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!

Obviously, I'm a big fan of mixing math and fun. It's time to give a little credit to one group that's responsible.

27 years ago this month, Square One TV, a PBS show teaching math with the use of comedy skits, music videos, and guest stars, premiered!

2 years ago, on Square One's 25th anniversary, I posted a tribute to this show, including some of my favorite segments.

At the time I was unable to provide links to complete episodes. Since then, however, several complete episodes have been uploaded to YouTube! Not every episode is available (yet?), but the complete episode guide will give you an idea of what's missing.

I've arranged the full episodes I can find into YouTube playlists by season, with the individual episodes arranged in order of broadcast. The season 1 playlist begins with the original IBM show promo, and then moves on to the very first episode. Here are all the YouTube playlists:

• Season 1
• Season 2
• Season 3
• Season 4
• Season 5

If you watch at least 1 full episode, you'll note that roughly the last third of each episode is dedicated to continuing segment called Mathnet, a sort of mathematical Dragnet parody. Square One TV originally aired Monday throughly Friday each week, so these segments always started a new adventure on Monday, and continued through with the conclusion reached on Friday's episode.

One of the downsides of not having every episode of Square One available is that it's difficult to watch complete runs of the Mathnet adventures. Fortunately, fans have solved that problem by posting 26 of the 30 episodes on YouTube, which you can find in this playlist! The complete Mathnet episode guide, which includes spoilers, can help you catch up on the ones which still aren't available.

I hope you enjoyed this mathematical walk down memory lane. I'll leave you with my favorite segment of Square One TV, a video about how to solve almost any type of problem title “Change Your Point of View”:

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)!

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!

As many regular Grey Matters readers already know, I'm a big fan of James Burke's documentaries, especially Connections and The Day The Universe Changed.

So, I was naturally thrilled to discover the recent updates on JamesBurkeWeb's YouTube channel!

JamesBurkeWeb has added new versions of Connections and The Day The Universe Changed, as well as some of his other series, that are a big improvement over the previously-posted versions.

The most noticeable improvement is that each episode is now available as a single video. For example, the first episode of Connections, “The Trigger Effect”, was originally posted as five 10-minute videos. The new version is just a single 49¹⁄₂-minute video. This makes each episode much easier to watch and embed.

There's another, more subtle change, as well. If you add up the segmented versions of the videos, you'll notice each episode runs a total of about 45 minutes. The single version videos, however, are roughly 49-50 minutes each!

This is because the segmented versions were taken from broadcasts on The Science Channel, who edited the episodes to allow for commercials. The newly-uploaded complete episodes seem to be from the original broadcasts, and have all the original footage.

As an example, below is a segment from part 1 of the “Trigger Effect” playlist, starting from 7 minutes into the episode, and playing for about 30 seconds:



From the newer version of the same episode, here's a segment starting and ending at roughly the same points as the earlier one. Note both that it starts almost a full minute later into the episode (7:58, as opposed to 7:00), and contains an entire missing segment not shown in the version above:



Even if you've watched the episodes available before, take the time to watch all the newer episodes. You may discover some new surprises, or, if you're old enough to remember the original airings, even re-discover some forgotten segments.

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)!

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!

While I often teach how to perform mental feats, especially over in the Mental Gym, it's also important to get an idea of how professionals present them.

In the past, you've seen how performers including Maths Busking and Dr. Arthur Benjamin. Today's post features some other professional performers of mental feats.

First, we have Anton Zellman, who has made a good long living as a trade show performers, using his mind to entertain and educate prospective clients.

Because it's one of my favorite mental challenges, check out his Day For Any Date video. Watch it once just to see the performance, then watch it again to pick up on the finer details. For example, note that he only gives dates in two recent years (1990 and 1991 in the video). I have no doubt he could handle many more years, but this limitation makes the feat current, quicker to do, and easier to verify using only the 2 calendars hanging behind him.

Also, notice that he teaches how to do the effect. He's not performing a magic trick, and he's not trying give the impression that he's superior to you. He's simply showing that the boundaries of the human mind are much bigger than we may think. The multiplication table analogy is a wonderful tool to get the idea across of how something that seems hard can quickly become much easier with practice.

Among Anton Zellman's other videos are ones on remembering names and memorizing lists. There are also other videos on Zellman's own website. Watch these, keeping in mind that he's working in an environment where you often have only seconds to attract and keep the attention of attendees. If you don't engage it, that's a potential lost sale.

