The position:
Survival Puzzle
KG3 is not possible due to D8H4 after KH2
*Its entirely possible there was an oversight in the design and surviving 6 rounds is impossible*

White had the good sense to resign anyway, after allowing black that pretty picture. It is not a composed puzzle but the final position of La Bourdonnais - McDonnell, 62nd match game, 1834.

1. h3 exd1=Q+ 2. Kh2 Qg1+ 3. Kg3 f1=Q 4. Qf3 I think I will hold on. You might not like 2...Qg1+, but then come up with something better. Hopefully without using an engine.
I came across this "puzzle" in the game Watch Dogs. One of the mini-games that the player can participate in is playing chess. You are able to play full games as well as a series of "puzzle" games. In this case its to try and survive a certain number of rounds. For this set up if the player is able to survive 4 rounds he gets 1 star, 5 gets 2 stars, 6 gets the full 3 stars. So far I have not been able to get 3 stars. I used "next chess move" and at best i can go 5 rounds.
This video at the 1050 mark http://www.youtube.com/watch?v=UnCYyCkTo9s has been the only one to get 3 stars. However, this player has been the only one who has gotten the AI to behave that way. Everyone else has been getting the Bishop to move to H4 after KH2.
This is the most common 5 round "solution"
1.h4 exd1=Q+ 2. Kh2 f1=Q 3.Qd4 Bc7+ 4. Qd6 Bxd6+ 5. Kh3 Qh1#
Some new info: The engine being used is Chiron Chess Engine.

Okay, so 2...Bh4 is the key. I caved and asked the engine. Indeed it's a mate in 5 from the starting position, so the OP's puzzle is impossible to solve.
The idea is to survive for 6 rounds as white.
3b2rk/3P2pp/8/p7/8/2Q5/PP1pppPP/3R3K w KQkq - 0 1
White to move. So far the best I've seen is 5 rounds.