh8 Kg2 13. Ke2 Kxg1 14. Bh2+ Kg2 15. Bg1 Kxg1 16. Bh2+ Kg2 17. Bg1 Kxg1 18. Nh3+ Kg2 19. Nf4+ Kh2 20. Bh8g7 Kh1 21. Kf1 Kh2 22. Nd3+ Kh3 23. Nf2#

Final edit:

It is enough to remove three bishops. And then it is mate in 28, but you have to begin with the king, not the knight:

Just one line for mate in 28, and a few variations that lead to mate in 27:

[Variant "From Position"]
[FEN "1B1B1Bk1/B1B1B1B1/1B1B1B1B/B1B1B1B1/1B1B1B1B/B3B1B1/KB1B3B/BNB1B1B1 w - - 0 1"]

1. Kb3 Kf7 2. Nc3 Ke6 3. Kc4 Kd7 4. Kb5 Kc8 5. Ka6 Kd7 6. Ne2 Kc6 7. Bd2c3 Kd5 8. Kb5 Ke4 9. Kc4 Kf3 (9... Kf5 10. Be1d2 Ke4 11. Bge1 Kf5 (11... Kf3 12. Kd3 Kg4 13. Ke4 Kh5 14. Ng3+ Kg4 15. Bgf2 Kh3 16. Kf3 Kxh2 17. Bg1+ Kh3 18. Bh2 Kxh2 19. Kg4 Kg2 20. B3f2 Kh2 21. Nf1+ Kg2 22. Ne3+ Kh1 23. Bg1 Kxg1 24. Bh8 Kh1 25. Kh3 Kg1 26. Nf5+ Kf1 27. Ng3#) 12. Ng3+ Ke6 13. Kb5 Kd7 14. Bh8 Kc8 15. Ka6 Kd7 16. Kb7 Ke8 17. Kc6 Kf7 18. Nh5 Ke6 19. Ng7+ Kf7 20. Kd7 Kg8 21. Ke8 Kxh8 22. Kf7 Kh7 23. Bhf2 Kh8 24. Bgh4 Kh7 25. Bh2g3 Kh8 26. Ne6+ Kh7 27. Ng5#) 10. Kd3 Kg4 (10... Kg2 11. Be1d2 Kh1 12. Bge1 Kg2 13. Bgf2 Kf1 14. Ng3+ Kg2 15. Ke2 Kh3 16. Kf3 Kxh2 17. Bg1+ Kh3 18. Bh2 Kxh2 19. Kg4 Kg2 20. B3f2 Kh2 21. Nf1+ Kg2 22. Ne3+ Kh1 23. Bg1 Kxg1 24. Bh8 Kh1 25. Kh3 Kg1 26. Nf5+ Kf1 27. Ng3#) 11. Ke4 Kh3 12. Be1d2 Kg2 13. Bge1 Kf1 14. Ng3+ Kg2 15. Bgf2 Kxh2 16. Bg1+ Kh3 17. Kf5 Kg2 18. Bh8 Kh3 19. Bh2 Kg2 20. Kg4 Kxh2 21. Bg1+ Kg2 22. Bh2 Kxh2 23. Nf1+ Kg2 24. Ne3+ Kg1 25. Bh8g7 Kh1 26. Kh3 Kg1 27. Nf5+ Kf1 28. Ng3#