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!
• 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:
• 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!
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.
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
Remember, that's just a starting point. From here, you may need to adjust the amount of letters, but only by adding or subtracting 1! What happens with which amount of letters?
If the value spells with 4 letters (four, five, nine, jack or king): Don't make any adjustment to the number of letters.
If the value spells with 3 letters (ace, two, six, or ten): Subtract 1 from the number of letters.
If the value spells with 5 letters (three, seven, eight or queen): Add 1 to the number of letters.
Once you've made that adjustment, you now know how many letters are in the card's full name! It seems difficult at first, but gets much easier with practice. 5 of Hearts? Hearts is 12 letters, and no adjustment needed, as F-I-V-E spells with 4 letters. 7 of Diamonds? Diamonds is 14, plus 1 for a 5-letter value (S-E-V-E-N), that's 15 letters. 10 of Clubs? Clubs is 11 letters, minus 1 for a 3-letter value (T-E-N), that's 10 letters.
From here, there are 6 ways the trick can go, so you have to quickly recall which out to use. There's really only 2 substantially different outs, with 12 and 13 letters. All the other outs are just modifications of those two. First, how do you handle cards whose names spell with 12 and 13 letters?
For 12 letters: Spell the name, and take the top card of those still in your hand.
For 13 letters: Spell the name, and take the last card that was dealt off.
How do you adjust this process for 14 or 15 letters? It's simple, you spell the value and suit without spelling O-F in the middle. This reduces any 14-letter card names to 12 letters and reduces any 15-letter card names to 13 letters. If you're keeping track, we've already covered 4 of the 6 possible outs!
The last two possibilities involve 10- and 11-letter card names:
For 10 letters: Spell T-H-E before the card name (such as T-H-E-A-C-E-O-F-C-L-U-B-S), resulting in 13 letters.
For 11 letters: Deal the turned up card aside, and spell its name with the next 11 cards, resulting in 12 cards being used.
Between determining the number of letter and which out to use, it can all seems a little confusing. However, like any good magic trick, it does take practice. The smoothness with which you can make this trick flow is the key to its deceptiveness.
For those who are wondering how the math of this trick works, the first deal is obvious. The selected card starts at the 10th position, of which 4 are dealt off, so it winds up at the 6th position. It's the second deal that is highly counterintuitive. In fact, watch the video starting at the 3:30 mark, and when they realize that the card winds up as the 13th card despite the two different spellings, Matt (the gentleman with the long beard, who has created his own original magic, as well!) comments, “My brain's breaking a little bit now!”
To explain, imagine you're doing this trick with cards numbered from 1 to 18, in order, with card 1 on top. If you deal 7 cards, as in the R-E-D-S-U-I-T possibility, as calculated on Wolfram|Alpha, you see that the 6th card from the top winds up being the 6th card from the bottom. If you deal 9 cards, as in the B-L-A-C-K-S-U-I-T possibility, Wolfram|Alpha tells us that, once again, the 6th card from the top winds up as the 6th card from the bottom.
It only seems like the different amount of letters should change the location of the card, but it actually has the same effect, as long as you deal past the selected card! If you have any further questions about this routine, or anything else on this blog, let me know in the comments below.
It involves playing cards and dice. Because it's mathematically based, however, you might just fool yourself while performing it. You'll really only fool yourself if you don't analyze the math behind the trick, which is exactly what we're going to do in this post.
There's quite a bit going on here, so let's break this down piece by piece.
CARDS: Let's ignore the dice for the time being, and focus only on the playing cards. During the dealing process, cards fall into 1 of 2 categories: Either they're dealt individually, or they're in the remainder that is placed on top of the stack. Let's look at each of these categories separately.
DEALT CARDS: We'll start with the first example from the video, where 10 cards were used, 7 of which were dealt. What happens to those 7 cards. The card in position 1 is dealt first, and will obviously become card 10. The card at position 2 will wind up as the 9th card, and so on. Take a look at where these 7 cards end up in the final stack:
Starting position Ending position
----------------- ---------------
1 10
2 9
3 8
4 7
5 6
6 5
7 4
See the pattern? The starting position plus the ending position always add up to 11, in this example! Why 11? What would happen if we used, say, 18 cards instead? Well, 1 would become 18, 2 would become 17, and so on. In that example, everything totals 19. The resulting total will always be 1 more than the number of cards involved.
