the idea of mining for bitcoins with puzzles lost me. https://www.bing.com/search?q=mine+bitcoin+with+puzzles&FORM=AWRE
Mining is a euphemism for getting a reward for securing the Bitcoin system through "proof of work." The network must demonstrate the a certain amount of computational work was performed in order to be able to write the next group of transactions into the ledger.
If someone wanted to go back and change a transaction, they would have to redo the computational work. Since the network in whole has much more computational power than any single user, its not feasible to go back and rewrite the ledger.
As for "solving puzzles" to describe the computational work, this is a bit misleading. But all that matters is that the work can be performed in parallel by the entire network and that the amount of work performed can be easily adjusted.