Age Guessing: Looking at the Roots

Published on Sunday, April 29, 2012 in , , , ,

Tyne & Wear Archives & Museums' photo of Lee Bennett guessing a woman's ageIn 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

Let's say you perform this, and the resulting number on the calculator is 1,248,695, and you announce that the person's age is 26, which is confirmed by them.

Mathematically, all they have is the formula 9x + y = 1,248,695. With two variables, that equation (known to mathematicians as a Diophantine equation) has an infinite number of solutions. How is it possible that you could narrow down the possibilities so quickly, and in your head?

How it works

When you're shown the total, you first add the digits of the answer up in your head. Next, using the age-guessing skills you learned in the previous post, ask yourself if the person could be that age.

If the person seems older than that, add 9 to the number you got and ask if that seems to be a more reasonable age. If that doesn't seem right, move up or down in 9-year increments, and keep doing that until you find an age that seems right.

In our example, you'd see the answer 1,248,695, so you add 1+2+4+8+6+9+5=35. Ask yourself if the person could reasonably be 35. Let's say they look younger than that, so you subtract 9. 35 - 9 = 26, so you consider 26, which we'll say seems more reasonable, so you guess that number out loud.

Why it works

You start with a long random number, and then multiply it by 9. What happens when you multiply any number by 9? Square One TV's Nine, Nine, Nine song explains:

When you sum up the digits, the result is known as a digit sum. The digit sum of 99 is 18 because 9+9=18. In the video above, notice they keep repeating the process of taking the digit sum until they get a 1-digit number. If you do this, the 1-digit number you get is called the digital root. The digital root of 99 is 9 because 9+9=18, and 1+8=9. The point of the above video, of course, is that any number multiplied by 9 will have a digital root of 9.

What happens when you add a number to a multiple of 9? Let's take 5 as an example. 9+5=14, and the digital root of 14 is 5 (1+4=5). 18+5=23, and the digital root of 23 is 5 (2+3=5), and so on. Let's 18+14, which is a multiple of 9 plus a number with the digital root of 5. 18+14=32, and 32's digital root is 5! Also notice that the answers remain spaced by multiples of 9: 5, 14, 23, 32, and so on. In short, any time you add a number to a multiple of 9, the answer will always have the same digital root as the number you added, and you'll always be a multiple of 9 away from another number with the same digital root.

Applying this to the trick, when you multiply by 9 and add the age, the digit sum (1+2+4+8+6+9+5=35 in our above example) will not necessarily be their age, but will have the same digital root as their age, and be some multiple of 9 away from the correct age (even if that multiple is 0).

Try this out for yourself. Get a calculator, enter any 5-, 6-, or 7-digit number, multiply that by 9, then add your age. Take the result, and enter it into the widget below, then click Submit. A window will pop up showing all the possible ages (listed as the variable a) between 0 and 100 you could be, based on the number you entered.

Sneakier ways of getting to a multiple of 9

If someone is familiar with the effects of multiplying by 9, they might suspect what you're doing. There are other less obvious ways of getting to a multiple of 9:

• From a sidebar in in Karl J. Smith's Nature of Mathematics (available at Amazon.com): Mix up the serial number on a dollar bill. You now have two numbers, the original serial number and the mixed-up one. Subtract the smaller from the larger. Assuming you didn't create two identical numbers, the result will have a digital root of 9, because you're subtracting 2 numbers with identical digital roots (More about this principle here).

• Also from the same sidebar in in Karl J. Smith's Nature of Mathematics: Using a calculator keyboard or push-button phone, choose any 3-digit column, row, or diagonal, and arrange these digits in any order. Multiply this number by another [3-digit] row, column, or diagonal. As it happens, most numeric keypads are arranged in such a way that any row, column, or diagonal of the numbers 1-9 will make a multiple of 3. Multiplying two multiples of 3 together will always result in a multiple of 9.

• You could also adapt Scam School's first Pi Day Magic Trick (YouTube link). Have them multiply 1-digit numbers together as shown in the video, until you get to a number somewhere between 1 million and 1 billion. Instead of having them remove a digit as in the original routine, however, have them add their age instead. As you see in the video, though, it is possible to get a number like 8,100,000,000. Adding their age to that would be obvious (assuming the guy in the video is 22, he'd get 8,100,000,022). To prevent this, tell them to avoid pressing the 5 and 0 keys, as this will just result in a lot of zeros at the end (or just one in the case of multiplying by 0).

These aren't the only secret ways to get to a multiple of 9, but are varied and interesting enough to get you started.

If you'd like an age-guessing routine that has the precision of math, but without the appearance of math (or even use of a calculator), I think you'll enjoy the next post, which will be the final installment in our series on how to guess people's ages.


Age Guessing: Judging Appearances

Published on Thursday, April 26, 2012 in , ,

Tyne & Wear Archives & Museums' photo of Lee Bennett guessing a woman's ageWhen 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

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.


Age Guessing: Pure Math

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

Tyne & Wear Archives & Museums' photo of Lee Bennett guessing a woman's ageBack 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.