We can use this to work out a formula for the dealt cards. The total number of cards plus 1, minus the starting position of a dealt card, will give you the ending position. So, in our example with 10 cards, of which 7 are dealt, we can work out the total number of cards (10) plus 1 (equals 11) minus the original position (say, 3, so 11 minus 3 equals 8) gives us the ending location of that card (so, we can easily say that, in this example, the card that started at the 3rd position will wind up at the 8th position).
A simpler way to say this is to use S for starting position, E for ending position, and N for the total number of cards. So, our formula for dealt cards could be written as E = (N + 1) - S
That's OK for the dealt cards, but what about the undealt cards?
UNDEALT CARDS: If you have 10 cards with 7 cards dealt, cards 8, 9, and 10 will not be dealt. They are simply placed on top of the dealt cards as a group.
In this case, the original 8th card becomes the first card, the 9th card becomes the 2nd card, and the 10th card becomes the 3rd card. Let's chart these positions and find a pattern:
Starting position Ending position
----------------- ---------------
8 1
9 2
10 3
This pattern is even simpler! The starting position, minus the number of dealt cards, gives the new ending position. Using D for the number of dealt cards, and the same variables from the first formula above, we have E = S - D for the undealt cards.
OK, we've got 2 formulas to handle our 2 cases, so let's bring the dice back in.
DICE: Since the dice are used to choose random numbers, we'll refer to them with the letter R, and since high, low, and medium numbers are important, we'll use R1 for the dice with the lowest number, R2 for the dice with the middle number, and R3 for the dice with the highest number.
FOLLOW THE TOP CARD: As explained, knowing the top card is the key to this trick, so we're only going to follow that particular card.
How many cards are used? In the first performance, the dice rolled with R1=2, R2=3, and R3=5, which means that 10 cards are used. More generally, the dice total determines the number of cards used, so N (total number of cards) = R1 + R2 + R3. In our example, this was 5 + 3 + 2 = 10 cards.
Step 1: Note that, when the cards are dealt initially, the top card ALWAYS becomes the bottommost card. So, the starting position for the predicted card is always at the bottom as well. In other words, S (the starting point) = R1 + R2 + R3, as well. So, the predicted card starts at 10 in the first example. In other words, S = 10.
Step 2: For the next deal, R2 is removed (3 in the first example), and R1 + R3 dice (D = 7) are dealt. The card as starting position 10 (S=10) is obviously not going to be dealt, so we'll apply the undealt card formula (E = S - D). E = 10 - 7 = 3. So, the predicted card winds up at position 3 in the example.
Let's look at this more generally. The card starts out at position S, which is also R1 + R2 + R3. You're dealing D cards, and D = R1 + R3. So, E = S - D can be re-written as E = R1 + R2 + R3 - (R1 + R3), which simplifies to E = R1 + R2 + R3 - R1 - R3, which further simplifies to E = R2.
In other words, after the first dealing of D cards, the predicted card will wind up at the position denoted by R2, the removed dice! That's interesting and unexpected.
Step 3: So, now we want to see what happens to the card at position R2 on the next deal. Because R2 must always be less than the total of R1 + R3 (removing the middle number ensures this), R2 will always be among the cards dealt in this phase. This means we need to follow the dealt card formula from above (E = (N + 1) - S).
In the 10 cards total/7 cards dealt example, we're now tracking the 3rd position, so E = 10 + 1 - 3, which simplifies to E = 8, so our 3rd card winds up in the 8th position.
The current general starting position is, as we already know, is R2. We can turn the formula, then, into E = (N + 1) - R2. Further, since the total number of cards, N, is R1 + R2 + R3, we can change the formula into E = R1 + R2 + R3 + 1 - R2. This simplifies into E = R1 + R3 +1.
Interpreting that general formula, that means the predicted card has now moved to the position denoted by the remaining dice plus 1. Sure enough, in our running 10/7 example, 7 remains, and the predicted card has moved to the 8th (7 + 1) position!
