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!

I recently ran across a number of videos I figured would be interesting to regular Grey Matters readers, so I thought I would share them.

We'll start things off with a little math magic, courtesy of Tom London and his appearance on America's Got Talent earlier this week:



Yes, I could explain the method, but I don't want to ruin the fun and the mystery. Just enjoy the magic of the prediction for what it is, since that's how it's meant to be enjoyed.

If you want mathematical explanations, however, I highly recommend checking out PBS Digital Studios' Infinite Series. These are videos on assorted advanced mathematical topics, yet they're taught in a very accessible way. Back in March, I discussed a puzzle which required the understanding of Markov chains to solve. Compare that to their video Can a Chess Piece Explain Markov Chains?, which also happens to employ my favorite chess piece, the knight:



If you enjoy Grey Matters, you may also the work of 4-time USA memory champion Nelson Dellis, who focuses on both mental and physical fitness. He has a series of memory technique videos, as well as interviews with masters of mental skills. Both of these are available on his YouTube channel, as well. As a taste of his skilll, watch his video, Memorizing 28 names in less than 60 seconds!:



Curious how he's able to do that? He explains in the next video in the series, HOW TO // "Memorizing 28 names in less than 60 seconds!".

At this point, I'll wrap things up so you can get started on a potentially mind-expanding journey.

It's now been a few months since Grey Matters was back, so now it's time to bring Quick Snippets back!

This time around, we have plenty of mathy goodness, so it's best to just jump right in!

• Besides the Clay Institute's famous selection of Millennium Problems, which will make you a millionaire if you prove or disprove any one of them, there's a lesser known set, known as John Conway's $1,000 problems. Not long ago, the 5th conjecture, which claims that working through a certain procedure (described in the link an video below) will always end in a prime, was disproven by physicist James Davis. The Numberphile video below details the problem and James Davis' counterexample:



For more about the million-dollar Millennium Problems, watch the BBC's Horizon documentary, "A Mathematical Mystery Tour of Unsolved Mathematical Problems."

• Speaking of fun discoveries in recreational mathematics, check out Allan William Johnson's "Magic Square of Squares", discovered back in 1990, and just recently posted over at Futility Closet.

• James Grimes introduces us to a different sort of "Square of Squares", in his latest Numberphile video, "Squared Squares". The challenge here is to make a perfect square shape from a set of smaller square shapes:



• Presh Talwalkar, of Mind Your Decisions, posted an interesting puzzle recently. It's titled, "The Race To 32,768. Game Theory Puzzle". Read the article up to the point where you're challenged to work it out yourself, or watch the video below up to the 2:12 mark, and try and figure it out for yourself. If you get stuck, try going over my Scam School Teaches the Game of 15 post for inspiration.



• Late last month, mathematical video maker 3Blue1Brown posted a must-watch video on the visualization of all possible Pythagorean triples. Even if you remember everything from you math classes about Pythagorean triples, this video is both eye-popping and an eye opener:



• We'll wrap this set of Quick Snippets up with help from Mathologer. His videos are always interesting, but his latest one is one of those unusual approaches to math that makes you appreciate its beauty. This video is titled, "Gauss's magic shoelace area formula and its calculus companion", and it teaches an simple but unusual method for working out the area of any polygon that doesn't intersect itself. The host even goes on to show how this approach can be adapted in calculus to work out the area contained by curves!

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.

Blame Marilyn vos Savant. Back in 1990, Craig F. Whitaker of Columbia, Maryland wrote to her with a probability puzzle, and found he'd kicked up a hornet's nest! He asked, “Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, 'Do you want to pick door #2?' Is it to your advantage to switch your choice of doors?”

Marilyn replied that, if you switch, your odds of winning the car are ⅔, and if you don't switch, your odds are only ⅓. It was difficult for many to believe. Even subsequent discussions about these probabilities, such as the Scam School episode on the Monty Hall Dilemma, find that the belief in a 50-50 chance prevails.

Despite all the numerous ways there are to explain it, practical demonstration if often the most effective way to see that the ⅔ odds of winning is correct.

