In the past 2 posts, we've looked at the precision of the purely mathematical approach to age guessing, and the skillful approach of estimating age by appearances.

In this post, you'll learn an approach that seems to be a math trick, yet seems impossible to explain in that way.

You starting by asking a spectator to put any 5-, 6-, or 7-digit number in their calculator. You then tell them to multiply that number by 9. The last steps are to add their age to that number, and then show you the resulting number.

You examine the number for a few seconds, and instantly announce their age!

Why this is deceptive

When it comes to age guessing, very few people think of the calculator feats such as the one in the previous post.

The first thing that usually comes to mind is carnival age-guessers. In this post, we'll take a closer look at age-guessing as a skill.

The best tips I've found on determining someone's age are in the article How to Guess Ages More Accurately. Since men tend to put less effort into hiding their age than women, here are a few extra tips on guessing a man's age.

Just knowing these tips isn't of much good without practice. Thankfully, there are several sites where you can practice guessing the age of random people:

• How Old Are You?
• Guess my Age
• Match>Age

Even though carnival age-guessers aren't having you put any numbers in a calculator, they're still able to use some very subtle math tricks. For example, instead of advertising that they'll hit your exact age, you'll usually see a margin of error such as, “I'll guess your age within 3 years!” That sounds quite close to most people.

If you think about it, however, a 3-year margin of error really isn't that close. If someone is 35, a guess of anywhere from 32 to 38 would be considered correct. In other words, all they have to do is be within the decade you were born in, and they'll be considered correct. The more experience they have, the smaller margin of error they can offer. For example, professional age-guesser Lee Bennett used an impressive 1 year margin of error.

Even more central to an age-guesser's actual purpose is the simple economics of the situation. Let's assume that the cost to have the carny make a guess is $3, and the cost per stuffed animal to the carnival is $.25 (since they buy them in bulk). If we assume the guess is wrong every time, perhaps to keep every customer flattered, they're making an 1100% profit on each prize!

As the guesser becomes more skillful, the profit margin goes up! If we assume the age-guesser can correctly guess the ages of 4 out of 5 people (an 80% success rate), then that's 5 people times $3/person or $15 they're taking in. Only 1 wrong guess out of those 5 means that they're giving up $.25 for every $15 they take in, a staggering 5900% profit margin!

So, when it comes down to it, age-guessing as a skill is all about the margin of error and the profit margin. And that's assuming they don't employ standard scams like writing two ages and then covering up the one that's farther away, using magician's techniques to write down a close answer after you state your age, or simply pickpocketing your wallet and looking at your ID.

Guessing ages is a skill, but only ever an approximate one at best. The mathematical approaches, as we've seen, offer precision. Perhaps the best approach is to develop the skill of age-guessing, and use math in a way that doesn't detract from the skill.

That's the approach we'll start developing in the next post in this series.

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.

I teach quite a few fun mental challenges over in the Mental Gym.

While I teach methods in as simple and straightforward a manner as possible, there isn't always just one approach. In this post, I'll take a look at new approaches to feats in the Mental Gym.

In my tutorial on Squaring 2-Digit Numbers Mentally, I already teach two methods - a mathematical approach, and Jim Wilder's pure memory approach.

NumberSense's approach takes advantage of an algebraic pattern. The number is separated into 2 variables, a being the 10s digit and b being the 1s digit. The problem then becomes (a + b)², whose expanded form shows how to make the problem easier:



Besides making the squaring of two digit numbers easier, this video also illustrates a good point about algebra. Algebra lets you see patterns of which you may not have been previously aware, and help you see a shorter, and possibly better approach.

Another mathematical challenge I tried to simplify over in the Mental Gym was the unit circle and its associated trigonometric functions.

These lessons are especially handy for students taking trigonometry. Here's a handy approach to memorizing the unit circle, especially useful for tests, that works solely by taking advantage of several simple patterns:



We'll wind up this post by focusing on two of the puzzles.

First, there's the Sudoku. I already link to instructions on Sudoku strategy, but if you find those hard to understand, e-How has a series of excellent instructional videos on the Sudoku-solving techniques that you may find helpful.

In the Towers of Hanoi, the seemingly-simple task of moving disks from 1 peg to another quickly gets complicated. Here's a short, direct tutorial that helps make the solving pattern much clearer:



If you've come across an alternative way of doing any of the feats over in the Mental Gym, I'd love to hear about it in the comments!

April's snippets want to run free. They range from our usual topics like math and memory, to games, and even a little law and politics!

• If you enjoyed my previous post on Notakto, but you can't play the iPad app, Thane Plambeck has an online version you can play. Like the app, you start with one board, and work your way up to 5-board play. You can only move to the next level after winning 3 consecutive games on your current level.

• Speaking of strategy games, I've come across some new work on a classic. Ever play Hangman, and always use the same letters to guess in the same order, such as E-A-T-O-N? There is a better Hangman strategy, described over at DataGenetics. Instead of just giving you a new strategy, though, they go the extra step and explain the detailed thinking behind it, so you can understand it more completely.