Step 4: This should be pretty clear. The card we're following is at position R1 + R3 + 1, and we're going to deal R1 + R3 cards off of it. In our example, the card we're tracking is at position 8, and we're going to deal 7 cards from above it. Either way, the card will be moved to the first position!
SHORT VERSION: If you read the above carefully, you can start to see WHY this card trick works. The predicted card starts at the bottom of the pile. Next, it moves to position R2, followed by a move to position R1 + R3 + 1, and finally to position 1.
Returning one last time to our 10 card pile (N=10) with 7 cards dealt (D=7) example, you simply set N and D, and the calculation will do the rest. The output you get from this run is {8, 9, 10, 7, 6, 5, 4, 3, 2, 1}. The numbers are the starting positions (S), and each number's placement is their ending position (E). 8 being placed first means that it started in the 8th position and moved to the first position. 9 being in the second position means that the cards which started out 9th has been moved to the 2nd position, and so on.
As a matter of fact, since the total cards (N) and the number of dealt cards (D) stay constant in most routines, you can use this one output to track the card through multiple deals. I'll show you what I mean by following through 3 deals of 7 cards from a pile of 10.
What does {8, 9, 10, 7, 6, 5, 4, 3, 2, 1} tell us about the original 10th card? One quick glance tells us it became the 3rd card (because there's a 10 in the 3rd position. Where does the 3rd card go from there? Another quick glance tells us that, since 3 is in the 8th position, the 3rd card must move to the 8th position next. Finally, what happens to the 8th card? It winds up on top, because we can see the 8th card at position 1!
That's how an entire routine with multiple deals can be explored using only one simple mathematical result!
Thanks for creating and sharing this routine, Presh! I'd love to hear about any variations my readers develop in the comments, as well!
This time around, we've got a round up of math designed to amaze and surprise you!
• @LucasVB is the designer behind some of the most amazing math-related graphics I've ever seen. You can see some of his amazing work at his tumblr site, and even more at his Wikimedia Commons gallery. Even if you don't understand the mathematics or physics behind any given diagram, they're still enjoyable, and may even prompt your curiosity.
• Our old friend @CardColm is back with more math-based playing-card sneakiness! In his newest Postage Stamp Issue post, he presents a sneaky puzzle that you can almost always win. After shuffling cards, the challenge is to cut off a portion of cards, and see how many of the numbers from 1 to 30 you can make using just the values of those cards. It seems very fair and above-board, but the math behind it allows you to win almost every time!
Don't worry, there's no complicated sleight-of-hand in these tricks. Not only does math make them easy, but you don't even have to do any math during the routines, since all the math involved has been worked out ahead of time.
I'll start with the simpler of the tricks. In this first one, you have someone think of any hour of the day, and you tap numbers on an analog watch while they silently count up to 20. When they reach 20, they say “Stop!”, and your finger is on the hour they secretly chose!
At first, the workings may confuse you, but a little experimentation with different numbers will help you understand it. Obviously, this is also true for anyone for whom you perform it, so don't treat this as a big mystery, but rather as a simple and interesting experience.
The next trick, courtesy of Card Colm, is a little more involved. You have someone name any card suit, have a regular deck of cards shuffled, and then the number cards (Ace through 9) are removed in the order in which they're found in the deck. You then make an unusual bet based on divisibility of various numbers formed by those cards.
This trick is called the $36 Gamble, and the method is found in Card Colm's post, The Sequence I Desire. Magic: When Divided, No Remainder. Beyond just the mathematical method, there's plenty to explore under the hood of this routine, including Arthur Benjamin's method for determining divisibility by 7, and a very deceptive shuffling method, which appears fair.
If you enjoy the deceptive shuffles discussed in the above post and its links, you also might enjoy Lew Brooks' book Stack Attack, which features the False False Shuffle. The false shuffle and the routines in Stack Attack mix well with the principles behind the $36 Gamble. In my 2006 review of the DVD of the same name by the same author, you can get a better idea of the contents.
Even though I've only linked to 2 tricks here, practicing them, understanding them, and digging in to the variations I've mentioned is more than enough to get your mental gears turning, so have fun exploring them!