Oxford Mathematics Professor Marcus Du Sautoy shows the effectiveness of practical demonstration to English comedian Alan Davies when it comes to the Monty Hall Dilemma:



While the practical demonstration in this video is effective, it's a little surprising that the switch approach won 4 times as often as it lost. This is one of the classic problems with using a small sample size (such as playing this game only 20 times). Over at Epanechnikov's blog entry on the Monty Hall Dilemma, he features a graph of repeated simulations that shows the problem with just 20 runs:



The probabilities don't even really start settling down to the calculations until about 300 trials have been run! To help see the true odds, why not use a computer to run thousands of simulations very quickly? Inspired by the spreadsheet approach used by Presh Talwalkar to simulate trials for a different probability puzzle, I decided to do the same for the Monty Hall Dilemma.

How do you set it up? The first column states the door which holds the car, and this is generated as a random integer ranging from 1 to 3. The second column states the door chosen by the player, and this is also generated as a random integer ranging from 1 to 3.

The next column is a little trickier. It's going to hold the door which is shown by the host, but there are restrictions on which door can be shown by the host, so we can't just randomly generate a number. The host will only show a door that was NOT chosen by the player and that the host knows will contain a goat. How do we communicate these restrictions to a spreadsheet? There are 2 cases to consider here. First, what happens when the door which contains the car DOESN'T match the door chosen by the player, such as when the car is behind door #1 and the player initially chooses door #2? In this case, the host can only show door #3. In fact, since 1 + 2 + 3 = 6, we can simply subtract the number of the door with the car and the number of the door chosen by the player from 6 to get the number of the door shown by the host.

That only works when the door chosen by the player and the door holding the car are different. What do we do when those two doors are the same? If the player chooses door #1 and the car is behind door #1, we can have the computer choose randomly between door #2 and door #3. A similar approach is used for the other 2 doors, of course. The final spreadsheet entry reads this way:

=IF(A6=B6,IF(RANDBETWEEN(1,2)=1,CHOOSE(A6,3,1,2),CHOOSE(A6,2,3,1)),6-A6-B6)

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

Duels are a classic staple of westerns, but it took Sergio Leone's 1966 film “The Good, The Bad and The Ugly” to immortalize the truel (neologism for a three-way duel) in the popular mind!

Back in 1975, a high school teacher named Walter Koetke took the truel into outer space in a puzzle from the May/June 1975 issue of Creative Computing. I first ran across it in the 1980s, in The Best of Creative Computing.

This puzzle concerned 3 planets, Mutab, Neda and Sogal, who have agreed to an unusually rational war. Each planet would fire planet-destroying missiles at one another in turns, with the order of the truns being determined by random drawing. The rocket launching continues in order until only 1 planet remains. However, each planet has different capabilities. Mutab is the most advanced, and when fired at a target, have a 100% chance of destroying that target. Neda is the next most advanced, but their missiles only have an 80% chance of destroying their target. Sogal is the least advanced of the three, with missiles that only have a 50% chance of destroying their target. The puzzle is this: What are each planet's chances of surviving a war?

Even when I first ran across the problem as a teen, I remember thinking how difficult this problem must be. I realized that any calculations between the less-than=100% chances of battles between Neda and Sogal would be difficult to deal with, as they could go on for several turns, much like encountering long heads-only or tails-only runs in coin-flipping. The problem did stick in the back of my mind, though, as I thought it would be interesting to analyze when I knew more about math.

That day arrived when I first learned about Markov chains. If you're already familiar with Markov chains, you're ready for my solution. I've set up a site that takes you through the solution step-by-step here. Start by clicking the “Click to work out 2-world battles” button, and then read through the logic and tables until you get to the next button. Keep proceeding that way all the way to the end. It's written using Bootstrap, so it should be viewable using any size screen (the larger tables can be swiped left and right on smaller screens). There are lots of labeled tables, explanations of what they mean, and even several links to calculations via Wolfram|Alpha. All the math is being worked out by your browser as the program runs, so you may need to be patient with slower computers.

If you're still reading because you're unsure what Markov chains are, they're not that difficult a concept, as long as they're explained clearly. The rest of this post will deal with the basic concepts. Below is an excellent and clear (and short!) introduction to the concept.



What I realized is that I could represent the targeting of one planet by another as one state, and each of the possible outcomes as individual states. From this, I could build a transition probability matrix, as explained in the video and this visual explanation of Markov chains.

Now, the video discusses the idea of a starting probability vector as a starting point. In the Mutab/Neda/Sogal puzzle, however, we're really only interested in the long term probabilities themselves, the effects over the long term, and not so much a particular starting point, referred to as a stability distribution matrix.

As an example, imagine a 2-planet battle between Mutab (100% deadly) and Neda (80% deadly). If we list starting states from top to bottom (yes, I know this is opposite of how Markov chains are usually done) as, “Mutab fires against Neda”, “Mutab wins”, “Neda fires against Mutab”, and “Neda wins”, and label the resulting states the same way from left to right, then we can assign corresponding probabilities. The probability of going from “Mutab fires against Neda” to “Mutab fires against Neda” is 0, the probability of going from “Mutab fires against Neda” to “Mutab wins” is 1, and the probabilities of going from “Mutab fires against Neda” to “Neda fires against Mutab”, or “Neda wins” are both 0, so our first row would read (0 1 0 0). Continuing through all the probabilities between just Mutab and Neda, we get the following matrix:



Notice the 1s in the 2nd and 4th rows. Those are absorbing states. The 1 in the 2nd row basically says, the probability of going from “Mutab wins” to “Mutab wins” is 1 (100%). The 1 in the 4th row says that the probability of going from “Neda wins” to “Neda wins” is 1. They serve as an endpoint.

Where does this all wind up? Well, if we run these probabilities over 1,000 generations to get the long-term outlook (the stability distribution matrix mentioned earlier) by raising the above matrix to the 1,000th power, we get the following results:



Read with the original labels, this tells us that, over the long term, if Mutab is the first one to fire, then it's probability of survival is 1, and Neda's is 0. Conversely, if Neda is the first one to fire in that two-world battle scenario, then it's probability of survival is 80%, while Mutab's is only 20%. Yes, we figured that much out early on, but when it comes to tougher probabilities, such as the long-term probabilities between Neda and Sogal, this is a very helpful tool.

Once you've understood all that, you're ready to go through the step-by-step explanation I developed!

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.

Thanks to a Singapore math exam, the internet is being driven crazy by the biggest problem in birthdays since the birthday paradox!

Here's the problem: Albert and Bernard want to know Cheryl's birthday, but Cheryl isn't willing to tell them directly. Instead, she gives them a list of 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, and August 17. She then whispers only the month to Albert and the date to Bernard. The following discussion then takes place between Albert and Bernard:

Albert: "I don't know when Cheryl's birthday is, but I know that Bernard does not know, too."

Bernard: "At first, I didn't know when Cheryl's birthday was, but I know now."

Albert: "Then I also know when Cheryl's birthday is."

When is Cheryl's birthday? We'll look at how to find the answer in this post!

The simplest and most direct explanation of this puzzle I've found is in Presh Talwalkar's post, When Is Cheryl’s Birthday? Answer To Viral Math Puzzle. The included video makes the answer seem so straightforward:



Another helpful approach is Mark Josef's interactive Cheryl's Birthday page, on which you can click each of the dates to see why that the logic determines that date to be right or wrong. Both Cahoots Malone and The Washington Post have also featured simple and straightforward video explanations of this puzzle.

For a more detailed look at the solution, check out Numberphile's thorough explanation, as well the extra footage:



Ever the intrepid explorer, however, James Grime takes an even closer look at Cheryl's Birthday, and finds that the intended answer may not necessarily be the right answer:



Has this puzzle driven you crazy? Did you manage to solve it? If so, how? I'd love to hear your answers in the comments below!

February's snippets are here. Thanks to some old favorites, and some new favorites, we have a good selection to share with you this month.

• Just 2 days ago was Friday the 13th, so MindYourDecision.com's Presh Talwalkar decided it was a good time to teach how to divide by 13 in your head:



This is a handy technique, and you really only need to learn how to do this up to 12, which isn't too difficult.

If you'd like to learn similar tricks for dividing by 2 through 15, check out the Instant Decimalization of Common Fractions video.

• Like me, Presh seems to have plenty of fun with mental math techniques. Here's a mathematical magic trick of sorts, in which you apparently divine a crossed out digit:



Are you curious as to why this works? Presh has a detailed proof on his blog.

For those who are worried that just multiplying by 9 may seem too obvious, scroll down to the end of my Age Guessing: Looking at the Roots post. The section entitled “Sneakier ways of getting to a multiple of 9” has several useful and clever ways to disguise the method.

• IFLScience just posted 21 GIFs That Explain Mathematical Concepts. More than a few of these will be familiar to regular Grey Matters readers. Many are from LucasVB's tumblr gallery, and others are from videos I've shared over the years. Nevertheless, it's nice having all these in one place.