Scott Flansburg, also known as the Human Calculator, performs amazing mathematical feats for business meeting, schools, and, fortunately for us, the occasional TV appearance. Here's his appearance on a Discovery Channel program called More Than Human:



A few of the feats you see on there can be learned right here on Grey Matters, including the long division feat and the cube root feat.

Again, the attitude here is important. Just like with Anton Zellman, he's sharing, not showing off. Indeed, the most recent tweet (well, retweet) from Scott Flansburg at this writing says:

RT @haleymillsap: Work for a cause, not for applause. Live life 2 express, not 2 impress. Don't strive 2 make your presence noticed, mak ...

— Scott Flansburg (@HumanCalculator) September 13, 2012

On August 25, 2012, the world learned the sad news of the death of Neil Armstrong, the first man to walk on the moon.

In this post, we'll take a closer look at the Apollo 11 mission, and just what it took to get Neil Armstrong and Buzz Aldrin to the moon and back.

James Burke, whom you probably know from his documentary series Connections and The Day The Universe Changed, was the main presenter and science reporter for the BBC back in 1969. His style of reporting really brought the Apollo 11 challenge to life. As seen in the videos in this playlist, he shows the cramped, yet functional, command module, the workings of the spacesuit, the escape plans, what zero gravity testing is like, and more!

As James Burke explains in more detail below, what NASA was really trying to do was launch a 36-story building at the Earth in such a way that it doesn't come back down:



One of the most impressive parts of the Apollo 11 mission was the workings of the computer itself. Compared to today's commercially available computers or mobile devices, the specs were amazing primitive. According to Grant Robertson's article, How powerful was the Apollo 11 computer?, the Apollo 11's computer only had 2K of memory and 32K of read-only storage. Simple, but quite capable of doing everything it needed to get to the moon.

For an even more detailed look at the computing power of Apollo 11, check out the Apollo Guidance Computer episode of “Moon Machines”, available on YouTube (Part 1, Part 2, Part 3). If you enjoy that, check out the rest of the “Moon Machines” documentary series.

Part of the whole triumph of landing men on the moon was the way the world seemed to stop and hold its breath together. The video below shows not only the landing itself, but also the world reaction to the feat at the time:



Sadly, the Apollo 11 mission, and even the subsequent Space Shuttle program, are now history. Yes, at the time it fired the imagination. Returning to James Burke again, he explains how imagination is not only the source of the program, but also its demise:



The fact that Neil Armstrong survived to return to Earth and live another 43 years is impressive itself. When trying something new and untested like sending a manned mission to the moon, you have to accept that there's a strong possibility of failure. It's not pleasant to think about, but the Nixon administration did have a speech prepared, in the event that Neil Armstrong and Buzz Aldrin found themselves trapped on the moon.

Neil Armstrong, you were a hero to the U.S. and the world. We are truly saddened to hear of your passing, and will continue to honor your memory.

Playing around with the math and graphics inGeogebra, the computer algebra system I mentioned in my previous post, I was reminded of a series of educational programs I used to watch on public television long ago.

I couldn't even remember the name, but a little digging eventually turned it up. It was called Project Mathematics!, and produced by Caltech from 1988 to 2000. I've found some of the episodes online, and will share them with you.

According to the Project Mathematics! homepage, there were 9 episodes total, plus a teacher's workshop tape. The teacher's workshop tape included brief segments of each of the episodes to give a general idea. You may have seen the Pi segment of the teacher's workshop tape on YouTube:



That's only a short clip (amusingly, it's about 3:14 long). The full Story of Pi episode is about 25 minutes long, and goes into much more detail about Pi.

The earliest episode I could find online was The Theorem of Pythagoras, at the Internet Archive's Moving Image Archive, courtesy of A/V Geeks.com. Even though you might not have seen the previous episode on similarity, the prerequisites on the video catch you up well. BetterExplained.com's post Understanding Why Similarity "Works" can help fill in the rest of what you might have missed.



After this episode cam the Story of Pi episode mentioned above. The next 3 episodes all deal with the nature of sines and cosines. Part 1 explores the reason sines and cosines are important when examining functions that repeat at regular intervals. Part 2, below, examines how sines and cosines are used in trigonometry. Part 3 delves into the nature and use of the addition formulas for sine and cosine.



All the episodes on YouTube are posted by NASA, so they should remain available there for the foreseeable future.

A site called GhanaTubes (presumably based in the West African nation of Ghana) hosts the Polynomials episode. Unfortunately, the sound quality isn't very good on this version.

If you have the RealPlayer plugin, you can watch some clips from the episodes, including the missing first episode Similarity, as well as the final two episodes, The Tunnel of Samos and Early History of Mathematics.

