How do they find this number? By guessing at random. The hash function makes it impossible to predict what the output will be. So, miners guess the mystery number and apply the hash function to the combination of that guessed number and the data in the block. The resulting hash has to start with a pre-established number of zeroes. There's no way of knowing which number will work, because two consecutive integers will give wildly varying results. What's more, there may be several nonces that produce the desired result, or there may be none (in which case the miners keep trying, but with a different block configuration).
Majority consensus in bitcoin is represented by the longest chain, which required the greatest amount of effort to produce. If a majority of computing power is controlled by honest nodes, the honest chain will grow fastest and outpace any competing chains. To modify a past block, an attacker would have to redo the proof-of-work of that block and all blocks after it and then surpass the work of the honest nodes. The probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added.
No. 1: Paper wallet or other cold storage. A paper wallet is simply a document that contains all the information you need to generate the bitcoin private keys you need. It often takes the form of a piece of paper with a QR code that can be scanned into a software wallet when you so desire. By storing your bitcoin offline, trusting nothing and no one but yourself, and if you have all the information you need to control and access your bitcoin, you're using the strongest "cold storage" method out there.
In January 2009, the bitcoin network was created when Nakamoto mined the first block of the chain, known as the genesis block. Embedded in the coinbase of this block was the following text: "The Times 03/Jan/2009 Chancellor on brink of second bailout for banks." This note has been interpreted as both a timestamp and a comment on the instability caused by fractional-reserve banking.:18