The Two Numbers You Are Really Carrying
You were handed a recipe with one knob on it: change the 8, and the answer becomes half of whatever you chose. Every other step stayed exactly where it was.
So here is the obvious next thing to do. Change one of the others.
Multiply by 3, add 12, divide by 3, take away your number. Try it on 7: 21, then 33, then 11, then take away 7, and you land on 4. Try 100 and you land on 4 again. A completely different recipe, forced onto the same answer.
Multiply by 3, add 12, divide by 2, take away your number. One character different from the last one. Try it on 4 and you get 8. Try it on 10 and you get 11. It is not a trick at all — the answer follows their number around, and they would notice at once.
Two recipes that look like each other, and only one of them works. You could test every idea you have on three different numbers and see which survive. It is quicker to know in advance.
What you are actually holding
At every moment during a trick, whoever is doing the arithmetic is holding two things at once, and it helps enormously to count them separately.
- How many secrets they are holding — how many copies of their own number.
- How many ones they are holding — how much plain, ordinary number.
At the start they hold 1 secret and 0 ones. That is what "pick a number and tell me nothing" means.
Now every kind of step does something plain to those two counts:
- Multiply by a number — both counts multiply.
- Divide by a number — both counts divide.
- Add a number — only the ones change. Adding 8 cannot give you more secrets; you have no idea what their secret is.
- Take away their number — the secrets drop by one, and the ones do not move.
That last pair is the whole reason a trick can work. Adding and subtracting plain numbers never touches the secret count, and taking away their number never touches the ones.
The original trick, counted
| Step | Secrets | Ones |
|---|---|---|
| Pick a number | 1 | 0 |
| Multiply by 2 | 2 | 0 |
| Add 8 | 2 | 8 |
| Divide by 2 | 1 | 4 |
| Take away your number | 0 | 4 |
Read the last line. No secrets left, and 4 ones. Their number is not in the answer any more — not hidden in it, not cancelled out at the end, simply not there — so the answer cannot possibly depend on what they picked. It is 4.
That gives you the rule, and it is a short one:
The answer is forced exactly when the secret count reaches zero. Whatever ones are left at that moment is the answer.
Now predict instead of testing
Multiply by 3, add 12, divide by 3, take away your number. 1 and 0 → 3 and 0 → 3 and 12 → 1 and 4 → 0 and 4. Zero secrets. Forced, and the answer is 4. You knew before you tried it.
Multiply by 3, add 12, divide by 2, take away your number. 1 and 0 → 3 and 0 → 3 and 12 → one and a half, and 6 → half a secret, and 6.
Half a secret is still a secret. Dividing 3 secrets by 2 does not clear them out, and taking one away leaves half of one behind — so the answer is half their number plus 6, and it moves whenever their number moves. The recipe was doomed at the divide, two steps before anyone would have noticed.
Sort six recipes without testing them
Run the two counts down each one. If the secrets reach zero, the answer is forced. If anything is left, the answer follows their number.
Six recipes. Only the secret count decides.
- Double it, add 8, halve it, take away your number
- Triple it, add 12, divide by 3, take away your number
- Double it, add 14, halve it, take away your number
- Triple it, add 12, halve it, take away your number
- Double it, add 6, halve it
- Add 5, double it, take away 10, halve it
That fifth one has no "take away your number" step anywhere, so nothing ever removes the secret. It cannot be forced, however tidy it looks.
You run the counts and end holding 1 secret and 6 ones. Can you announce their answer?
- Yes, it is always 6
- No — a secret is still in there, so the answer moves when their number moves
- Only if they picked an even number
The other kind of trick
There is a second family, and you have almost certainly seen one: the sort where the magician hands their number back to them.
Double it, add 10, halve it, take away 5. 1 and 0 → 2 and 0 → 2 and 10 → 1 and 5 → 1 secret and 0 ones.
One secret, no ones. That is not "an answer" at all — that is their number, on its own, with everything else stripped away. Which is why you can look them in the eye and say it.
So the same two counts run both families, aimed at different targets:
| Where you want to land | What you can say |
|---|---|
| 0 secrets, some ones | "Your answer is 4." |
| 1 secret, 0 ones | "Your number was 23." |
Anything else — half a secret, two secrets, one secret and some ones left over — is not a trick. It is just arithmetic, and your friend will see it coming.
Go and build one
Pick your target, then work backwards to a recipe that lands on it. Multiply by 5 and you will need to divide by 5 sooner or later, or take four of their numbers away one at a time. Add 30 before the halving and the ones will arrive as 15.
Run the counts first. If the secrets do not reach zero, no amount of showmanship will save it, and you will have found that out on paper instead of in front of somebody.