• Steve Sobek, who has a wide variety of videos on his YouTube channel, has recently made several mental math-related videos that are worth checking out. For example, here's his video teaching a trick for mentally subtracting large numbers:



You can find more of his mental math tricks at AmysFlashcards.com.

A recent question on the Mathematics StackExchange about mentally the compound interest formula caught my attention.

It got me thinking about good ways to work out a good mental estimate of compound interest.

Part of what makes it so tricky, is that compound interest doesn't work in a straight line, like much of the math with which we're familiar. Compound interest builds on itself exponentially (not surprising since the formula is a exponential expression). This is a good point to re-familiarize ourselves with the basics of the time value of money:



For a more detailed guide to interest rate mathematics, I suggest reading BetterExplained.com's A Visual Guide to Simple, Compound and Continuous Interest Rates.

BINOMIAL METHOD: The Mental Math wikibook suggests the following formula: To estimate (1 + x)ⁿ, calculate 1 + nx + ^n(n-1)⁄₂ x².

It is an interesting formula, especially considering that the first part, 1 + nx, is basically the simple interest formula. However, after using Wolfram|Alpha to compare the actual compound interest rate formula to this binomial estimation of compound interest method, you see that it really only gives close answers when x is 5% or less, and n is 5 time periods or less.

If your particular problem qualifies, that's not bad, but what about longer times?

RULE OF 72 AND OTHERS: Last July, I wrote another post about estimating compound interest discussing the rule of 72 for determining doubling time, as well as the rules for 114 (tripling time) and 144 (quadrupling time). Note especially that you can work out the effects of interest of long time periods with a little simple addition.

Yes, the rule of 72 has been explained many places, such BetterExplained.com's Rule of 72 post, and critically analyzed, such as MindYourDecision.com's Understanding the rule of 72 post and the related video, but as long as you understand its proper use and caveats, it's an excellent tool.

Understanding where the rule of 72 comes from, you can actually work out other rules for other multiples of your original amount, which is how the rules of 114 for tripling and 144 for quadrupling came about. If you want to estimate how long your money takes to grow to 5 times the original amount (quintupling), there's the rule of 168. Similar to the other rules, you can work out quintupling as ¹⁶⁸⁄_time ≈ interest rate (as a percentage) and ¹⁶⁸⁄_interest ≈ time.

To help increase the accuracy of needed estimates, you can also remember the 50% increases between each of the above rules. For a 50% increase, for example, there's the rule of 42. For 2.5 times, there's the rule of 96, for 3.5 times, there's the rule of 132, and for 4.5 times, there's the rule of 156.

This may seem like too many rules to remember, but there are a few things that help. First, keep the rules of 72, 114, 144, and 168 in mind as primary markers for 2×, 3×, 4×, and 5× respectively. Note that these are all multiples of 6, and that the “half-step” rules are also multiples of 6, and fall between the other numbers. So, if you forget how to work out 2.5×, you can realize that the rule is somewhere between 72 and 114, and then recall that 96 is the rule you need! Here is a handy Wolfram|Alpha chart for which rules go with which amounts.

“RULE” EXAMPLES: In the video above, Timmy needs to find out how long it will take to get 10 times his money at a 10% interest rate. Since we only have rule up to 5, how do we work this out? Well, 10 times the money is basically the same as quintupling the money, then doubling that amount of money. So, we can work out the quintupling times and doubling times at 10%, then add them together!

For quintupling, we use the rule of 168 to find that ¹⁶⁸⁄₁₀ ≈ 16.8 years. Since these are estimates, you can usually round up to the nearest integer to help with the accuracy. So, his money will quintuple in about 17 years. How long will it take to double from there? We use the rule of 72 for doubling, so ⁷²⁄₁₀ ≈ 7.2 years, and rounding up gives us 8 years. Since it will take about 17 years to quintuple, and about 8 years to double, then getting 10 times the amount should take roughly 17 + 8 = 25 years. In the video, they note that it will take 26 years, so we've got a good estimate!

In his Impress by doing compound interest in your head post, Martin Lewis describes the following interest problem: “What is the APR, ie annual interest rate, if you borrowed £80,000 and had to repay £200,000 six years later?”

Since £200,000 ÷ £80,000 = 2.5, we can use a simpler approach than Martin Lewis did in his article, as we know that the rule of 96 with the 6 year time span to work out the annual interest rate. ⁹⁶⁄₆ ≈ 16% interest, the same answer Martin worked out!