Besides Project Mathematics!, Caltech also used the same style of animation in their series The Mechanical Universe, which was a course in college-level physics. As mentioned in my science documentary post last year, Google Video still hosts the full series online.

Take the time to watch a few of these in full. Even if you're not interested in the subject matter itself, I think you'll find that they're fun and engaging, making them great lessons in how to teach effectively.

I've been practicing and re-examining the Knight's Tour lately, mostly due to my recent work on Notakto.

In the process, I've run across a few new and fun Knight's Tour videos I thought you might enjoy.

The first clip is from a late '90s British game show called The Moment of Truth. Contestants are given one week to practice some impressive feat, and then perform it before a live audience, often under time pressure, in order to win exotic and expensive prizes.

One of the best things about the video below is that it's a wonderful example of how to create suspense. Between the time clock, the live audience, the player's immediately family, and the possibility of winning prizes, this Knight's Tour has plenty of tension. The anticipation created can be felt strongly.



The next Knight's Tour video is more informational. The basics of the Knight's Tour are explained, and then a example solution is shown. This Knight's Tour video has the rather unusual feature of being shot in 3D! Instead of embedding it, I'll link to the video, so you can use the 3D menu to choose your preferred method of viewing the effect. If you prefer, it can also be viewed as a standard 2D video instead.

The final video in the set is a school lecture about the approaches and history of the Knight's Tour. Just listening to the terminology they use, I can't help but wonder whether these two ran across my Knight's Tour lessons in their research.



While there's not as much tension as the first video, there is plenty to learn from it. As a bonus, you have to love how they chose to end the lecture.

I've posted about memorizing the periodic table of the elements before, but understanding is just as important.

You might think trying to understand the basics of the elements would be a chore, but it can actually be quite fun.

Surprisingly, one of the best introductions to the atom I've ever seen is not from a documentary, but an episode of WKRP in Cincinnati. In this episode, Venus is trying to help a friend whose son has dropped out of school. In the following scene, Venus explains the basics of the atom in an effort to help get the son to go back to school:

It's time for March's snippets!

This time around, we have a selection of new and unusual approaches to using memory techniques:

• Back in January of this year, British Channel iTV premiered a new game show called The Exit List. Contestants answer trivia questions as they proceed through a “memory maze”, but there's a twist! Contestants must memorize an ever-growing list of the answers in order to exit the maze with the money. Correct answers add only one item to the list that must be memorized, while incorrect answers add four items to the list!

To make the game even more exciting, there's a hidden room in the last row containing $100,000 (the others contain $10,000) for correct answers, and also panic rooms, which can add lists of random letters (as opposed to the usual words or phrases) to your memory list. You can find episodes online to get a better idea of this Indiana Jones-meets-Simon game show.

This game show may be coming to America on ABC, as well, courtesy of the same people who brought Who Wants To Be A Millionaire? to the US.

• Joshua Foer, author of Moonwalking With Einstein, has given a talk on memorization techniques at TED 2012. At this writing, there is no video to accompany that article, but it will likely be available in the long run.

• If you enjoyed my posts on the MIT Blackjack Team, there's a new independent film out on a similar topic. It's called Holy Rollers, and is about a team of card counting Christians. From the trailer alone, it looks like it could be an interesting movie:



• If you've enjoyed my posts on the game of Nim, there's now a commercial desktop version of multi-pile Nim available. It's called Abacan, and is played by sliding beads along different bars from one side of the frame to another.

You can use what I've taught in my Nim posts to try and work out the strategy on your own, or you can just go to the multi-pile Nim Strategy Calculator, select 5 rows, and enter 1-3-5-7-9 for the sizes of the rows to get the winning strategy.

One final note: Ordinarily, my next post would be on Thursday. I'll be posting on Wednesday instead, as Wednesday is Pi Day (3/14)! Pi Day is also Einstein's birthday, and Grey Matters' blogiversary!

25 years ago today, Square One TV debuted on PBS! As a budding math geek, this show was a must-watch for me.

If you're not familiar with Square One TV, it was a show teaching math using comedy skits, music videos, guest stars, and whatever else to teach mathematical concepts. Everything from basic arithmetic to geometry to somewhat-advanaced algebra was covered in a way that was fun and interesting.

My favorite music video they ever did is a great example of this. It was called “Change Your Point of View.” Although largely about solving math problems by looking at the problem from different perspectives, it's also great advice for any type of problem:



To give you an idea of the comedy skits they used, here's a skit called The Adventures of Spade Parade, in which they have to figure out which consultant is which:



Magician Harry Blackstone, Jr. even had his own recurring segment, in which he would perform and teach mathematically-based magic:



Like many PBS shows, Square One TV was 30 minutes long (no commercials meant 30 minutes of content), and broadcast 5 episodes a week. The show itself had a rather unusual format, however. The first 20 minutes would consist of skits, songs, and other segments like the ones above.