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!
Grey Matters favorite Werner Miller is back, and he's brought more of his amazing mathematical wizardry with him!
If you're not already familiar with him, he's a retired mathematics teacher in Germany who has created some of the most original and compelling magic routines I've ever come across. He is the author of several magic books, including Ear-Marked, which is available in the Grey Matters store.
Starting off, we have a couple of good routines that are perfect as promotional tools, since you can print them on your business cards or brochures, and have people perform them for themselves or others without understanding how they work. There's Vive Le Roi!, which includes several variations of routines where you move your finger from card to card, eventually winding up on a predicted card. You can have two people do this together, as they'll be on different cards until the last card.
Werner Miller also created a very sneaky calculator trick, called You Push the Button... that seems to be a mathematical trick, but isn't. The use of the calculator helps conceal the outright sneaky method.
Getting back to his specialty of mathematical magic, he offers a great routine with dice. It's called Lined Up, and has two different phases, both of which begin with different-colored dice arranged with the numbers 1 through 6 in numerical order. In Phase 1, you have someone choose a die and turn that number face down. After getting the new total of these dice, you announce which color die has been turned over. Phase 2 is similar, except that you have someone choose a die and turn over every die EXCEPT for the chosen one!
I've saved my favorite for last! It's called Ghost Rider, and uses a chess knight and some file cards. One of the file cards is signed, then mixed into the pile and dealt out into a 3 by 3 square. The spectator then uses the knight and their own free choices to find their own signed card! Part of the principle is taught here on Grey Matters in my Knight Shift post, as mentioned generously in Werner Miller's article. His added touches, however, make this a very impressive trick.
• While playing around with memorizing the speed of light in meters per second (299,792,458 m/s), I was originally using the method of using words of a given length to represent a given number (3-letter word to represent 3, etc.). I noticed that it was taking longer to count the longer words, and thought it might be better to combine that technique with words that rhyme.
In other words, use 9-letter words that rhyme with the word NINE, 8-letter words that rhyme with the word EIGHT, and so on. Using sites like WordHippo and RhymeZone, here's what I came up with for the speed of light: “To recombine, valentine, shorten storyline. Do more alive, soulmate!”
That's all for this month's snippets. I hope you enjoyed them!
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.
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!
It seems like Wener Miller just never stops creating!
He's just released E-Z Square 6, the latest in his series of magic square books!
E-Z Square 6 is a bit different from the previous works. Vols. 1-5 each focused on magic squares with a particular theme, such as birthdays, playing cards, and so on. What makes E-Z Square 6 different is that it goes back and updates and improves the methods and routines from past books.
The first routine is an update on the birthday magic square from E-Z Square 1. You start by putting the spectator's age in the center square of a 5 by 5 grid, and then you fill the remaining squares in a seemingly random way. When you're done, the magic total of every row, column, diangonal, and even several cross patterns, total the year the spectator was born! While the effect is the same, the method is greatly improved. Once you have the first few numbers, which is easy enough, the rest isn't much harder than counting.
The next routine is also an update on a bonus, this time on the magic square routine involving a measuring tape from E-Z Square 2. This one is a little sneakier than most of the routines, so it manages to pack an extra punch.
In E-Z Square 5, Werner Miller focused on magic squares with playing cards. The main problem with one of the feature routines, however, is that the resulting 4 by 4 squares usually featured duplicate numbers. In this volume, Werner Miller shows how to solve that problem once and for all, with a little inspiration from Richard Wiseman's The Grid, which also feature playing card magic squares.
Just when you think you've seen everything, the author goes on to teach other playing card magic square ideas with 3 by 3, 4 by 4, and 5 by 5 grids!
This ebook then rounds out with some fun magic square puzzles. One set of puzzles challenges you to cut an existing magic square into 2 smaller magic squares. The other set of puzzles require you to complete magic squares with only a few numbers with which to start. These very same puzzles, I'm proud to say, were first shared by Werner Miller to Grey Matters readers back in 2010 (puzzle 1, puzzle 2, puzzle 3, puzzle 4, answer to puzzle 4).
