How to prove you've solved something without giving away the solution

A brief introduction to zero-knowledge proofs

Imagine you’ve just finished solving every puzzle in a Sudoku book. Dozens of pages, all filled in correctly – or so you claim.

You tell a friend but they’re skeptical. “Prove it,’ they say.

The obvious way to prove you’re telling the truth is to simply show the completed puzzles. But what if your friend has a copy of the same book and doesn't want to know the actual solutions? How can you convince your friend that you’ve solved the book without revealing any of the actual answers?

This is the idea behind a zero-knowledge proof: you want to demonstrate that you know something, without giving away what that something is.

In the case of Sudoku, one way to achieve this would be to set up a blank 9×9 Sudoku grid, and place small counters over the empty cells. On the underside of each counter, you write the number from your solution. The top is blank, with nothing visible.

Your friend picks a row and asks to verify it. You gather the counters from that row (still face-down), put them in a bag, and hand it over. They look underneath each one, privately.

They find the numbers 1 through 9. No repeats, nothing missing.

That doesn’t tell them the exact layout of the row – just that it’s valid. And if you repeat this across many puzzles, with different rows or columns each time, your friend can build confidence that you’ve really solved them – without learning any of the actual solutions.

One check isn’t conclusive, but repeated enough times, the chance that you’re bluffing becomes tiny.

Outside the world of puzzles, zero-knowledge proofs are used to protect sensitive information online. For example, you might want to prove that:

• You’re over 18, without showing your birthdate.

• You have a valid ID, without revealing your name.

• You own enough cryptocurrency to make a payment, without exposing your entire wallet history.

Traditionally, proving these things involves giving away more than necessary. Zero-knowledge systems let you avoid that tradeoff.

It’s a simple but powerful shift in how we think about trust and verification. By harnessing probability, you can provide someone with very high confidence that your claim is true, while keeping the actual answer private.

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If you're interested in emerging work on zero-knowledge proofs and other forms of probabilistic proofs, I expect you'll like my new book Proof: The Uncertain Science of Certainty.