I apologize for the irregular posting over the past few months. I've had to deal with some personal issues (don't worry, everything is fine!). The good news is that, with this entry, everything should start returning to normal.

Having said that, let's dive into August's snippets!

• James Grime and Katie Steckles made a video about a seemingly simple game:



First, if it's on Grey Matters, you know all is not always what it seems. Long-time fans of Grey Matters may remember this when I described it under the name Wythoff's Nim. It winds up having some very interesting math behind it. James went on to make a solo video explaining the mathematics behind it in more detail:



• We can't ignore Katie Steckles' game video after all that! Katie teaches 2 games (or does she?). The first one involves numbered fishes, and the second one involves cards with stars and moons on them:



It's a little bit surprising that these are actually the same game! Back in June's snippets, there was a multiplication version of this. Like this and Scam School's game of 15, they all go back to Tic-Tac-Toe. If you want to see some other interesting variations of this same idea, read Martin Gardner's Jam, Hot, and Other Games column.

• There's usually more than one way to use your knowledge. In my tutorial about mental division, I teach a simple method for mentally dividing the numbers 1 through 6 by 7. Presenting it as an exacting feat of mental division is one thing. How else could you present it? Take a look at how Scam School presents the same feat:



If you watch the full explanation, you'll notice another difference between the way I teach it and the way Brian teaches it. He puts emphasis on the last digit, which works well for performing the feat this way. In my version, I teach how to work out the first few digits, as you'll need those first when giving the answer verbally. This is a good lesson in the benefits of changing your point of view!

This week, Diamond Jim Tyler demonstrates a new take on an old trick. Regular Grey Matters readers won't be surprised to learn that I like it because it's based on math, and it's very counterintuitive. We'll start with the new video, and then take a closer look at the trick.

This week's Scam School episode is called The Collective Coin Coincidence, and features Diamond Jim Tyler giving not only a good performance, but also a good lesson in improving a routine properly:



Brian mentions that this was an update from a previous Scam School episode. What he doesn't mention is that you have to travel all the way back to 2009 to find it! The original version was called The Coin Trick That Fooled Einstein, and Brian performed it for U.S. Ski Team Olympic gold medalist Jonny Moseley. It's worth taking a look to see how the new version compares with the original.

Brian and Jim kind of rush through the math shortly after the 4:00 mark, but let's take a close look at the math step-by-step:

Start - The other person has an unknown amount of coins. As with any unknown in algebra, we'll assign a variable to it. To represent coins, change or cents, we'll use: c

1 - When you're saying you have as many coins (or cents) as they do, you're saying you have: c

2 - When you're saying you have 3 more coins than they do, the algebraic way to say that is: c + 3

3 - When you're saying you have enough left over to make their number of coins (c) equal 36, that amount is represented by 36 - c, so the total becomes: c + 3 + 36 - c

Take a close look at that final formula. The first c and the last c cancel out, leaving us with 3 + 36 which is 39. If you go through these same steps with the amount of coins (in cents, as it will make everything easier) as opposed to the number of coins, it works out the same way. This is what Diamond Jim Tyler means when he explains that all he's saying is that he has $4.25 (funnily enough, he says that just after the 4:25 mark).

As long as we're considering improvements, I have another unusual use for this routine. If you go back to my Scam School Meets Grey Matters...Still Yet Again! post, I feature the Purloined Objects/How to Catch a Thief! episode of Scam School, which I contributed to the show. It's not a bad routine as taught, but my post includes a tip which originated with magician Stewart James. This tip uses the Coin Coincidence/Trick That Fooled Einstein principle to take the Purloined Objects into the miracle class! I won't tip it here, so as not to ruin your joy of discovery.

Would you believe tha another of my contributions has made it on to Scam School again? It was 2 other recent Scam School submissions that spurred me to restart Grey Matters, so it's looking like that was the right move.

Even if you've seen this week's Scam School episode, you may want to take a look at this post, as I'm going to give a few tips that may make this routine easier to learn.

Let's get started right away with this week's Scam School episode with a trick I dubbed “Out of Control”!



Quick side note: On one hand, I love being promoted as "the genius". On the other hand, I can't help but think of “genius” in this context.

