As the last post of 2011, I thought this would be the perfect time to look back on Grey Matters, and take a look at some of the standout posts of the year.

January

NOTE: Since this coming Sunday is Christmas, I will be spending that day with my loved ones, and there will be no post that day. Happy Holidays!

I've spent much of the year teaching you about memory feats, and making you work and think.

It's the holiday season now, and it's time to relax. So, for a change, sit back, relax, and watch other perform the types of feats I've discussed on this site!

We'll start off with Bob Miller, who recalls 25 randomly selected cards, and even several that weren't selected!



Note that Bob Miller refers to remembering 40 names in the above video. If you're curious about that, check out this video in which he recalls 80 names!

I talk quite a bit about the day of the week for any date feat, so I was pleased to find a video of it being performed. Here's Greg Arce, performing the calendar feat the famed Magic Castle in Hollywood:



Harry Lorayne has been working with memory feats and techniques since before most people in his audiences were born. Here's an ad for his memory lessons from the October 1963 issue of Popular Science, and with further searching, I've found Billboard mentions going back to 1949! Here's Harry Lorayne performing the magic square in his inimitable style:



Among the lesser-known memory feats to the general public is the memorization an entire weekly magazine. The fact that the magazine contents change so frequently, and the number of pages involved, make this feat very impressive when well-presented. Watch Jonathan Levit recalling an entire Time Magazine from back in December 2007:



If you'd like to get an idea of how different performers present the same effect, compare the above performance to Nick Morton's magazine memory act here.

We'll round out this post with a memory act you may or may not have the stomach to watch. What do you get when you cross a memory feat with a nail gun? Naturally, you get Penn & Teller:



Do you have any favorite memory act videos? Let everyone know about them in the comments!

Magicians refer to what most would call “tricks” as “effects”. Anytime you have an effect, however, people are going to look for a cause.

The search for the cause naturally takes the audience from the effect itself to the performer themselves. In analyzing how to guide an audience's perceptions of cause and effect, superheroes often prove to be useful as an analytical tool.

In past times, ancient myths might have served the same purpose, as discussed in Joseph Campbell's Hero With A Thousand Faces. Indeed, summaries of his works can be found here (archived) and here (PDF). A fuller examination of this approach can be found in the Power of Myth 6-hour miniseries.

In an age where more people are familiar with Batman and Superman, as opposed to, say, Orpheus or Daedalus, superhero examples com much more readily to mind.

I was first made aware of this approach in Jon Armstrong's article “Superhero Theory”, published in the December 2004 issue of Genii. Sadly, there is no reprint of the exact article online, but it can be largely summed up with 4 major points:

• Superheroes are defined by their powers, to the extent that they're often named after them (e.g., Spiderman, the Flash).

• Audiences are familiar with what a particular superhero is capable of, so the heroes have certain expectations (without being made predictable), and they're made more memorable.

• Superheroes are limited by their powers (e.g., Batman doesn't have X-ray vision, Spiderman can't talk to sea creatures), creating focus, as well as opportunities for challenge.

• Speaking of limitations, many superheroes also have a weakness. How they deal with this weakness can be as engaging as how they use their superpowers.

December's snippets are here!

We're about to wind up one calendar year, and begin another one, so this edition is dedicated to the calendar!

• I've long been familiar with the use of knuckles as a mnemonic for the number of days in each month, but I've never seen this knuckle approach for determining the day of the week for any date in the current year! Unfortunately, it on works as written for non-leap years, so you'll have some major adjustments to make if you want to use this for 2012.

• Speaking of fingers and calendars, in my review of Speed Dating I hinted at, but never explicitly mentioned, the use of the finger abacus technique taught in the notes. It turns out that this technique was originated by Dr. Hans-Christian Solka, who teaches it in a free PDF download (English translation here). This really only makes sense if you're already familiar with the day for any date feat, but this technique does help eliminate the oft-dreaded "casting out 7s" calculation.

• I've seen plenty of perpetual calendars that you can print, but usually the printing is the final step. Over in Canon's Creative Park site, they offer 2 perpetual calendars that you print and assemble into a 3-dimensional desk accessory. It comes in classic black, or in a faux wood print. Since the calendar is always good, and is something you create yourself, this could be the last calendar you buy for a long time. It could also be yet another nice touch in your day of the week for any date presentation.