Mental Gym Updates

Published on Thursday, April 19, 2012 in , , , , , , ,

bastique's Wikipedia globe keychain photoI 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!


More Quick Snippets

Published on Sunday, April 15, 2012 in , , , , , , , , , ,

Luc Viator's plasma lamp pictureApril'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.


Secrets of Nim (Notakto)

Published on Thursday, April 12, 2012 in , , , , ,

NIM is WIN upside down!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

You should start by going first, but the worst possible opening move is to place your X on any of the edge or corner squares. Why?

Because your opponent can basically mirror your moves, and this strategy will ensure that you must eventually make a line of 3 Xs, as shown in the following animation:

As you can see at the end of the animation, when the first player puts their X on an edge or corner square, and the second player mirrors them, this leaves an open diagonal line on the first player's turn that forces them to complete a line.

I mention this strategy mainly so you can be aware of it, and make sure that it doesn't happen to you inadvertently. Should you let the other player go first and they place their first X on an edge or corner square, knowing about this becomes a winning strategy for you!

How To Win

To assure yourself the win, you start by placing your X in the center square. To play from there, Timothy Chow discovered the answer comes with help from a chess knight!

Knowing how a chess knight moves (2 squares horizontally and 1 vertically, OR 2 squares vertically and 1 horizontally) is all you need to win.

After the other player makes their move, mark your next X a knight's move away from where their previous move. Keep using this strategy and they'll always be forced to draw the losing X. Watch the following animation carefully, and you'll get the general idea:

When choosing your spaces using the knight's move strategy, you'll usually have more than one space that qualifies. Often, one of the spaces will complete a line of 3 Xs, while the other is safe, so you'll always want to double check that you don't inadvertently make a losing play when you don't have to.

You can find out more about the game from Bob Koca's original discussion or the MathOverflow discussion. For a deeper look at the mathematics of Notakto, you can also read Thane Plambeck's presentation in PDF form.

If you'd like to practice this and you have an iPad, Thane Plambeck has also developed a Notakto app which will let you practice this version, as well as more difficult versions!

There's a closely related game taught on Scam School, called Napkin Chess, which is won using a similar symmetrical strategy. It's interesting to see the similarities, even though it doesn't have a tic-tac-toe board's discrete spaces.


Hunting the Elements

Published on Sunday, April 08, 2012 in , , , , , ,

Kordas' Periodic Table photoI'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:

Earlier this week, NOVA aired a special called Hunting the Elements. The full special is about 2 hours long, and I recommend you make time to watch the entire thing.

Below are two short excerpts from that special, both roughly 8 minutes long. This first one discusses why the periodic table is arranged the way it is:

This second excerpt talks about the characteristics of the atom that gives each element gets its particular properties:

For more direct learning, NOVA has provided some wonderful teaching tools, such as their Name That Element Quiz. If you have an iPad, check out the NOVA Elements app (iTunes link). It not only includes the entire special, but also lets you play around with the elements by building atoms, putting them together in compounds, and much more!

Should you want to learn specific information about a given element, there's a great site called the Periodic Table of Videos. The periodic table on their homepage links to videos about the corresponding element. These videos are also available on their YouTube channel.

Of course, one of the things for which Grey Matters is known is teaching how to memorize just about anything. If you've been inspired to try and memorize the periodic table, check out my 2008 Elementary post. (Being 4 years old, some of the links are no longer available, but most of them are still functional.)


Gathering For Gardner 10

Published on Thursday, April 05, 2012 in , , , ,

Konrad Jacobs' photo of Martin GardnerJust 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.


National Poetry Month

Published on Sunday, April 01, 2012 in , , , , , ,

Natalie Roberts' magnetic poetry photoIn 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.
Some of the other reasons might not seem as impressive, such as the entries about keeping us connected and being a bridge among disciplines. If you take those lightly, check out Be a Man. Read a Poem. from the Art of Manliness site.

Once you appreciate the benefits, how do you go about doing the actual memorization? I've written quite a bit about memorizing poetry in past posts, but there are many more approaches. New technologies make it easier to memorize than ever before. In Essay on memorizing poetry - at the gym talks about using crib sheets while exercising, although these crib sheets could be recording or videos on a mobile device of poems you wish to learn, as well.

Mensa For Kids' A Year of Living Poetically lessons are a good selection with a great structure. The poem is presented, broken down, and once the poem is memorized, there are varying types of quizzes to test your knowledge.

A more adult version of this same approach is used in Shmoop.com's poetry section. For example, their guide to Poe's The Raven includes not just the poem text, but an intro, a summary, an analysis, a quiz and much more! Their poetry section also has plenty of classic choices, and is a great place to look for material.

Another good source is the book Committed to Memory: 100 Best Poems to Memorize. You can even find the full intro and a majority of the selections from this book at poets.org.

Remember, memorizing poetry should be fun. Looking for a fun short piece to memorize right away? How about this ironic choice, titled Forgetfulness by Billy Collins: