How many Different Mates on Move 3 for White are there?

There was an interesting post on reddit.com called “A counting exercise” that asked the question:

How many different ending positions exist where white delivers a checkmate on his third move? To get you started: there are 4 different ending positions where black delivers a checkmate on his second move. Can’t be too hard, can it?

I took a stab at answering this question.  I missed a couple of variations on my first attempt.

It is pretty fun, and you might want to try it yourself, before looking at my answer below.

Below is my updated count that I came up with, and each variation.  Note that the final position is what has to be different, not the move order.

These were my original assumptions in my mind for each one:

  • white has to move their e-pawn (e3 or e4)
  • black has to move their f pawn (f6 or f5)
  • black has to play g5
  • white has to mate on h5 with the queen or bishop on the third move. 
  • White has to kill one move to give black time to get their two moves out, which accounts for the differences in each variation.
  • White’s time wasting move cannot be Nf3, f3 or g4, as this would block the queen check on h5

After working on this for a while, and multiple additions, corrections and feedback, my assumptions were proven almost all wrong!

  • white has to move their e-pawn (e3 or e4)
  • black has to move their f pawn (f6 or f5)
  • black has to play g5
  • white has to mate on h5 with the queen or bishop on the third move. 
  • White has to kill one move to give black time to get their two moves out, which accounts for the differences in each variation.
  • White’s time wasting move cannot be Nf3, f3 or g4, as this would block the queen check on h5

Here are the answers I came up with:

Variations where white plays 1. e3, black plays f5, Queen Mates

Mate #1

  1. 1. e3 f5 2. a3 g5 3. Qh5#
  2. 1. e3 f5 2. a4 g5 3. Qh5# 
  3. 1. e3 f5 2. b3 g5 3. Qh5# 
  4. 1. e3 f5 2. b4 g5 3. Qh5# 
  5. 1. e3 f5 2. c3 g5 3. Qh5# 
  6. 1. e3 f5 2. c4 g5 3. Qh5# 
  7. 1. e3 f5 2. d3 g5 3. Qh5# 
  8. 1. e3 f5 2. d4 g5 3. Qh5# 
  9. 1. e3 f5 2. e4 g5 3. Qh5# 
  10. 1. e3 f5 2. f4 g5 3. Qh5# 
  11. 1. e3 f5 2. g3 g5 3. Qh5# 
  12. 1. e3 f5 2. h3 g5 3. Qh5# 
  13. 1. e3 f5 2. h4 g5 3. Qh5# 
  14. 1. e3 f5 2. Nh3 g5 3. Qh5# 
  15. 1. e3 f5 2. Na3 g5 3. Qh5# 
  16. 1. e3 f5 2. Nc3 g5 3. Qh5# 
  17. 1. e3 f5 2. Bd3  g5 3. Qh5#
  18. 1. e3 f5 2. Bc4 g5 3. Qh5#
  19. 1. e3 f5 2. Bb5 g5 3. Qh5#
  20. 1. e3 f5 2. Ba6 g5 3. Qh5#

Variations where white plays 1. e4, black plays f5, Queen Mates.  This one has an extra move, because of the possibility of playing exf5, that doesn’t exist when white plays 1. e3

  1. Mate #30 with exf5

    1. e4 f5 2. a3 g5 3. Qh5#
  2. 1. e4 f5 2. a4 g5 3. Qh5#
  3. 1. e4 f5 2. b3 g5 3. Qh5#
  4. 1. e4 f5 2. b4 g5 3. Qh5#
  5. 1. e4 f5 2. c3 g5 3. Qh5#
  6. 1. e4 f5 2. c4 g5 3. Qh5#
  7. 1. e4 f5 2. d3 g5 3. Qh5#
  8. 1. e4 f5 2. d4 g5 3. Qh5#
  9. 1. e4 f5 2. e5 g5 3. Qh5#
  10. 1. e4 f5 2. exf5 g5 3. Qh5#
  11. 1. e4 f5 2. f4 g5 3. Qh5#
  12. 1. e4 f5 2. g3 g5 3. Qh5#
  13. 1. e4 f5 2. h3 g5 3. Qh5#
  14. 1. e4 f5 2. h4 g5 3. Qh5#
  15. 1. e4 f5 2. Nh3 g5 3. Qh5#
  16. 1. e4 f5 2. Na3 g5 3. Qh5#
  17. 1. e4 f5 2. Nc3 g5 3. Qh5#
  18. 1. e4 f5 2. Bd3 g5 3. Qh5#
  19. 1. e4 f5 2. Bc4 g5 3. Qh5#
  20. 1. e4 f5 2. Bb5 g5 3. Qh5#
  21. 1. e4 f5 2. Ba6 g5 3. Qh5#

Variations where white plays 1. e3, black plays f6, Queen Mates

  1. 1. e3 f6 2. a3 g5 3. Qh5#
  2. 1. e3 f6 2. a4 g5 3. Qh5#
  3. 1. e3 f6 2. b3 g5 3. Qh5#
  4. 1. e3 f6 2. b4 g5 3. Qh5#
  5. 1. e3 f6 2. c3 g5 3. Qh5#
  6. 1. e3 f6 2. c4 g5 3. Qh5#
  7. 1. e3 f6 2. d3 g5 3. Qh5#
  8. 1. e3 f6 2. d4 g5 3. Qh5#
  9. 1. e3 f6 2. e4 g5 3. Qh5#
  10. 1. e3 f6 2. f4 g5 3. Qh5#
  11. 1. e3 f6 2. g3 g5 3. Qh5#
  12. 1. e3 f6 2. h3 g5 3. Qh5#
  13. 1. e3 f6 2. h4 g5 3. Qh5#
  14. 1. e3 f6 2. Nh3 g5 3. Qh5#
  15. 1. e3 f6 2. Na3 g5 3. Qh5#
  16. 1. e3 f6 2. Nc3 g5 3. Qh5#
  17. 1. e3 f6 2. Bd3 g5 3. Qh5#
  18. 1. e3 f6 2. Bc4 g5 3. Qh5#
  19. 1. e3 f6 2. Bb5 g5 3. Qh5#
  20. 1. e3 f6 2. Ba6 g5 3. Qh5#

Variations where white plays 1. e4, black plays f6, Queen Mates.

