Tomoyo
Fate
I don't understand my math homework at all, so I'll analyze your game instead.
1. e4 e5
2. Nf3 f6
This is a bad move, as shown by some guy named Damiano some number of years ago. Conceptually, it's bad because it blocks your g8 knight and exposes your kingside to attack. Tactically it's bad because it can lead to Nxe5, and if you take back, then 4. Qh5+ Ke7 (...g6 5. Qxe5+ loses the rook) 5. Qxe5+ Ke7 6. Bc4 and black is just kinda being owned.
3. Bc4
Lucky you. Your opponent didn't play Nxe5.
3. ... Nh6
As 2. ...f6 doesn't actually protect the e5 pawn at all, you probably should have played Nc6, d6, or even Qe2 to protect it. Also, developing a knight to the side of the board is usually bad.
4. d4
This threatens your e5 pawn and exchanging on h6. An exchange on h6 would be disastrous for you because after gxh6, an eventual Qh5+ would wreck havoc on your king's position. You should have played 4. ... g6 to avoid the doubled h-pawns, but your position is pretty bad at this point anyway.
4. ... Bb4+
This may appear to develop a bishop with tempo, but he can simply play c3 to attack your bishop, basically meaning that you did not address either of the two threats mentioned in the previous move.
5. c3 a5
Your bishop is worth more than his pawn. Retreat it.
6. cxb4 axb4
7. dxe5 fxe5
8. b3
I'm not sure why he doesn't just play 8. Bxh6 gxh6 9. Nxe5 followed by Qh5 which would certainly lead to a decisive attack.
8. ... c6
At this point you should have worked on your kingside defense, seeing how your f-pawn is missing and you have an undefended e-pawn. Best move probably would have been Qf6.
9. Nxe5 d5
10. Qf3 dxc4
11. Bxh6 gxh6
12. Qf7#
The mate was just the result of your open king. You probably could have prolonged it, but at this point, there was nothing you could do.
Basically, don't move the f-pawn if your opponent plays 1. e5. Not a hard and fast rule, but it's a fairly good one. Also, watch out for enemy threats and don't make hollow checks that have no purpose.
This is the last sentence of my post.
1. e4 e5
2. Nf3 f6
This is a bad move, as shown by some guy named Damiano some number of years ago. Conceptually, it's bad because it blocks your g8 knight and exposes your kingside to attack. Tactically it's bad because it can lead to Nxe5, and if you take back, then 4. Qh5+ Ke7 (...g6 5. Qxe5+ loses the rook) 5. Qxe5+ Ke7 6. Bc4 and black is just kinda being owned.
3. Bc4
Lucky you. Your opponent didn't play Nxe5.
3. ... Nh6
As 2. ...f6 doesn't actually protect the e5 pawn at all, you probably should have played Nc6, d6, or even Qe2 to protect it. Also, developing a knight to the side of the board is usually bad.
4. d4
This threatens your e5 pawn and exchanging on h6. An exchange on h6 would be disastrous for you because after gxh6, an eventual Qh5+ would wreck havoc on your king's position. You should have played 4. ... g6 to avoid the doubled h-pawns, but your position is pretty bad at this point anyway.
4. ... Bb4+
This may appear to develop a bishop with tempo, but he can simply play c3 to attack your bishop, basically meaning that you did not address either of the two threats mentioned in the previous move.
5. c3 a5
Your bishop is worth more than his pawn. Retreat it.
6. cxb4 axb4
7. dxe5 fxe5
8. b3
I'm not sure why he doesn't just play 8. Bxh6 gxh6 9. Nxe5 followed by Qh5 which would certainly lead to a decisive attack.
8. ... c6
At this point you should have worked on your kingside defense, seeing how your f-pawn is missing and you have an undefended e-pawn. Best move probably would have been Qf6.
9. Nxe5 d5
10. Qf3 dxc4
11. Bxh6 gxh6
12. Qf7#
The mate was just the result of your open king. You probably could have prolonged it, but at this point, there was nothing you could do.
Basically, don't move the f-pawn if your opponent plays 1. e5. Not a hard and fast rule, but it's a fairly good one. Also, watch out for enemy threats and don't make hollow checks that have no purpose.
This is the last sentence of my post.
).
) Bb2 