This trick is actually a combination of two idea from two men who have far more a right to be called genius than me. The dealing procedure comes straight from Jim Steinmeyer's routine Remote Control, as published in Invocation #43 and the May 1993 issue of MAGIC Magazine. If you check those sources out, you'll see that not much has changed, as the original involves spelling the word C-O-L-O-R, and using the 9th card.

I combined this trick with a technique from Simon Aronson's “Try The Impossible” called Simon's Flash Speller. It's this part that may help make it easier to work out what you need to do. First, you'll need to quickly work out how many letters are in the name of the turned-up card. Here's the starting point:

• For clubs, remember: 11 letters
• For hearts or spades, remember: 12 letters
• For diamonds, remember: 14 letters

Yes, I did take more than an extended break after my 10th blogiversary post, not posting at all in 2016, but with Pi Day coming around again, it seemed like a good time for a return! I probably won't have a regular posting schedule for sometime, as other commitments are keeping me busy, but I promise not to ignore this blog as I have been for about the past 1½ years.

While I've been thinking about returning on my own, one force that really pushed me over the edge to return to blogging is Brian Brushwood from Scam School. He recently used two of my submissions, and even encouraged people to go to Grey Matters!

The first of these two was posted back on January 18th. This was an update to Penney's Game, which was first taught on Scam School in their Sept. 15, 2010 episode (to which I also contributed!). Replacing cards with coins not might seem like such a big deal, it actually does have a significant effect. Coins can't run out of heads or tails, but cards can run out of reds and blacks. Watch the full episode and read more detail about the game here to understand it better.



Moving from cards and coins, we turn to dice. Here's an unusual dice scam from their March 1st episode. This one involves people picking how many dice of 5 they want, leaving you with at least 1 of them, and you're always able to predict the outcome. It seems like the odds are against you, but watch closely. At about the 3:30 mark, Brian asks the ladies how they think it's done. The theories included things such as particular numbers to pick, loaded dice, and changing the prediction based on previous evidence. The answer, as it so often is, is simple probability.



Even with all the troubles people have with most branches of math, probability is often the easiest to misunderstand and get wrong. As a result, many probability concept come across as counter-intuitive.

That's all for now. But I assure you, the time between this post and the next post will not be as long as the last post and this one.

Imagine having someone think of a number from 1 to 100, having them cube the number using a calculator, telling you only the result, and you're able to calculate the cube root of their result (the original number they put in the calculator)!

Learning to work out cube roots of perfect cube is an impressive feat, but it's far less difficult than it appears.

We'll get right to the method, taught in the video below. You can read the MindYourDecisions.coom post New Video – Calculate Cube Roots In Your Head for further details.



Over in the Mental Gym, I have a more detailed tutorial on working out cube roots of perfect cubes, including a cube root quiz.

Back in March of 2013, Scam School also taught the cube root feat in their own unique way. If you like this feat and want to take it a step further, check out Numberphile's fifth root feat tutorial. Surprisingly, this is even easier than the cube root feat!

It's time for October's snippets, and all our favorite mathematical masters are here to challenge your brains!

• I'm always looking for a good mathematical shortcut, in order to make math easier to learn. More generally, I'm always looking for better ways to improve my ability to learn. I was thrilled with BetterExplained.com's newest post, Learn Difficult Concepts with the ADEPT Method.

ADEPT stands for Analogy (Tell me what it's like), Diagram (Help me visualize it), Example (Allow me to experience it), Plain English (Let me describe it in my own words), and Technical Definition (Discuss the formal details). This is a great model for anyone struggling to understand anything challenging. This is one of those posts I really enjoy, and want to share with as many of you as I can.

• If you enjoyed Math Awareness Month: Mathematics, Magic & Mystery back in April, you'll love the 31 Tricks and Treats for October 2014 in honor of the 100th anniversary of Martin Gardner's birth! Similar to Math Awareness Month, there's a new mathematical surprise revealed each day. It's fun to explore the new mathematical goodies, and get your brain juices flowing in a fun way!

• Over at MindYourDecisions.com, they have a little-seen yet fun mental math shortcut in their post YouTube Video – Quickly Multiply Numbers like 83×87, 32×38, and 124×126. As seen below, it's impressive, yet far easier than you might otherwise think:



They've also recently posted three challenging puzzles about sequence equations that you might want to try.

