Offbeat Memory Challenges

Published on Sunday, March 31, 2013 in , , , , ,

Natalie Roberts' magnetic poetry photoWhen you first learn memory techniques, you also tend to apply them to a small set of standard lists, such as presidents, monarchs, countries, states, capitals, and so on.

It's not uncommon to start wondering whether there's any other type of lists to memorize, beyond just school standards. In this post, you'll find enough resources to challenge your memory skills for the rest of your life!

mentalfloss.com, which has long billed itself as, “Where knowledge junkies get their fix,” is a natural starting point. They also have a free iPad app (iTunes Link) in which you can read full issues, and many of these issues are themselves free thanks to sponsorship from Boeing. Earlier this month, mental_floss also started their own YouTube channel, including fun subjects such as 45 Facts About U.S. Presidents:

Not surprisingly, reddit.com can also be a good source, but there is the problem of too much information there. How do you find good sets of information to memorize there? Fortunately, reddit has done much of the work for you, with this network of subreddits, which I'll refer to as SFWP (take a look at any individual subreddit in the network, and you'll see why). Through the SFWP Network, you can easily find lots of fascinating information to challenge your memory. For example, in the map part of the network, you can see things like a map of the most common surnames in European countries in 2011, or even the most common European male and female first names in 2012. Try exploring, and you'll be amazed at the endless ideas these sections inspire.

Another fun subreddit is r/wordplay, where you can run across all sorts of weird and bizarre uses for words. Here's some 4 by 9 word squares, in which every horizontal and vertical line makes a legitimate word in English, and here you can find a rather nasty tongue twister.

Wordplay is a rich source of memory challenges. You might amuse yourself with this list of heteronyms and antagonyms, as well as a wide array of other word oddities and trivia!

When I originally created my free app Verbatim 2, I designed it for memorizing things like standard speeches, lyrics, and poems. However, when combined with some wordplay, memorizing AND repeating can be quite a feat. Matthew Goldman's Goonerisms Spalore site is a perfect example. The spoonerisms there range from the short and simple, such as Drain Bamage, to fully spoonerized stories, such as the many versions of Rindercella.

Spoonerisms aren't the only type of wordplay out there, though. Many variations of the classic “Who's On First” skit are fun to try and memorize and repeat.

As some closing inspiration, here's a collection of general wordplay videos, many of which have become classics in their own right:


Quick Snippets

Published on Thursday, March 28, 2013 in , , , , ,

Luc Viatour's plasma lamp pictureMarch's snippets may be a little late, but they are here!

This month, I present several classic geometry puzzles. Not all of them are solvable, but they are all interesting.

• Let's start this with one of the longest-running, and apparently most maddening, geometry puzzles in history! James Grime discusses “squaring the circle,” the challenge of constructing a square and a circle with the same area, using only a straight edge and a compass, in a finite number of steps:

Despite the impossibility, you can find many interesting approaches which have been tried over the years.

• One geometry puzzle that recently gained plenty of attention over at Gizmodo is the Winston Freer Tile Puzzle. You can purchase your own here, or a smaller version here, but ponder the seeming impossibility of it first:

James Tanton posted an interesting geometric challenge which can be presented in stages. The first challenge is just to determine the size of an arc without a protractor. This is usually solved by finding the center first, but can you do it without finding the center?

• Sometimes geometry itself is the puzzle! Jeff Dekofsky, via TedEd, discusses Euclid's puzzling parallel postulate. This is another part of geometry in which the answer will be forever closed off to us, but will remain interesting to ponder:

• I'll wrap March's snippets with Emma Rounds' poem, “Plane Geometry,” a parody of Lewis Carroll's classic “Jabberwocky”:

‘Twas Euclid, and the theorem pi
Did plane and solid in the text,
All parallel were the radii,
And the ang-gulls convex’d.

“Beware the Wentworth-Smith, my son,
And the Loci that vacillate;
Beware the Axiom, and shun
The faithless Postulate.”

He took his Waterman in hand;
Long time the proper proof he sought;
Then rested he by the XYZ
And sat awhile in thought.

And as in inverse thought he sat
A brilliant proof, in lines of flame,
All neat and trim, it came to him,
Tangenting as it came.

“AB, CD,” reflected he–
The Waterman went snicker-snack–
He Q.E.D.-ed, and, proud indeed,
He trapezoided back.

“And hast thou proved the 29th?
Come to my arms, my radius boy!
O good for you! O one point two!”
He rhombused in his joy.

‘Twas Euclid, and the theorem pi
Did plane and solid in the text;
All parallel were the radii,
And the ang-gulls convex’d.


