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Age Guessing: Pure Math

Published on Sunday, April 22, 2012 in , , , , , , ,

Back in 2008, I wrote a post about guessing ages. Unfortunately, it was several approaches compacted into one long post and lacked clarity, as a few readers have noted.

I've decided it's time to update the post. I'll break age-guessing up across several posts in an effort to improve the clarity, as well.

In this post, I'll start with the methods for finding someone's age using purely mathematical methods.

The first type of mathematical age trick that usually comes to mind is the algebraic type, such as the kind listed on this page under “Guess Your Age.” In the first, you have the person put their age in a calculator, triple it, add 1, triple it again, add their age again, and then show you the result. While the process of performing (((x * 3) + 1) * 3) + x looks complicated, it's just a long way around of getting them to multiply their age by 10 and add 3 (See the alternate forms section).

The second trick, in which they multiply their age by 7 and then by 1,443, isn't so much mysterious as it is surprising and amusing. 7 * 1,443 = 10101, so any 2-digit number multiplied by that is of course going to repeat itself 3 times.

In the original Age Guessing post, I also linked to this age plus a secret number approach, which explains it's own algebra, and these two algebraic approaches, one of which breaks up the age into two different numbers, and the other that makes use of the year the person was born.

These types of tricks can be very impressive for an audience unfamiliar with the basic concept of algebra, and can also be a great way to introduce new students to algebra. Anyone beyond that stage, even if they can't work it out at the moment, will recognize that there's some simple pattern that will get you the answer. Since this is the case, perhaps there's a mathematical approach that is more deceptive.

A deceptive approach that's long been a favorite of magicians is one known as the Age Cards. You can find an interactive version of it at this link. Look for your age in each group. If you see your age in a given group, click the checkbox for that group. Once you've checked all six groups for your age, and clicked where appropriate, click on the CALC button. The computer will tell you your age!

It works simply by adding up the smallest number (the one on the upper left corner) on each card on which the age was seen. If your age was 27, you would only click the boxes of Group One (smallest number is 1), Group Two (2), Group Four (8), and Group Five (16). Adding 16 + 8 + 2 + 1 gives 27, so the chosen age is 27.

See if you can follow how the secret number of 38 is determined in this video:

That's how the trick is done, but why does it work that way?

The method here is better hidden than the algebraic methods because instead of using our usual base 10 numbering system, which uses the digits 0 through 9, the Age Cards trick is based on the base 2 numbering system, better known as binary, which only uses the digits 0 and 1. Working with a different number base can seem scary and confusing, but BetterExplained points out that you work with different number bases more than you might think.

Even though binary is limited to using 0 and 1, it can represent any number our more familiar 0 through 9 system can. The PDF and the first video on Computer Science Unplugged's binary numbers page explain how clearly and quickly. The number 27, for example, converted to binary becomes 11011. In base 10, we only need 2 places (2 tens and 7 ones) to represent the number, but in binary, we need 5 places to represent the same number (1 sixteen, 1 eight, 0 fours, 1 two, and 1 one).

How does this all relate to the Age Cards? Note that there were six Age Cards used. Each card acts like one of the places in the binary number. Note that the smallest number on each card corresponds to one of the binary places, as well: 32, 16, 8, 4, 2, and 1.

To find out where a given number goes, we use it's binary code. As mentioned, 27 converts to 11011. We're working with 6 cards, though, so just like our regular base 10 numbering system, we can add zeroes to the left side without changing the value. Doing this, 11011 becomes 011011.

The rightmost spot in binary is the 1s spot, and if there's a 1 there, as there is in our 27 example, we put that number on the 1 card. There's a 1 in the twos place, so we also put 27 on the 2 card. There's a 0 in the 4s place, so we don't put 27 on the 4s card. The 1 in the 8s place and the 16s place indicate that the 8 and 16 cards do have 27 put on them. Finally, the leftmost 0 in the 32s place tells us not to put 27 on the 32 card.

In the video above, 38 only appeared on the 32 card, the 4 card and the 2 card because 38 in binary is 100110, which only has 1s in the 32s place, the 4s place, and the 2s place. Get the idea?

The Age Cards is well-known among magicians, so even this routine could benefit from a better disguise. Fortunately, Werner Miller has come up with some very creative work on the Age Cards!

First, there's his ingenious Age Cube, which is presented as a giveaway with five magic squares on it. You ask someone who is 31 or younger (because we're only working with 4 binary places) on which magic squares they see their age, and thanks to your secret addition of the numbers in the upper left corner of each magic square, you can magically divine their age!

His other approach comes as a webapp that works in any modern browser, and also as a Windows executable file. It's called Age Square, and builds impressive from the Age Cube. It only uses 4 binary places, but thanks to a secret better described in the original Age Square post, it still manages to cover ages from 30 to 85! Instead of giving the age directly as an answer, the app generates a new magic square, with their age as the total.

Divining someone's age purely using math can be interesting, but what about getting someone's age with some help from their appearance? That will be the topic of the next post in this series.