• If that's not enough, Scam School's latest episode (YouTube link) at this writing also involves three equations. If you have a good eye for detail, you may be able to spot the catch in each one before they're revealed:



That's all for this October's snippets, but it's more than enough to keep your brain puzzled through the rest of the month!

Many of you are spending today with your mother in honor of Mother's Day, so I won't strain your brain too much today.

In fact, I'll just leave a few free magazines on the table for your perusal when you have some time later.

I'll start with the brand new Recreational Mathematics Magazine. This magazine is available as a whole PDF, or as PDFs of individual articles. The first article that caught my attention here was “The Secrets of Notakto: Winning at X-only Tic-Tac-Toe”. It caught my attention because I'd written about Notakto strategy 2 years ago, including how to win playing on 1 or 2 boards, and then how to win when playing on 3 or more boards.

Don't let me rob you of the joy of discovery, however. The other articles, including the one about Lewis Carroll's mathematical side, the one about vanishing area puzzles, and others are all waiting to be discovered.

The next math magazine I'd like to draw your attention to is Eureka, published by the Archimedeans, the Mathematical Society of the University of Cambridge, since 1939. New issues are being made available online for free by mathigon.org. This is no minor mathematical publication, either. It was the Archimedeans' Eureka magazine that, back in October 1973, had the honor of being the first to publish John Conway's Doomsday Algorithm for calculating the day of the week for any date.

Generally, The College Mathematics Journal isn't available online for free, but they have generously posted the full contents of their January 2012 Martin Gardner issue online for free! It's full of the kind of recreational mathematics which Martin Gardner loved and Grey Matters readers are sure to appreciate and enjoy. There are too many articles to single any one out for special attention, so I suggest jumping in and seeing what catches your eye first!

The final magazine I'll set out for your perusal isn't a mathematical magazine, but rather a magic magazine called Vanish, which is free to download, or read online as a page-flipping e-magazine. The reason I'm including it here with math magazines is because of Diamond Jim Tyler's article on “The Game of 31”. This is variation of our old friend Nim. For a Nim variation, 31 has a surprising amount of its own variations, including a dice version, a finger dart version, and a version which you can still scam someone after teaching them the secret!

That's all for now, so I'll wish you happy reading!

Scam School has been scammed! Somebody slipped Brain Brushwood some counterfeit coins, and he needs your help to separate the counterfeit coins from the real ones.

OK, this is really just the start of a puzzle, but it's a rather fascinating puzzle. Just when you think you've got the hang of the puzzle, another version can come along and make things tougher.

We'll start by jumping right in to the counterfeit coin puzzles as presented on this week's Scam School (alternative YouTube link):



All in all, not a bad pair of puzzles. For the second puzzle, I would make sure to keep the weighed coins in separate piles, of course, so I can make sure to round up all of the counterfeit coins.

Let's add a new dimension to that second puzzle, just to challenge your thinking. What if, instead of 1 bag holding all counterfeit coins, there were an unknown number of bags? As in the original puzzle, each bag holds either all real coins, each weighing exactly 1 gram, or all counterfeit coins, each weighing 1.1 grams. You still have only one weighing to find out which bags, if any, contain counterfeit coins.

If you want to try and work this out for yourself, stop reading here, as I discuss the solution below.

.

.

.

.

.

.

.

If you think about it, what you really need is a series of yes-or-no answers for each bag in a way that allows you to get this information in a single weighing. How do you achieve this?

Our old friend the binary number system comes to the rescue! If you need a quick understanding of binary numbers, watch Binary Numbers in 60 Seconds.

The solution is take 1 coin (2⁰) out of the first bag, 2 coins (2¹) out of the second bag, 4 coins (2²) out of the third bag, and so on. For each bag, you double the number of coins taken from the previous bag, all the way up to 512 (2⁹) coins taken from the 10th bag.

You'll probably note that it's easier to denote the bags as 0 through 9, instead of 1 through 10. With the bags numbered 0 through 9, we can just remember that we take 2^x coins out of bag number x.

How does taking coins out in powers of 2 help? First, consider the weight we'd get if all the coins were real. We'd have 1,023 coins weighing a total of 1,023 grams. Any weight over 1,023 grams, then, can be attributed to the counterfeit coins.

