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The Secrets of Nim (Part 1)

Published on Thursday, July 29, 2010 in , , , , , ,

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Since I talked so much about Nim in my previous post, I thought it might be interesting to take a deeper look at the game. There are many places that discuss how to win at Nim, but not too many that discuss WHY these plays are the winning plays.

Terminology

There are actually several versions of Nim, so it's best to get our terms straight first.

Nim is a two-player strategy game in which players take turns removing objects from distinct heaps.

In what I'll call standard Nim, or simply Nim, the winner is the person who takes the last object. In what is known as MisÃ¨re (pronounced "Meez-air") Nim, the loser is the person who takes the last object.

There are also single-pile versions of Nim, in which you can only remove a limited number of objects in each turn, and multi-pile nim, where you can remove as many objects in each move as you wish, but only from a single pile. In this post, we'll be focusing exclusively on single-pile Nim, and save multi-pile Nim for my next post.

Winning Strategy And The WHY

The most oft-encountered version of single-pile Nim is MisÃ¨re Nim, where each player may take either one, two, or three objects in each turn, and the player who takes the last object is the loser.

This video below gets right to the point as to WHY the winning approach works:

It's easy to see why the “group of 4” strategy works with 5 objects, and note that it works with 9, 13, 17 or any other number that's 1 greater than a multiple of 4. The technical term for this is modular arithmetic, given a clear explanation here by BetterExplained.

Modular arithmetic is something you've been doing for most of your life, but without the fancy terminology. If two people are talking on the phone at 10 AM, and agree to meet at their favorite restaurant for dinner in 8 hours, when are they going to meet? That's right, they're meeting at 6 PM.

This post was put up on July 29th. My next post will be done on Sunday, 3 days from now. My next post after that will be next Thursday, 1 week from now. What are the dates of my next two posts? August 1st and August 5th.

Both of those examples required modular arithmetic to answer. In the first case, you mentally figure 10 + 8 to get 18. Because you're so used to a 12-hour clock, it was easy to work out that 18 - 12 = 6, thus the answer must be 6 PM. What you're basically saying here is that, ignoring 12s, 10 + 8 is the same as 6.

Similarly with the months of the year, ignoring 31s (the number of days in July), the 1st is the same as the 32nd (29 + 3 = 32), and the 5th is the same as the 36th (29 + 7 = 36).

So how does this help in MisÃ¨re Nim? First, the limit to the number of objects that either player can remove at one time is limited to less than 4 (1, 2, or 3 objects), so that the 2nd player can take advantage of the 1st player's move to get to total 4 objects.

Since, as we've already determined, that if we ignore 4s (referred to in mathematics as “modulo 4”), that leaving 17 objects (or 13, or 9, etc.) is the same as leaving 1 object, you can see why the game is won so far ahead of time.

Notice that, if you go first, or make a mistaken move, it is still possible to win by looking for opportunities to get the pile to 1 over any multiple of 4.

Let's say you have 17 objects, and the other player removes 2 (leaving 15). You should take 2, but let's say you take 3 by mistake (leaving 12). Now your opponent takes 1 (leaving 11) what do you do? You need to get to the next number that's 1 above a multiple of 4. 9 is that number, and you can get there by removing 2 objects!

Note that there's nothing special about using the number 4. We could use a number of objects that's one greater than a multiple of 5 (“modulo 5”), say 26 (25 + 1), and allow the each player to remove 1, 2, 3, or 4 objects (4 being 1 less than 5) in each round.

To take this to a silly extreme, we could have a pile of 501 objects (1 plus a multiple of 100), and allow anywhere from 1 to 99 (1 less than 100) objects to be taken each time. It might be hard to know how many your opponent takes in this case, but if you could determine that in some manner, just take enough to total 100 (Opponent took 47? You take 53! That's “modulo 100”). Now you see why the “modulo 4” caught on so much quicker than this version.

Notice that, in order for you to win, you must use information provided by your opponent's move, and therefore they must go first. What if your opponent gets suspicious of your constant winning, and wants you to go first? The age-old answer, as shown in Scam School's 8th episode, is to cheat by ringing in an extra match, and making your opponent believe that you're removing this extra match from the pile. The pile still has the same number of matches, and you can proceed as normal from here.

The alternative is get lucky and recover as described above.

