In our most recent Insights puzzle, I challenged readers to figure out how certain magic tricks work. The tricks in question were of the sort where the magician somehow divines the identity of your hidden numbers or playing cards. But how does the magician correctly deduce what’s in your mind based on seemingly no information? As we’ll see below, the secret is to extract just enough information to unravel the mystery through the inexorable logic of mathematics.
Puzzle 1
Our first puzzle was an elementary arithmetic trick suitable for captivating a child:
Ask the child to think of a three-digit number without telling you what it is. Then tell them that you will reveal the number by producing two copies of it side by side! First ask the child to multiply their number by 7. Then ask them to multiply the answer by 11. Finally — and at this point you can add feigned concentration and appropriate magical phrases — ask them to multiply the second answer by 13.
If the child has done this right, you might find a smile start to light up their face. The question for you, or for an older child, is — why does this work?
The reason, as many readers correctly pointed out, is that 7 × 11 × 13 = 1,001, which is the number that you are effectively asking the child to multiply by. That this will produce a copy of the original three-digit number (say, 457) is obvious when you write down the multiplication in the conventional way.