• Taking a quick look back at my own calendar-related 2011 posts, the single most-read post on the site would have to go to my revised day of the week for any date tutorial and its quiz. I first posted it back in March, but it quickly gained prominence in October, when Lifehacker featured the tutorial on their site! You can learn more via my Day For Any Date Toolbox post, and the added Odd + 11 post. If you'd prefer to cheat at this feat, there's always my dice calendar backgrounds (Just remember to use the set in “Leap_Year_Starts” > “0_Sunday” directory for 2012).

(NOTE: Check out the other posts in The Secrets of Nim series.)

In past posts on Nim, I've discussed multi-pile standard Nim, multi-pile Misère Nim, and how to better visualize your moves for either kind of multi-pile Nim.

I've even discussed Werner Miller's brilliant Wise-Guy Nim variation, in which you allow the other player to decide whether the game will be standard or Misère, after which you subtly alter the rules in your favor.

What would happen if, instead of deciding the goal beforehand, the goal was declared after the game had started?

This unusual variation was developed by John Horton Conway (seen on video in my Iteration, Feedback, and Change: Artificial Life post), who dubbed it “Whim”.

Whim starts just like any other multi-pile Nim game, but with no determination as to whether the last person to remove a match is the winner (standard) or the loser (Misère).

The big difference with Whim is that only once during the game (NOT once per player), one player may, instead of removing objects, make a "Whim move", in which that player declares the goal to be standard or Misère. At this point, the goal of the game is frozen, and no further Whim moves can be made by either player. What effect does this have on strategy?

Since the strategy for both versions of multi-pile Nim are the same up until the final moves, you might figure that this would make analyzing the strategy easy. A complication quickly arises, however, when you realize that not only must a goal be declared at some point during the game, but that the temptation to use the Whim move is itself a factor.

Think about by imagining that you're playing first. Let's say you analyze the layout as I've taught in previous posts, and you determine that no objects need to be removed. If you're applying the strategy from my Visualizing Multi-pile Nim post, this would mean that you noted that all the powers of two are “paired up”.

In this case, it's easy. Instead of making a move, you make the Whim move, declaring the game to be either standard or Misère as you wish (on a whim, as it were). This forces the other player into a losing move either way (their move must “unpair” a power of two), and you will win the game as long as you stick to the proper strategy.

Now let's look at the opposite situation. You analyze the layout, and the normal strategy for Nim says you need to remove some objects. You could remove those objects, but the other player could make the Whim move, and then you're left with another losing position AND no way out (assuming the other player understands and plays regular Nim strategy). The only other possibility here is to intentionally make a losing move (a move that still leaves one of the powers of two “unpaired”)!

It gets worse. The other player must use the same logic, and make another losing move, too. As soon as either player fails to make what would ordinarily be a losing move, the other player can use the Whim move, and win the game from there.

Consider also that the goal must be declared at some point in the game, and the closer you get to the end, the more you need to know the goal in order to win the game. If both players are well versed in multi-pile Nim strategies, this becomes a very big problem, and one that's not easily analyzed.

Despite the difficulty of the analysis, the proper strategy is surprisingly simple and elegant.

You need to treat the Whim move as a separate pile of its own. If there is any pile consisting of 4 or more objects, you imagine that this imaginary “Whim pile” contains 1 object, and analyze the play as if this pile were actually there. Once every pile contains less than 4 objects, you need to think of the imaginary Whim pile as containing 2 objects.

The imaginary Whim pile only gets removed when one of the players makes the Whim move (declares the goal of the game). After that, play is straightforward for anyone who understands multi-pile Nim.

Due to the fact that the rules of Whim create an unusual hole, yet don't provide any apparent way to fill it (until the idea of imaginary piles is properly considered), seasoned Nim players can be stumped by these new rules, even after playing multiple games!

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The old address, which began with http://members.cox.net/ is no longer in operation.

5 months ago, I posted my first Wolfram|Alpha trick. The time for the second one has come!

We start with a helpful lead-in from NumberPhile.

In mathematics, a factorial is simply the product of a positive integer (whole number) n multiplied by all the other positive integers less than n. 4 factorial, for example, would be equal to 4 times all the positive whole numbers less than 4, or 4 × 3 × 2 × 1 = 24. So, 4 factorial would be 24.

Instead of writing the word “factorial”, though, mathematicians use the exclamation point (!). So, 4! = 24 would be read verbally as “4 factorial equals 24”.

Do you have a guess as to the biggest number for which the average pocket calculator can determine the factorial? Here's the answer:



Thankfully, today we have tools like Wolfram|Alpha, so figuring 70! can be done in not only scientific notation, but in the form of regular numbers: 11,978,571,669,969,891,796,072,783,721,689,098,736,458,938,142,546,425,857,
555,362,864,628,009,582,789,845,319,680,000,000,000,000,000 (or, just a little over a googol).