Variations

Once you understand the basics behind single-pile MisÃ¨re Nim, you can also develop alternative versions. If your opponent either doesn't understand Nim, or knows the winning strategy for one particular version without understand the reasons behind it, this can help you get a new advantage.

What about turning MisÃ¨re Nim into standard Nim, where the person who removes the last object(s) is/are the winner? This is easy – start with 1 less object! Instead of playing with 17 or 13 objects, start with 16 or 12. Using the same modular arithmetic as above, ignoring 4s, leaving them with 16, 12, 8, or 4, is the same as leaving them with 0!

We keep focusing on objects, but you don't even need objects! MisÃ¨re Nim can also be played verbally. One of the more popular verbal versions is “21”. The first player starts stating either the numbers 1, 2, or 3 out loud. The other player then adds 1, 2, or 3 to that number and gives the new total. The first player then adds 1, 2, or 3 more to the running total, and gives that number. The person forced to say “21” loses (you're not allowed to give a total over 21).

Since the only difference here is cosmetic, this game is played exactly as if it were being played with 21 objects! If the other person goes first, you can always win just as in the original game. If you go first, you can really only recover by taking advantage of mistakes made by your opponent, as described earlier in this post.

You can mix these variations together, too. What about a Nim game where the winner is the first person to get to 100? Here's how that breaks down:

1) OK, the game is verbal, so we use no objects. As discussed before, this is only a cosmetic change, and doesn't affect play or strategy.

2) We'll take advantage of the fact that 100 is 1 greater than 99. 99 is a multiple of 3, 9, and 11. Using 3 would mean limiting the numbers added to 1 or 2 at a time. 9 wouldn't be bad, as the choice of numbers 1-8 would move the game along quickly, but 1-8 is an unusual range. How about 11 “modulo 11”? You could add numbers from 1-10 (a normal sounding range), and the game would move quickly, but be long enough to appear challenging!

3) Using 11 not only means that the choices are from 1-10, but also that our key numbers will be 1, 12, 23, 34, 45, 56, 67, 78, 89, and 100 (each 1 greater than a multiple of 11). These are pretty easy to remember, since each number except 100 has a sequential appearance (1 is after 2, so 12 is easy to remember, and so on).

4) You could also create a version where you played down to zero. However, changing direction does change the key positions you need to achieve. Starting from 100, only allowing the numbers 1-10, and playing down to 0 (where the person taking the last object wins), means your key positions are 99, 88, 77, 66, 55, 44, 33, 22, and 11. As learned with the version taught in the first video above, when counting down, your key positions are always multiples of your modulo number.

In their 116th episode, Scam School taught this version, along with a calendar version of Nim which I submitted:

Notice that, in the calendar version, you're working with two factors – a month and a date. In this case, to win you need to keep their relationship to a certain constant. Since your target date is 12/31, and 31-12 has a difference of 19, all the dates you get to must also have that difference of 19. Yes, the calendar race is a well-disguised modulo 19 version of Nim!

As noted in my post on that episode, going from 12/31 to 1/1 is even easier, as your target dates become 12/12, 11/11, and so on, down to 1/1! Believe it or not, this make the January 1st version a “modulo 0” Nim game.

I'll wrap up this part of my Nim series with 2 final interesting variations. The first is the card version, called “31”:

This card version which has several interesting layers to it. It's easy to understand that this is a “modulo 7” game, so that the numbers used in each turn are limited from 1 through 6. But why isn't the game played to 29 (as opposed to 31), which is 1 greater than a multiple of 7?

This is the final dirty secret of single-pile Nim. Your key states don't have to be limited to a number 1 greater than the “modulo” number you're using. As you as you play to move to numbers that are equivalent in modular arithmetic to that of your goal, you can use any amount.

In 31, ignoring 7s, the numbers 3, 10, 17, and 24, are all the same as 31. They're all 3 above a multiple of 7.

Let's say you're a Douglas Adams fan, and want to create a verbal version where the winner is the person who reaches 42. It's 1 above 41, which is prime, so it's only a multiple of 1 and 41. So how about using 40? We could set it up using 4 or 5 as our modulo number, but I like 8. So, each person can add 1 to 7 each time. Your goal number all have to be 2 greater than multiples of 8 in this case, just like 42. Therefore, the key numbers you play to are 2, 10, 18, 26, and 34 (and 42, of course).

