Mind Reading with Math

by Spencer Bowen

Here’s a great activity to impress students, friends, and family and show off what you can do with a little bit of creativity and some clever mathematics. This activity as presented was developed at the San Francisco Math Circle.

To start this activity you need four cards with numbers written out as so:

For the Math Mind Reading Trick, you’ll need a volunteer who’s willing to have their mind read. The person performing the trick holds out the four cards and askes their volunteer to pick a number (whole numbers only, no fractions allowed!) between 1 and 15 and keep it a secret. Next, the mind-reader asks the volunteer if their number is on the cards one-by-one. The volunteer answers the questions with yes or no answers, and with some magic and a little math, the mind-reader figures out their number!

Javier Haro performing the trick as an intro to a Math Monday sesson.

How it works

To get a better understanding of the trick, let’s look at an example. Say the volunteer decided on 13. Then their answers would be as follows:

Each number only appears on exactly one combination of cards, so “Yes, yes, no, yes” is the only possible set of answers to 13. If you’re performing this trick for students, they often immediately assume that the mind-reader memorized every card and is just mentally sorting through the numbers. Technically, this can be used to perform the trick, but the reality is a lot cooler (and easier).

It’s here that you might want to let students look at the cards and make observations. Looking carefully at the cards 13 appears on, the top numbers on the cards are 8, 4, and 1, and it just so happens that 8 + 4 + 1 = 13. Let’s see if that observation holds for another number: 7. It’s not on the first card but appears on the last three cards. And looking at the top number on these three cards, we see in fact 4 + 2 + 1 = 7!

This observation does work for any number 1 to 15, so to read someone’s mind, you just need to take the cards they answered “yes” to and add the number at the top of each card. But where does this pattern come from? Well, let’s look at the numbers at the top of the cards: 8, 4, 2, and 1. You may notice that these are powers of 2. Looking back at our example 13, we had:

It turns out this trick is entirely built around binary! Each card represents a place value in base 2. Whether or not a number then appears on the card determines if it has a 1 or 0 in that place value. This trick can be extended indefinitely! With enough cards, you can read minds for numbers up to any positive value. And, if you want to find more on this very topic, Dr. James Tanton and the Global Math Project use a version of this activity in their math phenomenon Exploding Dots. Here’s James’ lesson plan for the five card variant.

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