The double-spending problem: why digital money is so hard to make

You cannot hand the same banknote to two people. With money made of numbers, you can try. Who stops you when there is no bank?

1
Pay with a banknote, and it leaves your hand. You cannot give the same note to anyone else. The paper itself stops you.
2
Digital things are different. A photo can be copied a thousand times, and every copy is perfect. Money made of numbers could be copied the same way.
3
Miva sells you a lamp, also for 0.01 bitcoin. But you have only one 0.01. What if you sign two payments with it, one to Rono and one to Miva, and send both in the same second?
4
Both signatures are real, because the key is yours. So no node can refuse either payment for its signature. Spending the same money twice like this is called double spending.
5
One node alone can solve it: keep the payment that came first, and refuse the second. A bank does exactly this with its one list.
6
But the nodes are all over the world, and a message takes time to travel. The nodes near Rono hear his payment first. The nodes near Miva hear hers first. Now their ledgers do not agree.
7
This puzzle blocked digital money for years. Every earlier system needed one company in the middle to keep the list, like a bank. Then, in 2008, a paper signed Satoshi Nakamoto said it could work with nobody in the middle.
8
And there is no bank to decide. Thousands of nodes, no boss, and each one heard a different payment first. How can they ever agree on one ledger?

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