Another layer of the card/31 version is the use of a limited number of cards. Winning number 31 is 3 over 28, which is a multiple of 7, which is knowledge you use to win the first game. However, 28 is also a multiple of 4 (the number of cards in each pile), which helps in winning the 2nd game where you win while you're teaching.

When you teach the key numbers (3, 10, 17, 24, and 31) the first time, they naturally pick up the 3 first. You take the 4 to get to 7, so they have to take the 3 to get to the next key number (10). Taking all four 3s and all four 7s comes to 28. Now, it's no longer possible to choose 3 or 4. 5 and 6 would take you past 31, so the only real choices left are aces (1s) and 2s. If they take 1, you take the other, and you still win!

This is why the number 31 was so specifically chosen for this version of the game. It works in the first game, which is really “modulo 7”. When teaching the game, the game is effectively changed to “modulo 4”, which works until all the 3s and 4s are gone, at which point the game suddenly changes to “modulo 3” without warning! This is very sneaky.

Pop Quiz: What if you wanted to start from 31, and play down to 0, allowing only from 1 to 6 objects to be removed? What would your key positions be then? It's a modulo 7 game, so you would simply play down to the multiples of 7: 28, 21, 14, and 7.

The Importance of Understanding Nim

Our final video in this part of our Nim series shows why it's not just important to know the winning moves of Nim, but the why behind it. In this video, 2 men learn the simpler secret to winning Nim, but their failure to understand the reasoning behind it costs them financially:

In part 2 on Sunday, we'll begin our look into the secrets behind multi-pile Nim.

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Scamming with Androids and Apples

Published on Sunday, July 25, 2010 in , , , , , , , , , , , , ,

Learning about how scams work can be a fun gateway to learning surprising things about math and psychology. But what about when you're on the go? Get out your Android and Apple iOS mobile devices for this post!

Disclaimer: As with any column on scams on Grey Matters, I don't condone using these to rip people off, and simply present the information about scams here as an educational tool. Proceed with this in mind, and at your own risk.

Before we even get to apps, don't overlook the built-in features of your mobile device(s).

With the ability to view video on YouTube and/or subscribe to podcasts, you can check out programs such as Scam School (Scam School homepage), The Real Hustler UK, The Real Hustle US, and more!

The ability to surf the internet, as always, can also bring a wealth of information. Besides learning about scams here on Grey Matters, searching around forums can be excellent resources. There's the Scam School forums, as well as the various Magic Cafe's great forums, including The Gambling Spot, Pardon me, sir..., If right you win, if wrong you lose..., Betchas, Magical equations, and Puzzle me this....

A good blog to check out is Australia's Honest Con Man: Confessions, and his previous archives at The Honest Con Man's Guide To Life.

One of the most popular mathematical scams is the game of Nim. It has already made 4 appearances on Scam School (Nim, Advanced Nim, Thirty-One, Calendar Nim).

Watching those videos is one thing, but why not use your mobile device to practice and learn more about it? It's not surprising that Nim is so readily available for mobile devices. In the early days of personal computers, it was already popular. There were basic versions, such as 23 Matches, Batnum, and Nim. Some of the versions, such as the amazing Android Nim for the PET and TRS-80, did an amazing job of presenting this classic game!

On the Android you can get NimDroid and NimSwitch, both available for free!

For iPhone/iPod Touch and even iPad users (using the iPad's "2x" mode for these apps), there's plenty of free Nim versions, including myQuickGame Free, NeonNim: The Subtraction Game, Nim Game, and PYMINIM. Update: (August 17, 2010) Another Nim game, the Race To 100 app, was originally released for 99 cents, but has been free since August 8th. This is the version of Nim taught in the 116th episode of Scam School.

If you're willing to spend a little money, there's also some nice commercial versions of Nim for Apple's mobile devices. Dual Matches, Mind Nimmer, myQuickGame, and Nim.

Special mention should be made of the commercial Cannibal Muffin and Last Stone apps. Like the aforementioned 1970s Android Nim, the authors have taken extra time and care to present Nim in an extraordinary way.

Getting away from Nim, what about this interesting problem, known as the Monty Hall Problem?

Scam School also covered this, explaining how this fooled over 1,000 PhDs when it was discussed in Marilyn Vos Savant's column. Apple mobile device users use can try this counter-intuitive problem via the commercial Monty Doors or the free Monty Hall Paradox apps.

Although you can't win with this scam every time, you can win often enough that there are con men out there who use the Monty Hall Problem's counter-intuitive nature to their advantage.