Technbically, you don't need the previous volumes to get use of E-Z Square 6, but reading this volume will certainly attract your curiosity about all the other routines.
If you're looking for a different take on magic squares, E-Z Square 6, which is also available in German, provides plenty of great routines and food for thought.
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
Now, you can propose a game of face-up 21/blackjack, and people immediately get the idea the goal is a total of 21. It's recognizable, not some weird math game. You explain that the cards 3 through 9, a 10-value card (10, J, Q, or K) and an Ace will be laid out on the table face-up, and you and the other person will alternate taking cards, with the goal of getting exactly 3 cards that total 21.
You can even say that the Ace can be a 1 or an 11. Without the Ace, there are no combinations of 2 cards that add up to 20, so in practice it will always function as an 11 and never as a 1.
If you've learned the strategy for 15 as I teach it in my two posts linked above, there's some simple adjustments to make. Instead of 5, the center square is 7. The even numbers still represent the corners, and the odd numbers still represent the same remaining squares. The simple strategy taught by Brian in the video is also easily adaptable to the game of 21.
Those of you in the US are probably spending Mother's Day honoring your mom, so I'll just sneak a wide variety of snippets in today, and you can check them out later.
• Jan Van Koningsveld, along with Robert Fountain, has released a new book that will be of interest to Grey Matters readers, titled, The Mental Calculator's Handbook (Amazon link). If you're not familiar with Jan Van Koningsveld, he was able to identify the day of the week for 78 dates in 1 minute at the World Memoriad. I haven't had a chance to read this book myself yet, but his reputation does suggest the book is worthwhile.
• Starting back in 2008, I kept track of assorted online timed quizzes, the type of quizzes that ask you how many Xs you can name in Y minutes. I found these so fun, useful, and challenging, I even developed my own timed quiz generator, and even posted several original timed quizzes created with it. However, sporcle.com, home to numerous timed quizzes (despite starting out as a sports forecasting site) has gone and outdone this. Not only can you create your own timed quizzes, you can also embed them on your own site now! Find a quiz you like, for example, this landlocked states quiz, go down to the info box below the quiz, and click on Embed Quiz. A pop-up will ask whether you want a wide or narrow window (minimum width is 580 pixels), and you will be given the proper embed code, which can be used in a manner similar to YouTube embed codes.
• For those of you who do the Fitch-Cheney card trick, as taught on Scam School or YouTube, Larry Franklin has posted a simple tutorial on using Excel to practice this routine. As long as you understand your favorite spreadsheet program well enough, it's also not hard to adapt. It will take a while to create in the first place, but once it's ready, it's fairly easy to use.
• One of the most useful card memory feats to learn is memorizing basic blackjack strategy. Over in reddit's LearnUselessTalents section, user Tommy_TSW posted an interesting approach for memorizing this using your favorite video game, movie, or TV characters. Basically, you create a battle scenario for every possible situation, and when the various cards come up, you simply recall the corresponding battle (and result). Depending on the particular variation of blackjack you're playing, basic strategy can change, so you might want to calculate the right moves using basic strategy calculators at places like Wizard of Odds or Online-Casinos.
• Fans of the game Nim will enjoy this online version, playable even on all mobile devices. It's standard Nim, meaning that the last person to remove a card is the winner. It's simple, straightforward, and a good way to practice solo.
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
How would you even begin to code the chosen word? Even being familiar with the above methods, the particular coding isn't easy to work out here.
If you can't figure it out for yourself, Word Ways magazine wrote up “Remote Pager” here, complete with the explanation. As with all of Max Maven's routines, the approach is subtle and ingenious.
Play around with this routine, which is even more portable now thanks to smartphones. If you have any fun stories of performing this, I'd love to hear about them in the comments!
E-Z Square 5 is available as an ebook from Lybrary.com, available in English and in German.
As with previous books in the series, this one features a particular routine concerning magic squares. The major difference here being that these magic squares are created using playing cards, similar to Richard Wiseman's The Grid and Chris Wasshuber's Ultimate Magic Square, both of which are acknowledged in E-Z Square 5.