Besides getting larger at an ever-increasing rate (not surprising, consider how they're calculated), it's also easy to notice that there's lots of trailing zeroes (zeroes at the end). If you think about it, this isn't surprising, since roughly half the numbers in any factorial equation will be even numbers, and anytime one of those is multiplied by a 5, you'll get a multiple of 10 that adds another 0.

5! = 120, which is where we find our first zero. When will the next zero show up? Let's take a look:

6! = 720
7! = 5,040 (Hmmm...we have another zero, but it's not a trailing zero.)
8! = 40,320
9! = 362,880
10! = 3,628,800 (Aha! We just got our 2nd trailing zero!)

Note: This post first appeared on Grey Matters in 2007. Since then, I've made it a sort of annual tradition to post it every December, with the occasional update. Enjoy!

Since the focus of this blog is largely math and memory feats, it probably won't be a surprise to learn that my favorite Christmas carol is The 12 Days of Christmas. After all, it's got a long list and it's full of numbers!

On the extremely unlikely chance you haven't heard this song too many times already this holiday season, here's John Denver and the Muppets singing The 12 Days of Christmas:



The memory part is usually what creates the most trouble. In the above video, Fozzie has trouble remembering what is given on the 7th day. Even a singing group as mathematically precise as the Klein Four Group has trouble remembering what goes where in their version of The 12 Days of Christmas (Their cover of the Straight No Chaser version):



Just to make sure that you've got them down, I'll give you 5 minutes to correctly name all of the 12 Days of Christmas gifts. Those of you who have been practicing this quiz since I first mentioned it in last Sunday's post will have an advantage.

Now that we've got the memory part down, I'll turn to the math. What is the total number of gifts are being given in the song? 1+2+3 and so on up to 12 doesn't seem easy to do mentally, but it is if you see the pattern. Note that 1+12=13. So what? So does 2+11, 3+10 and all the numbers up to 6+7. In other words, we have 6 pairs of 13, and 6 times 13 is easy. That gives us 78 gifts total.

As noted in Peter Chou's Twelve Days Christmas Tree page, the gifts can be arranged in a triangular fashion, since each day includes one more gift than the previous day. Besides being aesthetically pleasing, it turns out that a particular type of triangle, Pascal's Triangle, is a great way to study mathematical questions about the 12 days of Christmas.

First, let's get a Pascal's Triangle with 14 rows (opens in new window), so we can look at what it tells us. As we discuss these patterns, I'm going to refer to going down the right diagonal, but since the pattern is symmetrical, the left would work just as well.

Starting with the rightmost diagonal, we see it is all 1's. This represents each day's increase in the number of presents, since each day increases by 1. Moving to the second diagonal from the right, we see the simple sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12, which can naturally represent the number of gifts given on each day of Christmas.

The third diagonal from the right has the rather unusual sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91. This is a pattern of triangular numbers.

But what can triangular numbers tell us about the 12 days of Christmas? If you look at where the 3 in this diagonal, it's southwest (down and to the left) of the 2 in the second rightmost diagonal. If, on the 2nd day of Christmas, you gave 2 turtle doves and 1 partridge in a pear tree, you would indeed have given 3 gifts, but does the pattern hold? On the 3rd day, you would have given 3+2+1 (3 French hens, 2 turtle doves and a partridge in a pear tree) or 6 gifts total, and sure enough, 6 can be found southwest of the 3! For any of the 12 days, simply find that number, and look to the southwest of that number to see how many gifts you've given by that point! Remember when figured out that the numbers 1 through 12, when added, totaled 78? Look southwest of the 12, and you'll find that same 78!

Let's get really picky and technical about the 12 days of Christmas. It clearly states that on the first day, your true love gave you a partridge in a pear tree, and on the second day your true love gave you two turtle doves and a partridge in a pear tree. You would actually have 4 gifts (counting each partridge and its respective pear tree as one gift) by the second day, the first day's partridge, the second day's partridge and two turtle doves. By the third day, you would have 10 gifts, consisting of 3 partridges, 4 turtle doves and 3 French hens.

At this rate, how many gifts would you have at the end of the 12th day? Sure enough, the pattern of 1, 4, 10 and so on, known as tetrahedral numbers (Java required, opens in new window), can be found in our Pascal's Triangle as the 4th diagonal from the right.

