really dumb question

[lorax2013]

Sorry total noob question.  I'm told the best way to safely store bitcoins is in a paper wallet.  I'm told to go to www.bitaddress.org and, for greatest safety, download and run their "wallet generating software" while offline.  So this will give me a wallet with a public and a private key - and supposedly nobody else can know the private key because the wallet was generated offline and it is only printed on paper. 

Then I'm told it is safe to start transferring bitcoins to the wallet by giving out the public key (bitcoin wallet address).  I'm told that at some future point, when I want to transfer coins out of this wallet, I can go to www.blockchain.info for example and then I'll have to enter the private key to access that wallet and get the coins out.

My dumb question is:  If nobody knows the private key associated with this wallet and it was never sent out to the cloud, how does the network know if the private key is correct once I finally enter it to access my bitcoins?   Or to summarize colloquially, "How do it know?"   

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[Shallow]

Think of it like the missing piece of a puzzle. The client can decide if what you're giving it fits with the data it already has.

[coqui33]

A mathematical operation can generate a unique address from a given private key, but no no operation can go the other way. So when you try to use a privkey to spend coins that were sent to an address, the software gens a trial address from the privkey that you submitted. If the trial address matches the one where the coins were stored, you're good. If not, you're not.