Let's say we try this out, and find that we have a total weight of 1,044.7 grams. Take away the weight of the real coins, and we're left with 21.7 grams extra. At 0.1 grams extra for each of the counterfeit coins, we now know there are 217 counterfeit coins among the 1,023 coins.

That's great, you might say, but we still don't know which bags are counterfeit. If you stop and think for a minute, you may have more information than you think. First, since you took 512 (2⁹) coins out of bag 9, the coins in that bag couldn't be counterfeit. If they were, you'd have a minimum of 512 counterfeit coins in the total.

The same argument could be made for bag 8, from which you removed 256 coins (2⁸). Still, isn't it difficult to work out all the possibilities for the remaining bags?

No, and I can explain why in a very simple way. In our everyday decimal system, how many ways are there to write the number 217? There's only one way, of course, and that's by writing a 2 in the hundreds place, a 1 in the tens place, and 7 in the ones place. The same is true for any other base, include base 2 (binary).

There's only one way to write the binary equivalent of the decimal number 217. To find out what it is, you can either do a binary conversion with the help of a tool such as Wolfram|Alpha, or, if you've been reading Grey Matters long enough, do the conversion in your head.

Done either way, the binary equivalent of 217 is 11011001, but what does this tell us? Each of these numbers represents one of the bags. To be fair and include all 10 bags, we should write it as a 10-digit binary number, 0011011001, and arrange each number under its corresponding bag like this:

9 8 7 6 5 4 3 2 1 0
0 0 1 1 0 1 1 0 0 1

In their recent 311th episode, Scam School featured an interesting scam that warrants a closer look.

After all, there's nothing fairer than a game board consisting solely of dots, right?

Check out Scam School's 311th episode, and see how fair it comes across versus the sneaky approach behind the game:



That's amazing and sneaky, but how would you even go about working out the math behind it?

Never fear, because Numberphile is about to come to the rescue! In a recently posted video, they take a look at a similar type of game called Brussel Sprouts. The math behind it is less complicated than you may expect:



As many regular Grey Matters readers realize, I love it when something fun like this can lead to a better understanding of the math behind it.

According to the above video, we can also expect a more detailed video about the Euler characteristic mentioned in this video. You can also find out more about Brussels Sprouts and other dot games in Martin Gardner's books Mathematical Carnival and The Colossal Book of Mathematics.

Have fun and play around with these games. You never know what else you may learn!

No, I'm not just putting random filler text in the title.

Ever hear the expression “the oldest trick in the book”? In this post, you'll learn about a card trick that certainly qualifies, as it's known to date back at least as far as 1769!

I'll start by letting Brian Brushwood perform and explain his version of this classic routine (YouTube link):



Now, you'll note that Brian teaches this with the words:

MUTUS
NOMEN
DETID
COCIS

BIBLE
ATLAS
GOOSE
THIGH

LIVELY
RHYTHM
MUFFIN
SUPPER
SAVANT

PILLAR
RHYTHM
MUFFIN
CACTUS
SNOOPY

MEACOCK
RODDING
GUFFAWS
TWIZZLE
RHYTHMS
KNUBBLY

If you watched even part of Dr. Arthur Benjamin's lecture in my previous post, you get an idea of the kind of joy and enthusiasm he has for mathematics.

In this post, we feature another new video from Dr. Benjamin, focusing only on Fibonacci numbers. This video is only about 6-7 minutes long, so you can enjoy it even if you couldn't find time to watch the complete video in my previous post.

This video is titled Arthur Benjmain: The Magic of Fibonacci Numbers, and is available on YouTube, as well as TED.com:



The presentation you see behind him in this video was created on Prezi.com, and Dr. Benjamin has made it freely available to view there. If you're viewing this on an Apple mobile device, the presentation can be viewed more effectively via Prezi's free apps.

Here on Grey Matters, we have a healthy respect for the magic and fun Fibonacci numbers make possible. Back in 2010, I posted about the classic Fibonacci addition trick, and other fun with phi. A little less than a year later, Scam School featured the same trick. Playing around with it on my own, I even found a little-used way to expand the routine to a list of any length!

Fibonacci numbers have even found their way into Nim, a favorite game here on Grey Matters. There's Fibonacci Nim, which is the standard take-away game with new rules that let you win with your knowledge of Fibonacci numbers. The Corner The Lady version of Wythoff's Nim is probably the most deceptive use of Fibonacci numbers.