The last 10 minutes of the show would always be an episode of Mathnet, a sort of Dragnet parody following the adventures of detectives Kate Monday (later replaced by Pat Tuesday) and George Frankly. A new adventure would start on Monday, and would be continued on each day, winding up on the following Friday.

To get a better idea, you can actually find full episodes online. Here's the very first episode of Square One TV. The very first skit, a song about the concept of infinity, recurs throughout the episode, as if it continued forever. The show's producers even convinced PBS to continue the gag even after that first show was over.

The Mathnet episode, “The Case of the Missing Monkey,” guest stars a young Yeardley Smith, better known today as the voice of Lisa Simpson. This adventure continues in the second episode, and the third episode. I can't find the fourth and fifth episodes online yet, but you can see the rest of the same case in the 39th episode and the 40th episode, when it was re-run.

The show had a good long run, and broadcast its last new show on May 6, 1994, seven years later. Happy 25th birthday, Square One TV!

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!

Although it's been around a while, interest in the Person-Action-Object, or “PAO” memory system has risen with the release of the Moonwalking With Einstein.

In this post, we'll take a closer look at the nature of the PAO system.

Despite the focus of the system's name, there's actually a whole other part of the system that needs to be learned along with it. This first part has been used since the days of Ancient Greece, and is know by several names, including "Loci", "Roman Room", "Journey", "Memory Palaces", and "Memory Theater". It involves taking a mental walk through a familiar location, such as your house, and placing your bizarre mnemonic reminders at key points throughout that journey.

In the “Matter of Fact” episode of The Day The Universe Changed, James Burke gave a simple example of this technique, helping you remember the 7 subjects of a medieval university arts course:

It's one thing to learn how to perform amazing mental feats, but it can also be very instructive to watch how others present their mental abilities.

For today's post, I've gathered some footage of some amazing math and memory demonstrations found on YouTube.

We'll start off with Ron White, "The Memory Guy". Why start with him? He's very generously posted and linked my work on memorizing US state flags and 400 Digits of Pi. Here's his appearance on Stan Lee's Superhumans:



Just a few days ago, he posted an excellent article on mind mapping that I highly recommend.

Also from his own segment on Stan Lee's Superhumans, we have human calculator Scott Flansburg.



It is interesting how often people with these abilities are presented with a superhero-type atmosphere about them. Another human calculator, Ruediger Gamm of Germany, was on a show lomg before Stan Lee's Superhumans, called Extraordinary People. Notice it's also given a comic book style. Here's part 1 of that appearance:



...and here's part 2:



Think I'm kidding about the superhero presentation? Check out 20/20's report on Daniel Tammet's amazing mental abilities (View at link - embedding disabled).

Even with the exaggerated superhero allusions, note how different all these people are in their own style of presenting their abilities. That's the short yet fun lesson for the day.

Back in July, I posted about people who cannot forget, in which I included a 60 Minutes report on the topic.

The idea of not being able to forget has certainly captured CBS' imagination. This Tuesday, they're premiering a show about an investigator who can't forget any day of her life.

There are a couple of emotional touches, of course. Most notably, her sister was murdered when they were children, and she can't seem to remember enough details about it to figure it out.

Here's the promotional preview CBS has released:



Since Numb3rs went off the air, I've been without a show I can enjoy in my own geeky way, and this looks like a good candidate. With the perfect memory premise, you can see why I'm naturally drawn to this show. When the preview got to the point where they give her the date March 27, 1998, I'm naturally trying to get the day of the week for that date too, and I shouted “Friday!” at the screen just a split second before she said, “...Tuesday.” You can see who was right by clicking here.

While I'm not crazy about yet another New York investigative drama, Numb3rs was able to bring a fresh life to L.A. crime investigation dramas, so maybe the perfect memory premise can do the same for this show. In this behind-the-scenes look at the show, star Poppy Montgomery mentions that she sees it as as a sort of superhero story. That's a good approach, and as long as they're able to keep the premise authentic, the show should do well.

Check this show out when Unforgettable premieres on Tuesday, Sept. 20th, at 10/9c on CBS. You can learn more about the show at CBS' Unforgettable homepage.

The sciences, especially math, are not often associated with fun. To combat that idea, I've gathered together links to many math and science documentaries online that challenge the notion that learning can't be fun.

