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Question about an analysis from a beginner.

I played a game, and after the match i watch the analyzation.
The match can be found here: hu.lichess.org/study/eV0jZvXm

I have two question, i guess they are dumb, but i am a beginner. So first of all, after the second step (d5) it said i should have hit the pawn (xd5). Why is it better? I guess he will just kill it with the queen (xd5), and i will lose place in middle.

However the move i absolutely dont understand is at turn 6, where i move my bishop to G2. The analytic sad the correct move would be H3. I have been thinking about this for 20 minutes, and i cant figure out why that is good. He will just simply kill it with his bishop, and i dont gain anything. Why is that a better move?

Thanks!
You're playing 1. e4 to have a control over d5 square so that when black plays d5 you can take it. You're right that black could take on d5 with the queen, but you can just kick it out with Nc3. That will force him to move the queen again and lose time.

The reason why computer prefers 6. h6 over Bg2 i'd say is because Bg2 doesn't address the pin of the knight that protects e5 and therefore loses material. After h3 black can either take, or play Bh5 (or Bf5 and abandon the pin). If black takes, he gives up 2 bishops and the pin (and e5 isn't hanging there because of b7). If white retreats to h5 it gives you an option to break the pin with g4 once black decides to take on e5.
@thalrador #1

2.e4e5 advances the white e pawn, but in some lines that pawn might get weak, and the advance of the e5 does not do much for white. In some lines black can get an advantageous version of French defense lines.
2.e4xd5 is the recommended move because it clears matters in the center, and challenges black to recapture the pawn, right away or later.

5.g2g3 was the introduction of the loss of a pawn because of the pinned Nf3, and the relative weakness of pawn e5.
5.h2h3 would have questioned the bishop on g4 : Either retreat or capture on f3.

Towards the end of the game you got lucky because your opponent blundered. You won on the light squares, which was possible by a blunder by black b5?? and the absence of the black light squared bishop.

As a reminder for the readers of this forum thread, here is also the game with analysis, apart from the study : en.lichess.org/TzmGhic9#3

Thank you. As i turned on the local analysis it turns out The f1h3 suggested by the computer is a very bad move. It local analysis offers h1h2 as the best solution.

The answer for my first question is very clear, thank you!
After 1. e4 d5 2. e5 c5 black has good control over the centre and the e5 pawn become weak eventually. Better then exd5 Qxd5 and Nc3 develops and wins a tempo driving the Q away.
The loss of the e5-pawn in some way demonstrates the weakness od the move 1. - e5 although white could've defended it more tough.
6. h3 Bxf3 Qxf3 Nxe5 Qxb7 wins back the pawn. 6. Bg2 just loses a pawn for nothing.

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