In a post a little over a year ago about the principle behind the classic Age Cards magic trick, there's even a James Grime video showing the Age Cards set up with Fibonacci numbers instead of the standard binary approach!

Take some time to explore and play with these magical numbers, and your time will be well rewarded!

Two of our old friends are back on Grey Matters: Scam School and the game of Nim!

This version uses a dart theme, but you don't even have to be good at the game!

Scam School's 294th episode features an impromptu version of Nim using your fingers as darts, and a napkin as a dartboard:



I'm not going to analyze the play too deeply, as this is simply stanard single-pile Nim, as defined in the first Secrets of Nim post. In addition, it's also essentially the same as Dice Nim, as taught in 250th episode of Scam School, but played to 31 instead of 50.

The Single-Pile Nim tab of the Nim Strategy Calculator can run through the strategy with the following settings:

• Player who makes the last move is the: Winner
• Maximum number of objects (limit): 31
• Nim Game is played: up to limit
• Number of objects used per turn ranges from 1 to: 6

After clicking the Calculate Nim Strategy button, the calculator will return the same strategy in the above video. You can even play around with higher or lower totals, and the numbers you're allowed to use on each turn. What would the strategy be if you allowed the numbers 1 to 7, or 1 to 8? How would it affect the strategy if you play to 32, 35, or 40?

The more interesting aspect of this version of the game, at least to me, is the use of a small dartboard instead of dice. One on hand, this change is merely cosmetic, as it doesn't affect the strategy in any way itself. The psychology on the audience, however, is completely different. Dice aren't as common as dartboards in bars and pubs, so the dart theme makes better sense. In addition, the impromptu nature of drawing on a napkin suggests fairness than someone who brings their own dice, which can suggest that the person suggesting the game has practiced it.

31 also seems to be a favored number for Nim players, most likely because it's high enough to allow you to use your winning strategy, but low enough to keep the game too short for anyone to catch on. In another 31 version using playing cards, the unique nature of playing cards allows you to catch even those who think they know the secret.

As you can see, just examining all the seemingly minor changes in Finger Dart Nim can give you a better understanding of the overall idea behind multiple variations of Nim. If you enjoyed this version, please explore the amazing variety of Nim games taught in other posts here at Grey Matters!

I've been having a rough week this week, between being offline for most of Sunday, and dealing with a family emergency the rest of this week.

A little magic always cheers me up, so that's the focus of today's post.

Let's start with a quick opener that's a little different from most Grey Matters fare.

It's this week's episode of Scam School, and there's no math or memory involved. It's just pure, classic sleight-of-hand, developed by Marcus Eddie, who is teaching his SPLINTER! routine:



Now that you're awake and got your splinters removed, let's turn to a little more traditional magic. Not only are playing cards involved, but there is a mathematical basis, so it's probably a little more what you're used to on this site.

Our old friend Card Colm has been experimenting with the Gilbreath Principle. His latest results are in his most recent column, Rosette Shuffling Multiple Piles. It turns out that using a special adaption of the Riffle Shuffle, known as the Rosette Shuffle, it is possible to mix 3 piles of cards together, and still get startlingly predictable results!

I do have to keep this short, due to all I'm dealing with this week, but I hope you found these magical tidbits as enjoyable as I did!

Ever try and solve a puzzle involving physical objects, such as cards, coins, or chips, with your eyes closed?

Despite the challenging conditions, this is just what you're going to learn in today's post! Even better, the solution is ingeniously simple.

Let's get right to the puzzle, shall we? It's presented in its basic puzzle form in this week's episode of Scam School. Before you play the video, grab 10 cards, and try and work out the puzzle before viewing the solution.



Did you solve it on your own? If so, congratulations! Even if you didn't, you should at least have viewed the solution before moving on from this point.

Now, let's try a similar puzzle, but we're going to throw a bit of a curve into it. Add another card to your group of 10, so you now have 11 cards. Just as before, mix them so that some are face-up and some are face-down. You challenge is to separate these 11 cards into 2 groups, so that both groups contain the same number of face-up cards. James Grime has a video puzzle version of this, using chips instead of cards.