• Eureka! - No, I'm not referring to the SyFy drama. Eureka! is a series of 5-minute animated shorts, each of which focused on an aspect of basic physics, such as intertia or mass. They're very creative, and great to help you catch up on the basics of physics.

• Game Theory 101 - Game theory is the study of human choices made in situations with defined rules, such as games. At this writing, there are several online courses teaching game theory, but William Spaniel's simple and direct videos make it easier to understand than most of the others. To help you follow along, he's even made a free spreadsheet calculator available!

• Nice Guys Finish First - While I'm on the topic of game theory, here's a BBC documentary on one of the most classic aspects of game theory – the Prisoner's Dilemma. At first, it seems like just a simple theoretical game. However, when you consider that it can model everything from business negotiations to international politics, it becomes much more important.

• Breaking Vegas: The MIT Blackjack Team - If you saw the movie 21, you saw the somewhat dramatized version of the story of the MIT Blackjack Team. This documentary provides a better understanding of exactly how the team came together and eventually fell apart. There's also a British documentary about the MIT team called Making Millions the Easy Way with more details.

• The Nature of Things: Martin Garder - Martin Gardner brought such a fun approach to mathematics for the masses, that it's said he turned a generation of kids into scientists, and a generation of scientists into kids. If you're not familiar with his work, this documentary is an excellent place to start.

• Algebra: In Simplest Terms - Sol Garfunkel hosts this series where Algebra is brought to life with real-world uses. This series is a great reply to the math student in all of us who is always asking, “When am I ever going to use this?” If you can catch his other series, For All Practical Purposes, before its removed from Google Video on April 29, 2011, I suggest you watch it, as well.

• James Burke's Documentaries - I can't say enough good things about James Burke's documentaries. While the zig-zag approach he took to teach history in his documentaries may seem run-of-the-mill to a generation that grew up with the internet, it was a breakthrough approach in its time. It still helps history come alive and feel more real, and thus more accessible. I've even gone to the trouble of annotating every episode of his first two documentary series with Wikipedia links.

• The Universe: Beyond The Big Bang - If you particularly liked James Burke's Infinitely Reasonable episode, Beyond the Big Bang should be your next stop. The two are very close, but this one spends two hours going into detail. Seeing Einstein's theories presented with a carnival ride analogy is a highlight of this show.

• Cosmos - For many in the 70s, Carl Sagan was the first to challenge their notions about the nature of the universe, as well as humanity's place in it. Thankfully, Hulu brings Sagan's delightful brand of astronomy right to your desktop.

• Scientific American Frontiers - This classic series, hosted by Alan Alda, and named after the magazine that brought you Martin Gardner, always seemed hard to catch on TV, with it's schedule of airing only once or twice a month. Now you can catch episodes on Hulu, or at PBS' own site.

• NOVA - NOVA is probably one of the longest running series anywhere on TV, having started in 1974. It was already in its 16th season when the Simpsons premiered! Many of its episodes are so classic, you may have seen them in school. To this day, it remains one of the best ways to keep up with the sciences.

• Hunting the Hidden Dimension - OK, I've already covered NOVA as a whole, but their documentary about Benoit Mandelbrot and fractals is a standout. It's also a fascinating lesson about how long simple ideas can remain hidden.

• A Brief History of Time - Stephen Hawking's classic work is a wonderful examination of the nature of the universe, especially concerning our understanding of it since the days of Einstein.

• CalTech: The Mechanical Universe - This 52-part series is a true college-level course in sciences, taught with recreations and computer graphics that really aid understanding. It covers everything from atoms to planets in a very accessible way, including some episodes on basic calculus that aren't hard to understand. The youtube edition is missing some episodes, but you can find a more complete set at Google Video until April 29, 2011.

Happy viewing!
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Answers to the puzzle in Sunday's post:

Besides 28 ÷ 7 = 13 and 25 ÷ 5 = 14, there are 20 other sets of numbers that can be substituted in the comedy routine:

12 ÷ 2 = 15
14 ÷ 2 = 25
16 ÷ 2 = 35
18 ÷ 2 = 45
15 ÷ 3 = 14
18 ÷ 3 = 24
24 ÷ 3 = 17
27 ÷ 3 = 27
16 ÷ 4 = 13
24 ÷ 4 = 15
28 ÷ 4 = 25
36 ÷ 4 = 18
15 ÷ 5 = 12
35 ÷ 5 = 16
45 ÷ 5 = 18
18 ÷ 6 = 12
36 ÷ 6 = 15
48 ÷ 6 = 17
49 ÷ 7 = 16
48 ÷ 8 = 15