Why is it hard to fake proof of work? Does someone understand this? 


  "Because of all this work, when a Bitcoin client signs on to the network it can trust the block chain that was most difficult to produce (since this is evidently the one that was being worked on by the most miners). If there was a "fake" blockchain competing with the real ones (say, where someone pretends that they didn't actually give Sofia those 8.4 bitcoins and they still have them), the fraudster would have to do as much work as the whole rest of the network to make their block chain look as trustworthy. So essentially, the intense work that goes into finding blocks through hashing secures the network against fraud. There is also, of course, some nifty code that figures out how to choose between conflicting transactions; and what to do if two people find valid blocks at the same time."
For each new block to be fixed to the end of the blockchain, a mathematical puzzle must be solved. In simple terms, the block is hashed, along with a hash of the previous block and a 'nonce', to create a new hash. The nonce can be thought of as the 'solution' to the block. It is a value such that, when all the information (previous hash, transactions and nonce) are hashed together, the resulting hash is below a certain value. (You can see that all the hashes of blocks have a long string of leading zeroes.)

The aim is for a new block to be completed every ~10 minutes, so the network agrees on a 'hash threshold'. The more computers that are competing to find the nonce, the lower the threshold becomes. How many computers are competing is gauged by whether blocks are regularly found in less than, or greater than, ten minutes on average.

So if your Bitcoin client finds a long blockchain, where each nonce is found according to a low threshold, it can surmise that a lot of work has gone in to creating that blockchain.

Even if another blockchain of the same length existed, for it to have such 'rare' hashes a lot of computer time would need to be dedicated to its construction. The longer the (real) blockchain gets, the harder it is to construct one that is equally long and has equally low hashes for each block.