Why is this version so much more challenging? There's no way to divide 11 into 2 piles of the same number of cards (or chips, coins, etc.), and that was such an important feature of the previous version.

Once again, try working it out for yourself with cards in hand, before peeking at the solution. When you're ready, here's James Grime with the solution to dealing with odd numbers of objects:



As you see, you need extra information to solve this latter version of the puzzle.

While this does make an interesting puzzle, it can also be turned into a magic trick quite easily. Bob Hummer and Jack Yates created a trick using this principle called Time Will Tell. That version uses 12 pennies, arranged in a clock formation, with some marker identifying one of the pennies as the 12 o'clock position. The performer turns away from the proceedings, and asks the volunteer to turn over any 6 pennies. Without looking at which coins were flipped and without asking any questions, the performer then separates the pennies into 2 groups, both with the same number of heads and tails in each group.

In the original version of the trick, you set up all 12 coins with heads up. As you've seen, however, this really doesn't matter. They can start with any number of heads and tails showing, and have them do several prescribed flips, and you can still separate the groups into 2 sets with equal amounts of heads and tails. Instead of simply having your back to the proceedings, you can even do this over the phone.

Whether you see it as a puzzle or a magic trick, this is a fun exercise to analyze and figure out why it works. If you develop any interesting versions of this routine, I'd love to hear about it in the comments!

Today, you'll learn an interesting new mathematical game in which the object is to obtain 3 numbers that total 15.

Well, it's not exactly new. As a matter of fact, it's something with which you are probably very familiar!

Let's jump right into the game. Watch the 274th episode of Scam School (YouTube link) below, and once they've played the game and the ad starts, stop the video and ask yourself if you can come up with any simple way to play and win the game. Once you've either figured it out or given up, go ahead and watch the remainder of the video.



Were you surprised? Yes, it's your old friend Tic-Tac-Toe (or Naughts and Crosses)! This is just one of numerous ways that have been developed to disguise the true nature this classic game. It's almost embarrassing how effective such a simple disguise can be.

Last August, I delved into strategy for the game of 15, with Part 1 teaching you the basics and how to win when you go first, and Part 2 teaching you how to win, or at least avoid losing, when you go second.

I created those posts so you can ideally play the game without ever referring to a Tic-Tac-Toe board. There's still one hitch with the game, however, and you can see it in the above video. When the game is introduced, it's explained as a mathematical game, and people immediately get apprehensive. The game is already unfamiliar, and the mathematical aspect often just adds stress.

Since you generally want to put people at ease, perhaps it's best to make the game seem more familiar. Instead of using 15 as the magic total, use 21! How would you do this? Simply increase the numbers in each part of the magic square by 2. Instead of the top row being 8, 1, and 6, you change it to 10, 3, and 8. The whole square should look like this:

10   3   8
 5   7   9
 6  11   4

The order of operations is one of those things in elementary school math that probably caused you great frustration.

The order of operations, as taught in the US, is: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. This is usually taught with a mnemonic, such as PEMDAS (the first letters), or “Please Excuse My Dear Aunt Sally” (the first letters with an easily remembered phrase). Elsewhere, you may have learned BEDMAS (B is for Brackets, same as parentheses), BODMAS (O is for Orders, same as exponents), or BIDMAS (I is for Indices, same as exponents).

Having dealt with them before, you may be glad that you finally mastered the order of operations. However, they do still hold quite a few surprises.

The first surprise is that, as explained in the minutephysics video below, the order of operations is wrong. More accurately, it's not so much wrong as it is a weak attempt to mechanize the logic of how to handle mathematical equations.



Once you think you've got a handle on the ideas behind the order of operations, it's time to put your understanding to the test.

In what they deem to be their hardest puzzle ever, Scam School challenges you in just this way. Below is a number puzzle in which all the numbers are provided for you. The challenge is to add in the right operations so that each set of numbers total 6. Watch the instructions up to about the 3:30 mark, and try and solve it without any of the hints given later.



I'm proud to say that I managed to get all 10 answers working on my own without hints. I did, however, come up with a different approach for the 8s. As long as you've already tried and either succeeded or given up, you can see my answer guilt-free.

As Diamond Jim Tyler and Brian Brushwood mention, this puzzle doesn't seem to have hit the states until now. It is a good one to have in your arsenal, especially once this episode is several months old. I can't be sure, of course, but I'd like to think Martin Gardner would've appreciated this puzzle.

