BrightKidz Library
Subjects
Three different starting numbers, one answer every time On the left three rows of counters, and each row holds a different amount: two, then five, then three. Arrows carry all three rows into a magician's top hat in the middle, brim and all. Out of the other side come three rows again, but now every single row holds exactly four counters.

Math as Magic: The Always-4 Trick

About 4 minutes

Here is a trick where you never learn your friend's number, never see their working, and still know their answer before they have finished. It works every single time, and once you know why, you can invent your own.

The Trick

Ask a friend to pick a number and keep it secret. Then tell them:

  1. Pick any whole number and do not tell me what it is.
  2. Multiply it by 2.
  3. Add 8.
  4. Divide by 2.
  5. Subtract the number you first thought of.
  6. Announce that their answer is 4. It always is.

Try it on paper with 7. Doubled is 14, plus 8 is 22, halved is 11, take away the 7 you started with and you have 4. Now try it with 60, or with 1,000. Still 4.

Prove It On Every Number At Once

Testing a few numbers is not proof — there are infinitely many you have not tried. Here is the trick written as a program. Press Run and it does the whole thing for four very different starting numbers.

The trick function does exactly the five steps above. Change the numbers at the bottom to anything you like and run it again.

function trick(secret) {
  var n = secret;
  n = n * 2;        // multiply by 2
  n = n + 8;        // add 8
  n = n / 2;        // divide by 2
  n = n - secret;   // subtract the number they started with
  return n;
}

console.log('They picked 7    ->', trick(7));
console.log('They picked 100  ->', trick(100));
console.log('They picked 1    ->', trick(1));
console.log('They picked 4213 ->', trick(4213));

Four different numbers, four identical answers. That is a strong hint — but it is still only four numbers.

Why It Actually Works

To cover every number at once, stop using numbers. Write x for your friend's secret, whatever it happens to be, and follow the steps:

Step What they have now
Pick a number x
Multiply by 2 2x
Add 8 2x + 8
Divide by 2 x + 4
Subtract the original 4

The important line is the fourth. Halving 2x + 8 halves both parts, giving x + 4. So at that moment they are holding their own number plus 4 — and the last step takes their number away again, leaving the 4 stranded on its own.

The 8 is the only thing that ever mattered. It got halved into a 4, and it was never attached to their number at all.

What would the answer always be if you told your friend to add 10 instead of 8?

Make Your Own

You now have the recipe: pick any even number to add, and the answer is always half of it. Add 20 and everyone gets 10. Add 6 and everyone gets 3.

Add an odd number and it still works, but your friends end up with a half in their answer — try 9 and see. Then perform it on someone and refuse to explain.