If you look at the 2nd rightmost diagonal, you'll see the number 2, and you'll see the number 4 two steps southwest (two steps down and to the left) of it, which tells us you'll have 4 gifts on the second day. Using this same method, you can easily see that you'll have 10 gifts on the 3rd day, 20 gifts on the 4th day, and so on. If you really did get gifts from your true love in this picky and technical way, you would wind up with 364 gifts on the 12th day! In other words, you would get 1 gift for every day in the year, not including Christmas itself (also not including February 29th, if we're talking about leap years)!

If you're having any trouble visualizing any of this so far, Judy Brown's Twelve Days of Christmas and Pascal's Triangle page will be of great help.

One other interesting pattern I'd like to bring up is the one that happens if you darken only the odd-numbered cells in Pascal's Triangle. You get a fractal pattern known as the Sierpinski Sieve. No, this won't tell you too much about the 12 days of Christmas, except maybe the occurrences of the odd days, but it can make a beautiful and original Christmas ornament! If you have kids who ask about it, you can always give them the book The Number Devil, which describes both Pascal's Triangle and Sierpinski Sieve, among other mathematical concepts, in a very kid-friendly way.

There's another 12 Days of Christmas calculation that's far more traditional: How much would the 12 gifts actually cost if you bought them? PNC has been doing their famous Christmas Price Index since 1986, and has announced their results. Rather than repeat it here, check out their site and help them find all 12 gifts, so that you can some holiday fun and then find out the total!

Since my Christmas spending is winding up, I'm going to have to forgo the expensive version, in favor of Miss Cellania's internet-style version of The 12 Days of Christmas. Happy Holidays!

This week's Scam School episode uses an iPhone, and it reminded me that quite a few episodes have focused on scams that either take advantage of iOS devices (iPad/iPhone/iPod Touch), or possibly could use these features.

I'll apologize to Android users right now, as this post will specifically focus on the iOS platform.

For starters, here's this week's episode:



Update: The required graphics for this routine have now been posted on Revision3's website here (scroll down to find them).

Once you've watched the episode, your first question will probably concern how to obtain the graphics. I couldn't find an official source, so I did the next best thing. I did some screengrabs from the HD version of the podcast:

• First Screen
• Second Screen
• Folded Card
• Blank

The iPhone's/iPod Touch's photo app, along with the built-in camera, was also featured in episode 180, in the teleportation scam. From what I've seen in in iOS-based magic, the photo app seems to be the focus of many original tricks.

Not long ago, episode 186 featured a probability scam in which Brian used the Wolfram|Alpha iOS app. This one is somewhat flexible, as you could also use your system's calculator along with online access to a search engine.

Speaking of images and search engines, the Mind Control Scam from back in episode 28 takes advantage of the iOS YouTube app, in conjunction with a search engine. It actually works better on a mobile device than it does on a desktop or laptop, as the standard YouTube layout can accidentally divulge an important part of the secret.

Getting back to the calculator app, it's probably the most used of the built-in iOS apps by Scam School. There was the Fibonacci addition trick (episode 170), episode 153's calculator trick, and the original Pi Day Magic Trick back in episode 112!

Of course, there's no need to limit scams to use of built-in apps on iOS. Many apps have been developed relating to Scam School episodes. Back in episode 164, they even released a custom app to help you practice the 2nd Pi Day Magic Trick!

Other have developed iOS apps that, by either luck or design, happen to work well with various Scam School episodes. Episode 159, Petals Around The Rose, works well with just about any dice-rolling app. My personal preference is for the 3D Dice app, but many others will work for this scam. Similarly, numerous tic-tac-toe apps could be used for episode 119's Tic-Tac-Toe prediction.

The Three Houses App allows you to try out the basic version of episode 155's scam, but you'll still need the all-important extra prop for the punch in that episode. The 4-coin puzzle from the 2nd episode was also developed into its own app. If you're having trouble believing or understanding the Monty Hall paradox from the 108th episode, you can try it for yourself with Monty Doors or the Monty Hall Paradox app.

Of course, the Nim-related apps are plentiful. If you liked episode 116's Nim With Nothing, but prefer to use props, you can do the calendar version with the Date Game app, or the 100 version with Race to 100 app, or the TenSteps app (HD version here), whose interface makes it seem more like a true game.

Regular single-pile Nim, as taught in the 8th episode, is really only found in Game 1 of Dual Matches, while multi-pile Nim from the 37th episode seems to be much more popular with developers. Besides being available in Game 2 of Dual Matches, there's My Quick Game (Lite version), Mind Nimmer, 12 coins, as well as the creatively-themed Cannibal Muffin and Mystic Pyramid apps.

If nothing else, you can at least practice what you learn from Scam School while you're on the go!