• The Major System is a great technique for memorizing numbers, but can be challenging to learn. Over at the memory basics page, I've just added a few new resources that may help those who want to learn it. First, I added the Great Courses free video How To Memorize Numbers, a free lecture video from their Secrets of Mental Math course which I originally mentioned in October's snippets.

Over at Math Dude :: Quick & Dirty Tips, they also have an excellent series of 3 podcasts that teach the Major System. Part 1, Part 2, and Part 3 are available on their web site, as well as from iTunes (episodes 92 through 94).

• Vi Hart fans may remember her video Oh No, Pi Politics Again, about someone who claimed to have copyrighted music based on Pi. Writing music based on Pi is hardly a new and original idea, but the copyright claim was used to shut down others' videos anyway. Judge Michael H. Simon was the Oregon judge who presided over this case. Read the article Can you copyright music of pi? Judge says no to learn more about this decision, and exactly why the claim was denied.

• If you'll forgive me, I'm going to wind this post up with a little boasting. Back in January, I released Day One, my approach to speeding up and presenting the classic Day of the Week For Any Date feat. I updated it in February to include some unusual calendar-related bonus feats, as well. I'm proud to announce that, according to Lybrary.com's hot list, Day One is currently their 3rd best-selling magic item at this writing! The response has been simply incredible, and I'd like to thank everyone who bought it and who helped make this possible.

Back in 2010, Backgammon giant Bob Koca was playing tic-tac-toe with his 5-year-old nephew, when the nephew whimsically suggested that they both play as X.

Being a mathematics professor, he used his knowledge to analyze this weird version of the classic game with various rules, boards, and objectives. It turns out that this all-Xs version of tic-tac-toe is a version of our old friend Nim!

To keep the game familiar, I'll stick to the standard 3-by-3 board in this post. The rules are as follows:

• Players alternate taking turns, and neither player may pass on their turn.
• A player marks any empty space on the board with an X on their turn.
• The loser is the first person to mark an X on the board that completes a horizontal, vertical, or diagonal line of 3 Xs.

This game is known to mathematicians as neutral or impartial tic-tac-toe, but I prefer the name given to it by Thane Plambeck, who lectured on this game at G4G10: Notakto (pronounced “No Tac Toe”).

As I mentioned, this is variation of Nim, more specifically a Misère version, so there must be some way to win it. I'll start, however, by explaining how to lose the game, instead:

What YOU Should NOT Do

I've posted about memorizing the periodic table of the elements before, but understanding is just as important.

You might think trying to understand the basics of the elements would be a chore, but it can actually be quite fun.

Surprisingly, one of the best introductions to the atom I've ever seen is not from a documentary, but an episode of WKRP in Cincinnati. In this episode, Venus is trying to help a friend whose son has dropped out of school. In the following scene, Venus explains the basics of the atom in an effort to help get the son to go back to school:

Just last week, there was a gathering honoring the late Martin Gardner in Atlanta, called Gathering For Gardner 10, or G4G10 for short.

I didn't go myself, but the people who did attend have already started sharing their experiences with us.

If you're not familiar with Martin Gardner, you can see posts relating to his work right here on Grey Matters. David Suzuki, in his documentary series Nature of Things, spent one entire program on Martin Gardner, and introduces it at an early Gathering for Gardner event:



The G4Gs are invitation-only events, and there's not much available from G4G10, the most recent get-together. There are, however, a few goodies already online.

Over on flickr, there are already many photos from G4G10 posted. Even if you don't understand the subjects of the photos themselves, they're still wondrous and amazing to behold.

One of the biggest treats from G4G10, however, has to be Colm Mulcahy's library lecture about the life and work of Martin Gardner. Regular Grey Matters readers will probably recognize Colm from his Card Colm column and his Colm's Cards page.



Here's part 2 of the lecture:



The searchable collection of Gardner's work, which Colm mentions in the lecture, is Martin Gardner's Mathematical Games CD-ROM, and is currently available for around $40 at Amazon.com.

If you attended G4G10, or even just have any personal stories to share about Martin Gardner's influence, I'd love to hear about it in the comments.

In the US and Canada, April is national poetry month! (Sorry, Great Britain, you'll have to wait until October.)

Since memory is one of my favorite topics, I'll take a look at memorizing poetry in this post.

To most people, memorizing poetry sounds like something out of 19th century schoolhouses or 1960s beatnik coffee shops. The truth is, there are plenty of good reasons to learn to memorize poetry, especially if it's something you want to do, as opposed to something you're being forced to do. In Five Benefits of Memorizing Poems there are the usual education reasons. If that's not enough, Ten Reasons You Should Memorize Poetry expands on this, including some reasons that are right down my alley, including:

1) It is a brain challenge. Got a kid with a strong memory? I’ve got some long poems for you. Interested in history? Learn a poem based on a historical event or some of the poetry of that period. For anyone seeking a way to challenge a gifted child in way that is free (!) and virtually unlimited, you’ve found it. Even copying poems down (or lines of poems) and illustrating them is a wonderful activity for younger children.
...
8) It’s a great party trick. If you’re ever stuck for a spur of the moment talent, you’re in luck if you’ve got a poem in your mind you can whip out and recite from memory. It’s easy, it needs no props, and you will not be doing the same tired trick as everyone else. Unless they read this blog.