Werner Miller's Sub Rosa 3 and 4

Published on Sunday, March 24, 2013 in , , , , ,

Cover of Werner Miller's Sub Rosa 1 and 2If you're familiar with Werner Miller's work, you're in for another treat from him!

If you're not familiar with Werner Miller's work, you're in for several treats!

For those not familiar with him, he's a creator of mathematical magic from Germany. What makes his work so special is his knack for ingeniously and effectively disguising the mathematical principles used in his routines. If you haven't already done so, go back through this site to explore some of his other effects. You'll quickly get an appreciation for his style of thinking.

Last October, he released the first two volumes in a new series called Sub Rosa (Latin for in strictest confidence). Earlier this month, Werner Miller released two new books in the series, Sub Rosa 3 and Sub Rosa 4.

Thanks to the math involved, none of the routines require any difficult sleight-of-hand. For example, one of the routines from Sub Rosa 4 is titled, “ESPecial Countdown,” and performed with ESP cards. After the ESP cards are mixed, an audience member chooses a card. This card is set aside face down, and no one, not even the audience member, knows which ESP symbol was chosen.

The performer then removes 8 cards, and asks for a number (8 or less) from which to countdown. The performer then runs through a procedure in which the cards are counted down. When the countdown reaches 0, the card at that position is removed. Interestingly, it proves an exact match for the card previously chosen by the spectator.

Generously, Werner Miller has allowed me to reveal the method to “ESPecial Countdown” here on Grey Matters. The method is explained below, and the PDF can be downloaded via this link.

I'd like to thank Werner Miller for his generosity in letting me share this routine with you. If you'd like to show him your thanks as well, you can buy Sub Rosa 3, Sub Rosa 4, and his other works over at Lybrary.com.


Cube Roots From Scam School

Published on Thursday, March 21, 2013 in , , , , ,

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

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

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

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

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

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

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

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

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


St. Patrick's Day Puzzles

Published on Sunday, March 17, 2013 in , , , , ,

SunOfErat's photo of 4-leaf and 5-leaf cloversHappy St. Patrick's Day!

In honor of the holiday, I thought I'd share some classic Irish-themed puzzles for you to ponder.

I'll start with a simple puzzle, which you can simply enjoy without choosing to solve it. Developed by Canadian puzzler and magician Mel Stover, this first one is called The Vanishing Leprechaun:

If you'd like to get a closer look at this page, there are several sites, such as this one, which feature the artwork in detail. On that linked page, you can click the top illustration to switch between the two modes yourself, of look at the 2 individual stages in the illustrations below that. The solution to the puzzle is also available on that page, but spend some time trying to figure it out for yourself first.

It seems the Irish have a knack for vanishing in tricky and amusing ways. Take the story or Casey, for example, who marched in many a St. Patrick's day parade. Sam Loyd's puzzle How Many Men Were In The Parade?, which comes to us courtesy of Martin Gardner, and can be read online (Part 1, Part 2), or as follows:

During a recent St. Patrick's Day parade, an interesting and curious puzzle developed. The Grand Marshall issued the usual notice setting forth that “the members of the Ancient and Honorable Order of Hibernians will march in the afternoon if it rains in the morning, but will parade in the morning if it rains in the afternoon.” This gave rise to the popular impression that rain is to be counted as a sure thing on St. Patrick's Day. Casey boasted that he “had marched for a quarter of a century in every St. Patrick's day parade since he had become a boy.”

I will pass over the curious interpretations which may be made of the above remark, and say that old age and pneumonia having overtaken Casey at last, he had marched on with the immortal procession. When the boys met again to do honor to themselves and St. Patrick on the 17th of March, they found that there was a vacany in their ranks which it was difficult to fill. In fact, it was such an embarrassing vacancy that it broke up the parade and converted it into a panic-stricken funeral procession.

The lads, according to custom, arranged themselves ten abreast, and did march a block or two in that order with but nine men in the last row where Casey used to walk on account of an impediment in his left foot. The music of the Hibernian band was so completely drowned out by spectators shouting to ask what had become of the “the little fellow with the limp,” that it was deemed best to reorganize on the basis of nine men to each row, as eleven would not do.

But again Casey was missed and the procession was halted when it was discovered that the last row came out with but eight men. There was a hurried attempt to form with eight men in each row; again with seven, and then with five, four, three, and even two, but it was found that each and every formation always came out with a vacant space for Casey in the last line. Then, although it strikes us as silly superstition, it became whispered through the lines that Casey's “dot and carry one” step could be heard. The boys were so firmly convinced that Casey's ghost was marching that no one was bold enough to bring up the rear.