While Nim and Monty Hall reign as scams from which you can learn great lessons, there are many other ways to learn about scams with your mobile device. For iPhones, iPod Touches, and iPads, Bar Tricks Free teaches you some basic scams, and its big brothers, Bar Tricks I and Bar Tricks II, both of which are paid apps, can teach you more. iDrink4Free and Gags are also available on the App Store.

Update: (August 17, 2010) The Author of the previously-mentioned Race To 100 app, also has another app called FourQuarters. From what I can tell, it's a version of the four-coin puzzle taught on the 2nd episode of Scam School.

If you're really serious about understand why these and similar scams work, Bruce Frey's book Statistics Hacks (also available in paperback) is a great, clear way to understand these often perplexing propositions. You can get a free preview of this book here, as well as the paperback link.

I'd love to hear about any insights you've developed by playing with these apps. Also, if you have any others that are relevant, I'd love to hear about them. Talk to me in the comments!

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Game Show Memory Feats

Published on Thursday, July 22, 2010 in , ,

One of the attractions of a trained memory is the fantasy of using it to win lots of money. Today, we're going to talk about 3 men who did just that on TV game shows!

Michael Larson

Who better to start off with than someone who was always looking for a way to make a quick buck? Michael Larson was an unemployed ice cream truck driver who, while dreaming about lots of money and watching daytime game shows, noticed a pattern to the seemingly random lights and prize spots on a 1984 game show called Press Your Luck.

After watching to get a detailed understanding of the patterns used on the board, he then memorized them, got on the show, and won roughly \$110,000. That may not sound like a lot in game show prize money now, but in the days where the top game show champions usually left with \$40,000 in prizes and cash, it was an astounding amount of money.

While you could watch Michael Larson play the game itself, you'd be missing much of the story. Back in 2003, GSN produced a 2-hour documentary containing the full story, narrated by the show's host, Peter Tomarken. The entire fascinating story is shown below:

Ken Jennings

Ken Jennings, best known for his stint as Jeopardy!'s longest running contestant, is hailed today as a trivia legend. His knowledge of trivia and his ability to recall it when needed is the cornerstone of his fame.

The big question is, how did he prepare for such a long run on such a notoriously difficult game show? For that matter, why not that show instead of another one? In his book Brainiac (free preview available here), Ken delves into those questions.

If you're looking for brief background on what led him to Jeopardy!, check out this Ken Jennings interview from current employer mental_floss magazine. For a more detailed look, however, check out the hour-long Authors@Google interview, shown below:

For those who wonder what he's been up to since his Jeopardy days, he's written several books (including the aforementioned Brainiac and Ken Jennings's Trivia Almanac), has his own website, and has even released a board game called Can You Beat Ken?.

On its 25th Anniversary, Jeopardy! caught up with Ken Jennings, and gave a brief look into his life since his championship run:

Terry Kniess

Terry Kniess isn't as well known as Ken Jennings, or even Michael Larson, perhaps because he's the most recent of the three game show contestants mentioned in this post.

Terry Kniess put his memory to work on The Price Is Right on a show that was broadcast on December 16, 2008.

Like Michael Larson, Terry noticed patterns in his game show of choice. In the case of The Price Is Right, he noticed that the same products were used over and over again (which makes sense due to the same sponsors), and were always priced the same. He studied the program, obtained all the prices he could, and used his knowledge to get up on stage, win his games up on stage, and eventually win the Showcase Showdown at the end!

In Esquire's article TV's Crowning Moment of Awesome, they detail the whole story from preparation, through a conclusion so surprising that not even Terry Kniess himself expected it.

Carey announced Sharon's bid first. Actual retail price, \$31,019. She had missed by just \$494 — a remarkably close bid, since trips were notoriously difficult to figure. Trips were budget savers.

Then came Terry. "You bid \$23,743," Carey said through his teeth.

Today, at his kitchen table, Terry says he'd seen all three prizes before. The karaoke machine was \$1,000. The pool table, depending on the model, he says, went for between \$2,800 and \$3,200. Terry went with \$3,000. The rule of thumb for campers, he knew, was about \$1,000 a foot, plus a little more; he says today he'd actually misheard the length of the trailer, thought Rich Fields had said it was nineteen feet long — so, \$19,000. That gave him \$23,000. And then, he says, he got lucky. He picked 743 because that was the number he and Linda had used for their PINs, their securitycodes, their bets: their wedding date, the seventh of April, and her birth month, March. Here's their wedding certificate, he says, and here's her passport: \$23,743.