Werner Miller explores the possibilities through 3 main routines, and a bonus routine. The first routine is the simplest, in which the spectator generates a total by selecting 4 cards out of 16, and you quickly deal a 4 by 4 square with 16 different cards whose rows columns and diagonal give the same total. The second routine, which is my personal favorite, has the spectator cut off about half the deck, and you as the performer are able to create a 4 by 4 grid whose rows, columns, and diagonals are equal to the number of cut-off cards.
In the 3rd routine, the spectator cuts off a group of cards, and deals them into 2 piles, while the performer uses the remainder of the deck to create a 5 by 5 grid of cards. When the magic total is revealed, it proves to be the same as a number created from the top 2 values on the spectator's piles!
The bonus routine may be familiar if you've purchased Werner Miller's da capo 3, as it is Squaring the Cards. In this 4 by 4 magic square routine, the magic square's total is equal to the total of the remaining cards not used in the routine!
If you're nervous about handling the various arrangements and calculations required in normal magic square routines, EZ-Square 5 is an excellent choice, as the routining and use of playing cards takes care of much of the work automatically. As any Werner Miller fan already knows, not much more than basic card knowledge is required in his routines. I recommend E-Z Square 5 highly!
This month, we're going to delve into math and memory techniques you may have thought were too dificult to develop. With sufficient practice, however, they become powerful additions to your mental toolkit!
• One of the main reasons people want to improve their memory is so they can recall names and faces. This appears difficult to many people, because of the social pressure involved, and the apparent difficulty of connecting a name with the face. As USA Memory Champion Nelson Dellis will show you, it's not as difficult as you may think:
• Another memory skill that comes across as impressive is memorizing the order of a shuffled deck of cards, especially when you can do it in under 60 seconds. Over at the Four-Hour Work Week blog, they have a wonderfully vivid tutorial on memorizing the order of a shuffled deck. They use the easy-to-understand analogy of a software purchase. They start your new brain software off will a trial version they call “Bicycleshop Lite,” where you get the basic process down of memorizing shuffled cards. Once you've done that, you're ready for “Bicycleshop Pro,” which improves your speed. Need some incentive to learn this feat? They're offering $10,000 to the first person who masters it from their tutorial!
Once you develop those skills, the following video will teach how to bring them together to take any small number to any small power in your head:
• Multiplying numbers by themslves repeatedly is one thing, but how about multiplying any 2 numbers together in your head, up to, say, 7 digits? YouTube user Joesph Alexander has a series of tutorials on how to develop your mental multiplication skills to this level. He starts by teaching how to handle 2- to 4-digit numbers (presentation, explanation), then moves you up to 5-digit numbers (presentation, explanation).
When you're comfortable with doing those type of problems in your head, you're ready to move up to 7-digit numbers (presentation - shown below, explanation):
Try picking just one of these skills to develop, and you just may amaze yourself at how far you can go!
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.
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!
Perhaps part of Neale's genius is in creating things that are so simple and yet so deceptive that those who aren't directly familiar with his work just think they're things that “have always been around.” Earlier this year, bicyclecards.com presented his Hypercard, first as a puzzle, then in a post explaining how it's made. They correctly credit Martin Gardner with popularizing it, but never mentions Robert Neale himself. Here's a video created by a reader of that tutorial:
If you really want to see the man himself at work, as well as learn some of his amazing thinking, the easiest place to begin is his Celebration Of Sides video. In this video, you see not only his amazing and ingenious approach to creating routines, but his warm and charming ways of presenting even the most abstract concepts. Here's a quick glimpse of some of the things shared on this video:
Hopefully, you've enjoyed this look at some of the work of one of the most creative thinkers in magic, and maybe even learned something new.
It seems British Magician and math professor Peter McOwan, with help from a few others, has been busy putting together an astounding array of mathematical magic in recent years.
They've made all these works available in the form of e-books and videos, and generously posted them all on the web for free! In this post, I'll let you know where you can find this amazing body of work.
This book is very well organized. The tricks are taught in sections by the type of math involved, such as arithmetic, algebra, geometry, and so on. Each trick is subdivided, as well, into the trick as it appears to your audience, the method behind the trick, the math behind the method, and even where to find the same type of math in your everyday life!