There's an old magic trick out there that's been in the public domain for so long, its origins seem to have been lost.

In the classic version of the trick, a card is chosen, and a mysterious person is called. Somehow, this person is able to name the correct card, despite not even being in the same room, or even the same state or country!

Magicians know this trick as “The Wizard”, as most of them learned the version by that name from the book, Scarne on Card Tricks. You can read that particular trick for free online (page 42, page 43).

As with many tricks, the presentations grow and change over the years. Some magicians also know this same trick as “The Phantom” or some other equally mystic name. When Scam School taught this routine (YouTube link), their figure of choice was a secret member of a government conspiracy:



If you think about it, any bit of data which can be identified by two simple pieces of information, in a manner similar to grid coordinates, can be coded in a similar fashion. It's quite obvious that playing cards can be broken down into 2 bits of information, their value (Ace through King) and their suit (clubs, hearts, spades, diamonds). What if the data to be coded didn't have 2 such obvious factors? If we could manage that, this routine could be even more deceptive!

Max Maven developed a version called “Remote Pager” in which a word is chosen from the following old letter

Impossible, but true! A demonstration of intuition, custom tailored for you by Mister Zulu. Cnoose any word in the paragraph of at least four letters. After you choose a word, contact me by phone. Believe it or not, I'll announce the word you are thinking of! Imagine tne surprise ~ but be on guard: I presume my demonstration is going to haunt you...

Mister Zulu

Scam School's newest episode is right down the alley of many regular Grey Matters readers.

Their 265th episode teaches how to memorize a list of 20 items very quickly. This is a classic feat and a classic technique, but it's a rare treat to see it actually performed.

You can find the episode on Scam School's own site, and on YouTube, as well. Let's get right to the memory feat and the explanation:



As you see it performed in the video above, it's a pretty bare bones technique. That's a great way to learn it, but there are other handy tips that can take it to another level.

First, as the items are called out, make sure to specifically ask for objects you can encounter in everyday life. That way, you don't get hard-to-picture images such as sickle cell anemia, as in the video. Hopefully, you don't encounter maggot-infested tacos, either, but at least it's easier to picture.

Also, ask for more details. If someone calls out a car, ask for a specific model of car, or even the color. This additional level of detail makes the feat seem more difficult, but actually makes the image more vivid, and thus easier to remember.

In the video above, you always see them calling out numbers, and having the item given in return. As long as you've formed your images effectively, there's no reason you can't have them call out the items and give the number in return, as well.

Once you've memorize the list, and they're starting to call out numbers or items, each time you recall the image, imagine your mnemonic frozen in a block of ice. If someone calls out 2, and you recall that's a unicorn, imagine the unicorn with a shoe for a horn frozen in a block of ice. Have them call out numbers or items until they've covered about 60% of the list or so. After that, you can recall the items and numbers that were never called by mentally going through the list from 1 to 20, and recall which images weren't frozen! This is a great finish!

You don't have to use rhyming pegs, of course. On my Memory Basics page, you can also learn shape-based pegs, or even the Major System, which allows you to turn any number into a vivid image.

If you've already learned pegs from something else, such as the pegs I teach for 1 through 27 in my Day One calendar feat, you can quickly adapt those, too.

To learn more about various peg systems, I have one YouTube playlist focusing on simple peg systems, and another focusing specifically on the phonetic peg system.

Yes, this feat requires a little more work than most Scam School feats, but it's worth it not only for the results, but also for practical everyday uses!

Scam School has just covered one of my favorite mathematical feats of all time!

Doing cube roots in your head is a skill that seems very impressive, but as you'll see in the video, you can learn the basics and start praciticing in less than 10 minutes.

If you take a number x and multiply by itself two more times, x × x × x, or x³ (x cubed), then x³ would be referred to as the cube of x, and x would be referred to as the cube root of x³. For example, since 3 × 3 × 3 = 27, we say that 27 is the cube of 3, and conversely that 3 is the cube root of 27.

Now that we're clear on the terminology, here's the Scam School episode about doing cube roots in your head:



Over at the Mental Gym, I've had a post explaining this feat for quite some time. Once you feel confident doing cube roots in your head in this manner, you can move on to 5th roots! The approach is similar, but strangely, doing 5th roots is actually easier than cube roots!

