Hardest Mate in 2 puzzle

If you find the answer to this puzzle you will get 5 trophies

ke6 and then pawn promotes to queen? 

Yeah I think

I have solved this puzzle before

There is two scenarios. 

1st is black can castle. In that case, the black's last move must be e5. So we would play dxe6. If they castle, then b7is mate. Otherwise g8 is also mate

2nd scenario is black can't castle. In that case, simple Ke6 and g8 would win

i didnt knew that white can castle?

I showed

ok @Fayez58

I will give you 5 trophies 

I got it

Move your king to e6. After the black's move your pawn to g8

This! If black can castle then by deduction the last move by black must be a pawn move, and it can't be e6-e5 since it would be putting white in check, so the only logical conclusion is e7-e5, making en passant legal.

It's easy! Ke6 and g8=Q (:

Variant with castling is amazing!

The answer depends on the logical model applied to the diagram - not on the rules of chess. Models are required when information is missing which is available in an actual chess game. So we are only talking composition chess here - not chess games.

Composition chess offers additional conventions to resolve the information missing from the diagram - i.c. blacks right to castle and white's right to e.p.. Skipping the analysis, the current convention dictates that you split this assignment in 2 parts (histories):

1. Black has castling right and white has e.p. right. White starts with 1. dxe6 e.p.!
2. White has no right to play e.p. and black has no right to castle. White starts with 1. Ke6!

So the complete solution contains both parts. Note there is also a third possible history where white has the right to play e.p. and black has no right to castle which would support both solutions but that history is discarded by the conventions.

Ke6 and pawn promotes to queen

did you give me?

oh sorry