USING e: Once you start getting too far beyond 25 time periods (25 years for annual interest rates), you should start using e (roughly 2.71828...) to estimate compound interest over the long term. Last March, I wrote Calculate Powers of e In Your Head! to help with this exact task. At this point, you're probably more concerned with the scale of the answer, rather than the exact answer, so working out just the equivalent power of 10 is all you really need.

SHORT VERSION: So, instead of providing one way to estimating compound interest, here are 3 methods for different scale problems. If the rate is 5% or less, and the interest is applied 5 or fewer time, the binomial method is the way to go. If your problem is larger than that, and covers less than 25 time periods, then use the rules approach (rule of 42, 72, 96, etc.). If the interest is applied more than 25 times, use e to get an idea of the scale.

It may not be the simplest estimation approach for compound interest, but if you're stuck without a calculator, this will help you get by until you can more accurately crunch the numbers.

Presh Talwalkar, from the Mind Your Decisions blog, recently shared a fun magic trick.

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.

First, let's take a look at Presh's video, from this January 2015 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

Starting position     Ending position
-----------------     ---------------
        8                   1
        9                   2
       10                   3

The first snippets of 2015 are ready!

This time around, I have some clever and fun approaches to math to share. I think you'll be surprised by them, even (or especially) if you don't usually like math.

• This January marks the 28th anniversary of Square One TV, an educational program that taught math with the use of skits, songs, and other fun approaches. While it's not on TV anymore, YouTube user Anton Spivack has been making full episodes available. I've been gathering them together in playlists by season if you want to experience this show for yourself:

Square One TV: Season 1
Square One TV: Season 2
Square One TV: Season 3
Square One TV: Season 4
Square One TV: Season 5
Square One TV: Mathnet

• While I'm thinking about YouTube channels, check out Funza Academy's site, as well as their YouTube channel. Being interested in math shortcuts, I especially enjoy their Math Concepts and Tricks playlist, as it teaches some impressive math shortcuts, including rapidly multiplying any 2-digit numbers together!

• Magic Cafe user RedDevil, author of the RedDevil Mentalism blog, recently shared a great tip for my Day One routine. Day One is my approach to minimizing the work required for the classic Day of the Week For Any Date feat.

RedDevil took this one step further by pointing out that you don't need to remember all the year information I teach in there. Instead, you can only memorize just the leap years, and move 1, 2, or 3 days forward as you go 1, 2, or 3 years ahead respectively.

If you have Day One, you'll understand this. If you don't have Day One, it's still available for only $9.99! If you're a member of the Magic Cafe with at least 50 qualifying posts, you can read his tip in more detail in RedDevil's original thread.

Yes, the snippets are short and sweet this month, but there's still plenty to explore in these links if you take the time to learn and enjoy them!

As we wrap up the year, it's natural for thoughts to turn to the calendar.

Yes, I've talked about calendar calculation many times before, but I'm always on the lookout for better methods and better teaching. MindYourDecision.com's Presh Talwalkar, whom you may remember from last week's post, is back this week with some great calendar calculation lessons!

Let's face it, calendar calculation can sometimes seem daunting. One of the best approaches to learning such a skill for the first time is to ease yourself into it. Presh's starting approach is to teach you how to work out New Year's Day for any date from 2000 to 2099:



Try practicing this skill for yourself. First, use Wolfram|Alpha to get a random year in the 2000s, workout the day of the week for New Year's Day as above, and then verify your answer with Wolfram|Alpha.

It's not difficult, and once you get the hang of this skill, you're ready to move on to the next step.

Using the New Year's Day skill as a starting point, Presh then introduces you to John Horton Conway's well-known Doomsday approach to calendar calculation:



Once again, practice is the key here. You can use Wolfram|Alpha to generate a random date in the 2000s, work it out using the above method, and then verify the correct date using Wolfram|Alpha again.

If you're interested in calendar calculation in general, I not only have several posts about it here on Grey Matters, but numerous lessons, including quizzes, about calendar calculation over in the Mental Gym. Once you get the knack, it's amazing where you can take this skill!

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 back in 2007 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, 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)! Below is the mathematical equivalent of this calculation:



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!

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!

I apologize for not posting the past 2 Sundays, but my internet connection was down.

Grey Matters is back this week, however, with some Sunday afternoon football...postgame conference, anyway. How does this relate to math or memory? Read on!

