Math magic

Shoe magic

I saw this in my Facebook feed, and instead of finding it cute, I found it a frustrating and disheartening display of innumeracy. The comments were full of "Oh wow! It worked for me," and "That's crazy!"

No, it's not crazy, it's not magic, it's math. Of course it worked for you because it's just an algebra trick, and not a particularly exciting one. Much more interesting than the trick is how it works and what its limitations are.

"Shoe magic" explained

If, instead of step-by-step instructions, it were written out as an algebra expression

20(5x + 50) + 1013 – y

and the directions were to simplify the expression, a lot of the people wowed by the "magic" would run screaming from the math. It's the same thing, people.

So let's do the math and simplify the expression

20(5x + 50) + 1013 – y  
100x + 1000 + 1013 – y  
100x + 2013 – y

Let's group the simplified expression into two parts:

100x + (2013 – y)

Hopefully two things become pretty clear:

  • If x is your shoe size, then 100x is your shoe size moved to the hundreds place; e.g. if you're a 7, then it's 700.
  • If y is the year you were born, then (2013 – y) is your age.

So you'll get a three- or four-digit number1 that has your shoe size in the thousands and hundreds places2 and your age in the tens and ones places.

What are the limitations of this trick?

This "trick" works best right about now. The "Add 1013" step sneaks in the current year. We're at the end of the year, so most people will have had their birthdays already. It will work fairly well at the beginning of 2014, but at some point3 it'll be wrong more often than it's right.

The trick works only for people under the age of 100. That's probably a good assumption. Comparatively, there aren't a whole lot of people over 100, and they're probably not the ones passing this trick around Facebook, anyway. To get it to work for centenarians, you'd need to change step 2. Multiply by 50, instead of 5. Now the shoe size is in the ten thousands and thousands places, and there are three digits for the age.4

The first step (rounding your shoe size) is important for a similar reason. If you use a half size, that half will end up overlapping with your age, and that will spoil the trick.

I've assumed US shoe sizes, but does this work with European sizing? Yep. Since the shoe size goes out front, there aren't limits on the number of digits.


The University of Cambridge has a variant that puts age on the left and shoe size on the right. Note that it's written for children; the example age is 11.

Math secrets

I explained this trick to the relative who had shared it, and she replied:

Hey, good magicians are never supposed to give away their secrets!

Sure, magicians may not give away their secrets to their audiences, but they do tell their apprentices. I think everyone needs to do at least an apprenticeship in mathematics. Maybe then they'd understand that it's not magic, it's math.

1: Depending on whether your shoe size is under 10 or not

2: If your shoe size is less than ten, then the hundreds place is 0. Typically, we leave leading zeros off, hence 700, rather than 0700.

On a totally unrelated note, I suspect the international man of mystery known as "007" would be a little less mysterious if he was just called "7."

3: It's tempting to say "halfway through 2014," but not all birthdays are equally likely, so I'm going to hedge and just say "sometime."

4: If even Methuselah didn't make it to 1000, it's probably a safe bet that three digits are enough.