There is a nice build to this manual. Early on, you learn a few routines that are simple, and you often find that later routines build on these simple skills.
• The Magic of Computer Science, Vols. I and II: With a daunting title like The Magic of Computer Science, you might worry about whether these volumes would be filled with complex math and special programs. Fortunately, both of these volumes are a lot more human-friendly.
Volume I, in fact, teaches computer concepts using only a deck of cards. It's amazing how you can start from a trick in which you predict the total of 4 cards chosen by the audience, and then learn that this trick's principle is what makes CAT scans possible!
Volume II also uses cards quite a bit, but also has tricks with other props. There's an amusing and amazing vanishing robot routine, using a downloadable graphic available on this page. My favorite routine in this book, however, is “The Power of Prophecy,” in which someone takes a number of cards from the deck without you peeking, yet you can apparently divine how many they've taken. Even if you're familiar with this routine from other sources, the related lessons in algebra and modeling systems are worth reading.
• Maths Made Magic: This book is clearly designed to appeal to the Harry Potter crowd, right down the visual design. It's made to look like it was printed on old, yellowed paper, and presents the routines as if they were spells and sorcery.
The methods, all real-world mathematics, are explained simply, yet still manage to teach advanced concepts quite well. As a matter of fact, Maths Made Magic uses more advanced math than many other math magic resources I've come across. It's not often you see magic tricks that help you understand sine and cosine, or simultaneous equations mixed with the Pythagorean theorem.
• Mathematical Magic:Mathematical Magic isn't an ebook. Instead, it's an iTunes U course, playable through iTunes on your computer, or the iTunes U app on an iOS-based mobile device.
Besides the medium, the Mathematical Magic course is different in that not every routine is explained (Don't worry, most of them are explained). Knowing that all the routines are math-based, though, I like the fact that not all the tricks are explained. It's a sort of test of your observation and critical thinking skills that have hopefully been developed by reading and trying out the other routines.
• Illusioneering: Clever Conjuring Using Secret Science & Engineering: If you're ready to get away from cards and pure mathematics, there's still plenty of magic to be had use scientific principles. Illusioneering features magic with liquids, wheels, balloons and other assorted props.
The math is still there, underlying the science, but these tricks tend to play bigger and be more visual than those in the previous books. The Illusioneering homepage also features videos of many of the effects (including their methods), as well. The Illusioneering book and videos are also available via iTunes U.
The price is certainly right, so take some time to download these resources, explore, and have some fun!
What are the chances of two named values being together in a deck of cards? Brian mentions his experience of the probabilities in his write-up:
Amazingly (and to just about everyone's disbelief), it seems that about 70% of the time, any two named values will just happen to be side by side in a shuffled deck of cards!
(by the way, math wizards: if you can figure out a way to calculate the exact odds on this, I'm all ears. After hours of playing with the numbers, I finally gave up and just did a brute force calculation: after 50 trials, I ended up averaging about a 70% success rate)
It's not easy to develop probability equations for this challenge. Just defining all the possible arrangements involved is a challenges. I don't doubt that this is why Brian gave up playing with the numbers, and turned to brute force calculations, otherwise known as the Monte Carlo method.
James Grime filmed a response video in which he explains the difficulty of calculating the odds via equations, and the result of his own Monte Carlo simulations:
The video shows a probability of 48.3%, and the information box in the video says that other experiments moved that closer to 48.6%.
After running 10 million trials of my own simulation, my results suggested a 48.63627% chance of succeeding, effectively the same 48.6% chance described above. In short, the person betting against the 2 values showing up next to each other will win roughly 51.4% of the time. With such a low probability of success, how did this bet manage to become popular?
The first thought I had about this was that perhaps it involved paying less than true odds. The odds of you winning this bet are roughly 1.056 to 1 against. In other words, as long as you can convince someone to bet at least $1.06 to every $1 you bet, you could still make money with this bet over the long term. That doesn't seem very likely.
Many bets hinge on a little wordplay. For example, there's a classic bet where you claim you can name the day someone was born, with an accuracy of plus or minus 3 days. Once they put up their money, you simply say “Wednesday,“ and take their money. Since every day of the week is plus or minus 3 days from Wednesday, you can't lose.