"Actual retail price, \$23,743," Carey said. "You got it right on the nose. You win both Showcases."

Yep, Terry did something no one had ever done before, hit the price of a Showcase Showdown right on the nose! Behind the scenes, the show became very subdued. The price was so exact, that the feeling was that Terry had somehow cheated. As a result, most people on the crew figured that the show would never air. When it was discovered that there was no cheating, the prizes were awarded and the show was aired.

In the following footage, watch after the bids are made and the show comes back from the commercial. Notice how subdued Drew Carey is? He's just walking through it because he believes he has to complete the show, but doesn't believe it will air:

After the show aired, but before the full story was finally told, there were many people who believed that the show was rigged. He was able to get his story out in this radio interview, not too long afterwards:

The fact that both Michael Larson and Terry Kniess were accused of cheating, despite their legitimate approaches, shows you just how powerful a trained memory can be. Even though it wasn't on a TV game show, I've had the same experience myself of using my trained memory and being accused of cheating.

It just shows you that the use of a well-trained memory is so unbelievable to most people that other explanations are more readily accepted.

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Yet Again Still More Quick Snippets

Published on Sunday, July 18, 2010 in , , , , , , , ,

Grey Matters is back, and the computer troubles are now over! I thought I'd return to blogging with July's snippets.

• There's a new group of British street performers that are right in the spirit of Grey Matters. They're called Maths Busking and they entertain with mathematical feats, in an attempt to get more people to see how interesting and engaging math (or “maths”, as they say in Europe) can be.

There's a great Maths Busking intro video over at Guardian.co.uk (Flash required), and an amusing video of BBC reporter Ruth Alexander's turn at Maths Busking (Flash required) on the BBC site.

• Speaking of things that fit well within the spirit of Grey Matters, there's now a blog called New Mental Magic. It focuses on teaching easy ways to do math in your head. Give it a look.

• For those who like to keep their MAGIC Magazine database up to date, I've just added the information for the July 2010 issue of MAGIC. On that same page, you can also get data for issue from November 2009 through June 2010, as well. All the other data is available at the MAGIC Magazine link at the beginning of this paragraph.

• The July 2010 issue of MAGIC features a very large tribute section to the late Martin Gardner, including a look at the mini-column he wrote in MAGIC for many years. Even if you don't regularly read MAGIC, this issue is well worth picking up for Martin Gardner fans.

• If you haven't been keeping up with my Twitter feed (as seen in the righmost column of this blog), I've been posting a series of videos (and one text page, so far) teaching various memory methods. You can find the posts at these links: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10

• Most readers of Grey Matters know that I have a special place in my heart for Pi, even going so far as to start this blog on 3/14. So, it was a little jarring to hear it suggested that Pi is Wrong! (PDF). The basic idea is that what we call 2*Pi should really be Pi, because of how most formulas involving Pi work out. Michael Hartl explores this idea further in The Tau Manifesto: No, Really, Pi is Wrong.

If all the talk of radians confuses you, check out BetterExplained's Intuitive Guide to Angles, Degrees and Radians.

• One last goody – the movie The Phantom Tollbooth is available online again for free! Instead of being in an unwieldy 14 parts, as it was previously, it's down to a more manageable 2 parts. Check out the movie, and enjoy!

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Computer trouble

Published on Thursday, July 08, 2010 in

Unfortunately, due to computer trouble, I'm not able to provide a full post today. I'm getting the trouble cleared up, and I'll be back with much more for Grey Matters readers soon.

In the meantime, you can still keep up with mental goodies by following my Twitter feed (at the link or over in the rightmost column.

Thanks for your patience!

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July 4th Brain Fun

Published on Sunday, July 04, 2010 in , , , , , , , ,

The United States of America is 234 years old today. In honor of that, we're going to challenge your brain's knowledge of the USA!

Don't worry. I'll start slow. Before any kind of exercise you need to stretch, don't you?

Start by imagining that Puerto Rico becomes the 51st state. According to the Flag Act of 1818, a new flag containing 51 stars would be released on the following July 4th.

