How do two friends build the same secret while a spy listens?

A spy hears every single thing Ana and Bo say across the line. Yet the two of them can still end up holding the exact same secret number — one the spy can't make. How? With a trick that's easy to do one way and almost impossible to undo.

1Meet the number clock

Hopping forward is easy. Going backward is a nightmare.

Everything happens on a clock with 17 spots. You start on a start spot and take a secret number of hops — each hop jumps you to a new spot using the same rule. Two things to feel first:

Easy forward ➜

Hopping is a snap. Tell me your secret number of hops and I'll land on a spot in a blink. The more hops, the further you skip around the clock.

Stuck backward ✋

Reading it back is brutal. Show me only the landing spot and ask how many hops it took — I'm stuck. There's no shortcut. I'd have to try every number.

2The two directions

A one-way street made of math

Forward = make your spot

The "count the hops" way

You pick a secret number of hops and follow them around the clock. You land on one spot. That landing spot is fine to shout out loud — it's your public spot. Doing this is instant.

Backward = guess the secret

The "which hop count?" way

Now you know only someone's landing spot and you want their secret hop count. There's no clever formula — every spot looks equally likely. The only way is to test hop counts one by one until one matches. That's the spy's whole problem.

3Run the handshake

Pick two secrets and watch both friends land together

The clock and the start spot are public — the spy sees them too. Ana and Bo each pick a secret hop count. They shout their landing spots across the line, then each one hops again from the spot they were handed. Slide the secrets and watch where they end up.

Ana's secret hops: 6
115
Bo's secret hops: 15
115
Shouted across the line (the spy hears these): Ana's spot A  Bo's spot B

Ana does

Hops 6 from start 3 → lands on 15

Then hops 6 from Bo's spot → ends on 8

Bo does

Hops 15 from start 3 → lands on 6

Then hops 15 from Ana's spot → ends on 8

Both friends now hold the same secret: 8
Try other secret hop counts — they always meet on one number.

4Now you be the spy

You heard everything. Can you make their secret? 🕵️

You're the spy in the middle. You wrote down the clock (17 spots), the start spot (3), and both shouted spots: Ana's A and Bo's B. You never heard either secret hop count.

Guess before you try it

You copied everything that crossed the line: the clock, the start spot, and both landing spots. Is that enough to figure out their shared secret number?