In a similar manner, perhaps we can use wordplay to give us a better margin of error for this bet. What if, instead of mentioning that the cards must be next to each other, the bet was that the two values would be within 1 card of each other? If the two cards show up right next to each other, as in the original bet, this sounds exactly like what you bet. In addition, it also covers the possibility of the 2 values showing up with 1 card between them.
I re-programmed my simulation to include the new possibilities, ran it another 10 million times, and came up with about a 73.6% chance of success, or odds of roughly 2.8 to 1 in favor of winning!
In today's post, I'll show you how to put together your own calendar prop inexpensively, and even some other directions you can take the basic idea.
To start, you'll need a magnetic dry-erase board with a blank calendar pre-printed on it. I used an 11 by 14 dry erase calendar from Expo, which comes with two magnets. The two important features in the board are that it be magnetic, as well as small enough to use and carry for a performance, while still being visible for your audience. You'll also need to make sure your chosen board has 5 weeks to mark.
You'll also need a dry erase marker (usually included with dry erase calendars), a dry erase eraser, and a permanent marker, such as a Sharpie. Optionally, you may want a ruler for making your marks consistent. If you choose a ruler, I suggest a cork-backed ruler to minimize damage to the board.
What you're going to do is use the permanent marker to write the dates in the squares from 1 through 28, similar to the way the calendar clipboard is laid out. Most dry erase calendars have a small space in a corner for the date, such as the corner notches seen on this dry erase calendar, but for better audience visibility, you'll want to use as much of each date's square to write the date.
If you prefer, you can use the ruler and a dry erase marker to create even guidelines for your dates first. Personally, I didn't do this. In performance, I need to write and/or erase 29, 30, and 31 on the board, and those will usually be written freehand, so they tend to stand out if the other dates aren't written freehand, as well.
Once you've written the dates from 1, in the upper-leftmost square, through 28, in the rightmost square of the 4th week, using large numbers as discussed above, you can put the permanent marker away.
Not surprisingly, most dry erase calendars have the days of the week permanently marked at the top. Yet, you need to be able to change the days of the week in routines like Day One. To solve this problem, I simply use magnets printed with days of the week. This is why I emphasized the importance of a magnetic dry erase calendar earlier. Even on larger boards, these days of the week magnets cover the pre-printed days, and highlight the days printed on the magnets.
Before each performance, you'll use a dry erase marker to write 29, 30, and 31 on the first 3 squares of the 5th week on the calendar, and have the days of the week magnets arranged in the remaining 4 days of the 5th week in the calendar.
When you're given the month and year, write them in the space for the month at the top, and use the Day One technique to determine where to place each day of the week magnet. After placing the magnets, erase any of the last 3 dates as needed (For example: If you're given a February in a leap year, erase the 30 and 31, leaving the 29) and your calendar should be arranged correctly!
After each performance, erase all the dry erase markings, and put the magnets back down in the 4 empty squares of the final week. If you're about to do another performance, write the 29, 30, and 31 back in. If you're not, you can simply put the board and magnets away until you're ready to perform again.
The basic idea of using permanent markers to create a custom design (and knowing how to remove it in case of mistakes), should start the gears turning for other ideas. Starting from a blank dry erase board, you could create things like a grid for a magic square or a chessboard for the Knight's Tour.
The cork-backed ruler I mentioned earlier is an essential when designing grid-based layouts, of course. For the chessboard, I recommend creating the board in a blue that's dark enough to be distinguished easily from the white squares, yet still light enough to contrast with dry erase numbers written in black.
Don't forget that using a magnetic dry erase board can also be a great way to display magnetic playing cards, either in full-size or in miniature.
For one last idea to inspire you, how about a Sudoku grid? You could use it to display your apparent Sudoku genius as taught in Werner Miller's Swindle Sudoku routine!
That should be enough to be enough to inspire you and get you thinking about different ways to customize and present the mental feats you've learned here on Grey Matters.
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.
Would you believe tha another of my contributions has made it on to Scam School again? It was 
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