What exactly would the US flag look like? Skip Garibaldi has examined our past flags, and used that knowledge to work out what the US flag would look like with up to 100 stars. He also discovered that we'll be faced with a quandry if we ever have 69 or 87 states.

The adoption of the Declaration of Independence on July 4th, 1776 is, of course, why we celebrate it as Independence Day.

The late Martin Gardner has actually managed to turn the first sentences of the Declaration of Indpendence into a magic trick, with a little help from Martin Kruskal.

We're passed the warm-up, and we're going to start with the big challenges. Sporcle is curious to know how many signers of the Declaration of Independence you can name in 10 minutes. Mental Floss, on the other hand, wants to know if you can identify the states that some of the lesser known Declaration of Independence signers were representing.

If you've read this far thinking, “Would the Founding Fathers really support celebrating Independence Day with these sorts of brain challenges?”, I'd like to think at least one would support this approach: Ben Franklin. He loved magic squares! A little over 4 years ago, the Sunday Times even posted Ben Franklin's Magic Square Challenge, whose goal was to complete this square so that it totals 2,056 in as many directions as possible.

From those original 13 states, the USA has grown to 50 states. To help you learn the 50 states in a fun way, explore the visual USA state mnemonics over at 50 States of Mind. They're fun, educational, and will help with the following quizzes.

The first quiz? Name all 50 states in 10 minutes, of course. If you know your states, can you name all 50 state capitals in 10 minutes?

Now that you have all that information, how well can you filter it out? One classic challenge is to name all 8 states that begin with the letter M in 1 minute. Did you find that tough? Study those states with Lou Ryder, and then go back and try again.

For a similar challenge, try and name all 21 states whose names end in the letter A in 4 minutes. No, I don't have any resources to help you learn this one.

Don't get too used to states with certain letter patterns. Geography is about physical features. Can you name the 23 states that border the Atlantic Ocean, the Pacific Ocean, or the Gulf of Mexico in 3 minutes?

I'd love to hear how well you did on any of these challenges. Let me know in the comments!

Have a safe and happy July 4th!

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Easy Magic Square Cheat

Published on Thursday, July 01, 2010 in , , , , , , ,

I've posted quite a bit about magic squares in the past, but most of them required some calculations to present.

This week, Scam School teaches the Easy Magic Square Cheat, where you look like a genius with far less work than it would appear!

Check out the video below, and after I'll delve into more detail about this presentation.

There's really two parts to this presentation: The first matrix, which forces the number 34, and the magic square you've simply memorized as a result.

The forcing matrix is actually quite a versatile tool, and can be developed for almost any size and almost any number. Doug Dyment's article, How to Construct a Forcing Matrix, is an excellent introduction to this topic. It even includes a downloadable Excel spreadsheet, so you can understand the ideas hands-on.

Martin Gardner gives an even more thorough examination of the forcing matrix in chapter 2 of his book Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games (see a partial chapter preview here).

This nice thing about working with a fixed total in this presentation, is that you can present a magic square without practicing any calculations. As long as you rehearse the arrangement taught in the video, you'll be able to look like a math whiz doing fast and difficult calculations.

Are there other calculation-free magic square presentations? Of course!

Here's one: Introduce the basic concepts of the magic square, and point out that it's long been thought by mathematicians that the smallest possible magic square is a 3 by 3 arrangement (Why are there no 2 by 2 magic squares?).

You then bet that you can show an arrangement of fewer squares that makes a magic square. This sounds impossible. How is it done? You show them this. At first, it doesn't even seem to be a square, but when you pull out the mirror, it not only becomes a square, but a magic square, as well!

Werner Miller is also responsible for another great magic square presentation, in which your Windows computer, iPhone, or iPod Touch handles all the needed calculations for you. It's called the Age Square:

iPhone/iPod Touch online version
Windows offline executable version

Here's the presentation and the method behind Werner Miller's Age Square:

In my recently-added 15-puzzle tutorial, I include a magic square solution that doesn't require any calculations, either. You do need to practice solving the 15 puzzle itself, though. Doug Dyment, who wrote the above forcing article, also helped me find the arrangement used in that feat.

Returning to Werner Miller, who must dream in magic squares, our final calculation-free magic square presentation is his Holey Number puzzle/paradox. As a matter of fact, I present this feat mixed with the previously-mentioned 15 puzzle, in a presentation I described here.

If you've tried any of these magic square presentations out, or simply have any questions or comments about them, please let me know in the comments!