What is the "double-spending problem," and why is it hard without a central authority?

A digital coin is just data, so its holder could copy and spend it twice — with no central ledger checking every payment, nothing inherently stops that.

Physical cash can't be double-spent: handing it over means you no longer hold it. A digital coin is a number that can be duplicated, so the same coin could be signed over to two different people. Banks solve this by being a central ledger that sees every transaction and decides which came first.

A peer-to-peer system has no such authority, so it needs another way for everyone to agree a coin was already spent. The key insight: the earliest spend is the valid one, and the only way to confirm a coin wasn't already spent is to be aware of all transactions — which is why they must be publicly announced and put in an agreed order.

Go deeper:

• Double-spending — Wikipedia — why duplicable digital money is hard to secure, and how proof-of-work addresses it.