In the November 30 Texans vs. Titans game, Texan player Ryan Fitzpatrick threw 358 yards for 6 touchdowns, setting a record for the franchise.

You'd think that would be the big talking point of the postgame press conference, but Fitzpatrick's son Brady winds up stealing the show with his mental math skills (starting about 1 minute into the video):



I'm not sure exactly how long Brady has been performing this feat, but I've found an excellent candidate. It seems that just 10 days before that conference, the Mind Your Decisions blog posted about how to perform this exact feat. You can learn it below, including how to handle numbers in the 80s:



With a little practice, you can multiply numbers like these as quickly as Brady Fitzpatrick. The next step, performing this on TV, is a little trickier, however. However, you can still perform this for your friends and family!

Sorry about missing a post last week. It turned out to be a busier weekend for me than I originally planned.

I'm back this week, and I've brought plenty of snippets with me to make up for the missing post!

• Back in September, in DataGenetics' Grid Puzzle post, an intriguing puzzle was posed: Imagine there is a grid of squares. You draw a line connecting one intersection to another intersection on the grid. The question is: How many squares does this line cross through?

The link works through the answer, including an interactive widget which may help you work it out. It turns out that the answer involves only simple arithmetic and being able to work out the greatest common divisor (GCD) of 2 numbers.

If you can work out the GCD of 2 numbers in your head, you can turn this into a mental math presentation. How do you go about that? Math-magic.com has the answer, in both webpage form and PDF form! Page 94 of Bryant Heath's Number Sense Tricks also has some great tips on finding the GCD of 2 numbers in your head.

• Also back in September, I wrote about Knight's Tours on a calendar, and included sample calendars for both September and October 2014. I figured it's time for the November 2014 version, included below. This month, 6 is most date-to-move matches I could find:



• OfPad.com seems to have a similar goal to Grey Matters, which is to improve your genius. There's plenty to explore as a result, but I'd like to draw your attention to their Mental Math Tricks posts in particular. There's lots of great tricks, some of which you may not have seen before.

• To wrap up this month's mental math snippets, I'd like to focus on one particular technique for multiplying any two 2-digit numbers together. I've seen this technique before, but the way dominatemath teaches this multiplication technique, it just made so much more sense than the other instructions I've seen for this approach.



That's all for this month's snippets. Have fun exploring them!

Earlier this year, I posted about calculating powers of e in your head, as well as powers of Pi.

This time around, I thought I'd pass on a method for calculating powers of a much more humble number: 2. It sounds difficult, but it's much easier than you may think!

BASICS: For 2⁰ up to 2¹⁰, you'll memorize precise answers. For answers to 2¹¹ and higher integer powers, you'll be estimating the numbers in a simple way that comes very close.

First, you must memorize the powers of 2 from 0 to 10 by heart. Here they are, along with some simply ways to memorize each of them:

Problem   Answer    Notes 
  20    =     1     Anything to the 0th power is 1
  21    =     2     Anything to the 1st power is itself
  22    =     4     22 = 2 × 2 = 2 + 2
  23    =     8     3 looks like the right half of an 8
  24    =    16     24 = 42
  25    =    32     5 = 3 + 2
  26    =    64     26 begins with a 6
  27    =   128     26 × 21
  28    =   256     Important in computers
  29    =   512     28 × 21
  210   =  1024     210 begins with a 10

“Martin has turned thousands of children into mathematicians, and thousands of mathematicians into children.” - Ron Graham

100 years ago today, Martin Gardner was born. After that, the world would never again be the same.

His life and his legacy are both well represented in David Suzuki's documentary about Martin Gardner, which seems like a good place to start:



As mentioned in the snippets last week, celebrationofmind.org is offering 31 Tricks and Treats in honor of the Martin Gardner centennial! Today's entry features a number of remembrances of his work in the media:

Scientific American — “A Centennial Celebration of Martin Gardner”

Included in the above article is this quiz: “How Well Do You Know Martin Gardner?”

NYT — “Remembering Martin Gardner”

Plus — “Five Martin Gardner eye-openers involving squares and cubes”

BBC — “Martin Gardner, Puzzle Master Extraordinaire”

Guardian — “Can you solve Martin Gardner's best mathematical puzzles?”, Alex Bellos, 21 Oct 2014

Center for Inquiry — “Martin Gardner's 100th Birthday”, Tim Binga