  1. 1. e4 f6 2. a3 g5 3. Qh5#
  2. 1. e4 f6 2. a4 g5 3. Qh5#
  3. 1. e4 f6 2. b3 g5 3. Qh5#
  4. 1. e4 f6 2. b4 g5 3. Qh5#
  5. 1. e4 f6 2. c3 g5 3. Qh5#
  6. 1. e4 f6 2. c4 g5 3. Qh5#
  7. 1. e4 f6 2. d3 g5 3. Qh5#
  8. 1. e4 f6 2. d4 g5 3. Qh5#
  9. 1. e4 f6 2. e5 g5 3. Qh5#
  10. 1. e4 f6 2. f4 g5 3. Qh5#
  11. 1. e4 f6 2. g3 g5 3. Qh5#
  12. 1. e4 f6 2. h3 g5 3. Qh5#
  13. 1. e4 f6 2. h4 g5 3. Qh5#
  14. 1. e4 f6 2. Nh3 g5 3. Qh5#
  15. 1. e4 f6 2. Na3 g5 3. Qh5#
  16. 1. e4 f6 2. Nc3 g5 3. Qh5#
  17. 1. e4 f6 2. Bd3 g5 3. Qh5#
  18. 1. e4 f6 2. Bc4 g5 3. Qh5#
  19. 1. e4 f6 2. Bb5 g5 3. Qh5#
  20. 1. e4 f6 2. Ba6 g5 3. Qh5#

Variations where white Bishop Mates

Mate #82 Bishop Mate

  1. e3 f5 2. Be2 g5 3. Bh5#
  2. e4 f5 2. Be2 g5 3. Bh5#
  3. e3 f6 2. Be2 g5 3. Bh5#
  4. e4 f6 2. Be2 g5 3. Bh5#

Variations where white Queen kills a tempo

  1. e3 f5 2. Qf3 g5 3. Qh5#
  2. e3 f6 2. Qf3 g5 3. Qh5#

Note that the Queen can kill a tempo with Qe2, Qf3 or Qg4, but these all result in the same end position.

When I originally attempted this, I missed the Queen kill a tempo moves, and the Bishop moves on move 2 like Bd3, Bc4, Bb5, Ba6, and the one where white can play exf5.

So I am coming up with a total of 89 different ending positions where white mates black on the third move :-)

Edit/Update:

OK I found 4 more after getting some feedback!!  Turns out my assumptions above were bad!  In the software world, these would be called the “corner cases:-)

Mate #88

  1. 1. d4 f6 2. Qd3 h6 3. Qg6#
  2. 1. d4 f6 2. Qd3 h5 3. Qg6#

So that brings my total now to 91 different positions!

 Third Update!  Bishop mates on g6.

  1. 1. e3 h6 2. Bd3 f6 3. Bg6#
  2. 1. e3 h5 2. Bd3 f6 3. Bg6#

So that brings my total now to 93 different positions!

Fourth Update! Forgot the Queen mates on g6 starting with an e pawn move!

  1. Mate #99

    1. e3 h6 2. Qg4 f6 3. Qg6#
  2. 1. e4 h6 2. Qg4 f6 3. Qg6#
  3. 1. e3 h5 2. Qg4 f6 3. Qg6#
  4. 1. e4 h5 2. Qg4 f6 3. Qg6#
  5. 1. e3 h6 2. Qg4 f5 3. Qg6#
  6. 1. e4 h6 2. Qg4 f5 3. Qg6#
  7. 1. e4 h5 2. Qg4 f5 3. Qg6#
  8. 1. e3 h5 2. Qg4 f5 3. Qg6#

So that brings my total now to 101 different positions! lol! :-D


Mate #100 Savielly's Mate

Fifth Update! D’oh!

Savielly on RHP writes:

I think you missed the most obvious one

  1. 1.e4 e5 2.Qh5 Ke7 3.Qxe5#

Thanks Savielly!

6th update

yosiman on RHP writes:

Hmm how about that same variation but with e3 instead?

Also, there are a few variations where you move the c pawn first, move the queen to c2 and then g6 mate.

Mate #102 Yosiman's Mate

 

  1. 1.e3 e5 2.Qh5 Ke7 3.Qxe5#
  2. 1. c4 f6 2. Qc2 h5 3. Qg6#
  3. 1. c3 f6 2. Qc2 h5 3. Qg6#
  4. 1. c4 f6 2. Qc2 h6 3. Qg6#
  5. 1. c3 f6 2. Qc2 h6 3. Qg6#

Thanks Yosiman!

7th Update, some cleanup

Bronstein, the original poster of the question, pointed out I have some duplicates.

These two result in the same position.  In one the queen kills a move, and the one the e pawn moves twice.

9. 1. e3 f5 2. e4 g5 3. Qh5#
87. e4 f5 2. Qf3 g5 3. Qh5#

and same with these

50. 1. e3 f6 2. e4 g5 3. Qh5#
89. 1. e4 f6 2. Qf3 g5 3. Qh5#

So I will remove the original 87 and 89, which makes the final total 105.






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