Once you master squaring numbers ending in 5, you can even handle square roots in a similar manner, as well.

I've always thought it was somewhat amusing that lower roots were more challenging to approach than higher roots with this approach.

How far have people taken this approach? There are several people who have practiced finding the 13th root of a 100-digit number, and beyond! For a feat like this, any time under 2 minutes is considered an excellent time. It is, of course, far more challenging to turn this one into an entertaining bar stunt.

Practice this one and have fun displaying your new-found skill for your friends!

Imagine you and a friend are performing some of the feats you've learned from, say, here on Grey Matters, on Scam School or from James Grime, and you're accused of witchcraft! You're hauled off by the authorities, and given a test to see if there's a supernatural connection between the two of you.

You and your friend are put in separate rooms, and each of you is given a single playing card taken from a shuffled deck. You must state whether you believe that your friend has been dealt a card of the same color or a different color. Naturally, your friend is asked the same question.

The judgement of the outcome, however, is very harsh. If both you and your friend make correct statements, that is considered sufficient proof of supernatural powers, and you are put to death. If both you and your friend make incorrect statements, you are released and the charges of witchcraft are dropped. If one makes a correct statement and the other makes an incorrect statement, the results are considered inconclusive, and the two of you are tested again, up to 26 times (since a pair of cards are used each time, and there's 26 pairs of cards in a standard deck).

Considering that, at any point when your guesses match, you would be put to death, it's probably a good idea at this point to consider probabilities of survival.

If you and your friend just make random guesses as to the other's card, what are the chances of surviving?

Technically, if the cards aren't returned to the deck after each round of guessing, you'd have to do some math to work out the probabilities of the distribution of cards remaining in the deck at each point, but we're not going to take that consideration. In our calculations, we'll assume that the cards from each round are returned to the deck, and the deck is shuffled again. In mathematical terms, the dealing of 2 cards in each round will be considered an independent event, as opposed to a dependent event.

If you and your friend make a random guess on the first round, there's a 25% chance of you both being correct, and therefore being killed. There's a 25% chance of you being set free, and a 50% chance of being required to take another test.

Note that this means you have a 75% chance of surviving the first round, but only a 50% chance of being required to being submitted to another round of testing.

That's simple enough, but how do we take the 2nd round into account? For the second round, you still have a 25% chance of dying, but you only have a 50% of getting to that test in the first place. You add the 25% of dying in the first round to the 50% × 25% chance of dying in the second round. We work this out as .25 + (.50 × .25) = .25 + .125 = .375, or a 37.5% chance of being killed by the end of a second round.

For a third round, the logic is the same, and the equation becomes .25 + (.50 × .25) + (.50² × .25) = .25 + .125 + (.25 × .25) = .25 + .125 + .0625 = .4375, or a 43.75% chance of dying after 3 rounds.

Notice that each time, the chance of dying is increasing, so already this test doesn't look promising. Also note that the 50% chance for any given step is effectively the number of the round minus 1. This is even true for round 1, since ¹⁄₂⁰ = 1 (Why is that, anyway?).

So, for 26 rounds, we have .25 + (.50 × .25) + (.50² × .25) + ... + (.50²⁴ × .25) + (.50²⁵ × .25). The mathematical shorthand for this is:



Running this through Wolfram|Alpha, we find that the probability of being put to death by the 26th round is 0.499999992549419403076171875, or a roughly 49.99% chance. That's not much better than a “heads you live, tails you die” coin flip.

Is it possible to improve on the 50/50 odds of survival? Fortunately, yes it is, and Scam School's Brian Brushwood, with the help of a friend, prove they can survive through all 26 rounds. Can you guess how they do it before their secret is revealed?



With all the complex math, it seems like the answer should be equally complex, but the answer is dead simple. When you think about it, they approach used in the video above must ALWAYS render the test inconclusive, thus allowing them to live through 26 tests.

You might notice that, contrary to the initial description of the problem, the cards aren't returned to the deck before the next round. However, the strategy nullifies the chances of dying in the first place, so the importance of the changing deck composition isn't important any longer.

If you have any thoughts about this puzzle or the math behind it, let's hear about it in the comments!