The Grand Marshal, however, was a quick-witted fellow who speedily laid out that ghost by ordering the men to march in single file; so, if Casey did march in spirit, he brought up the rear of the longest procession that ever did honor to his patron saint.

Assuming the number of men in the parade did not exceed 7,000, can you determine just how many men marched in the procession?
I'll keep you guessing until Thursday, when I'll reveal the answer to this puzzle.

If your puzzle tastes run more towards the jigsaw variety, try one of Jigzone's St. Patrick's Day online jigsaw puzzles! The ones with the all-over clover patterns are especially challenging.

That's all for now. I simply wished to share some quick puzzles for the holiday. If you have any favorite St. Patrick's Day puzzles you'd like to share, let me know about it in the comments!


Grey Matters' 8th Blogiversary!

Published on Thursday, March 14, 2013 in , , , ,

Mehran Moghtadaei's Pi Digit GraphicCan you believe it? Grey Matters is 8 years old today, March 14th! How long is 8 years? When I started this blog, YouTube had been formed as a company, but it would be another month before they would publicly unveil their website.

Besides being this blog's 8th blogiversary, it's also Pi Day (3/14) and Albert Einstein's birthday, so let's have a little fun, shall we?

Mental_Floss.com helps gets the party started by sharing 11 unserious photos of Einstein. Yes, of course the famous tongue picture is there, but there are more with which you may not be familiar.

For a Pi Day party, we need food, and what better food than pies? Matt Parker shows us how to calculate Pi using pies:

If you're concerned about food being used in this way, Matt Parker assures:

Your next concern might be about the accuracy of the measurement, which Wolfram|Alpha gives here to 10 places. At first glance, 3.138 doesn't seem as impressive as it should be.

However, if you remember last month's post on bringing pi digits to life, you'll recall that it only takes 38 digits to measure a universe-sized circle with an accuracy to the nearest hydrogen atom. Considering that, measuring a circle in terms of pies to 3.138 is less surprising, and is a considerably good result.

When I was doing research for my continued fractions post, I was thrilled to discover L. J. Lange's continued fraction of Pi, which he developed in his May 1999 paper An Elegant Continued Fraction for π:

As much as you hear about the randomness and unpredictability of Pi, this continued fraction has an astonishingly simple and regular pattern. The denominators, of course, are all 6. The numerators are the squares of all the odd numbers starting with one. In fact the numerator at any level n can be calculated with the formula (2n - 1)2. For example, the 6th numerator is calculated as (2 × 6 - 1)2 = (12 - 1)2 = 112 = 121. Using Gauss' Kettenbruch notation, we can then write this formula for Pi as:

How fast does this get us to the 38 digits for our universe-sized circle which measured to the nearest hydrogen atom?

We can use Wolfram|Alpha to get an idea. The first 10 levels of this fraction give us Pi to 4 digits (the integer part plus 3 decimal places). We get Pi accurate to 7 digits by the 100th level, and to 10 digits by the 1000th level.

Assuming this logarithmic rate of 3 digits for every order of magnitude continues, we would need to go tens of trillions of levels deep to get universe-level accuracy!

Alas, it seems the beauty of this formula's pattern is at the cost of slow convergence to Pi. Since the original formula takes the process to infinite levels, however, at least it gives Pi in the long run. If you're wondering how someone like Archimedes worked out Pi 2200 years ago, without textbooks, calculators, or even calculus, it's actually due to this ingenious approach described at BetterExplained.com. Note that after taking his own approach 96 levels deep, Archimedes also calculated Pi accurate to only 4 digits.

Naturally, many others are celebrating Pi Day today. Check out Ben Vitale's Some Musings on Pi, both part 1 and part 2. The Math Dude podcast also took some time to celebrate the world's best known mathematical constant. One of the more amusing moments in Pi history was the time that the Indiana almost legislated the value of Pi to be exactly 3.2, and James Grime tells the story well.

Thanks to all my readers for reading Grey Matters and keeping this blog going for 8 wonderful years! Now, it's time for you to keep an eye out for what I have in store for my 9th year.


John Conway's Rational Tangles

Published on Sunday, March 10, 2013 in , , , ,

Jkasd's image of Tangle operationsJohn Conway has brought many new mathematical recreations to the general public. Martin Gardner wrote about him quite often in his Scientific American columns, and I've referred to his works many times here on Grey Matters.

Today's post focuses on a puzzle Conway created. It uses 4 people, some ropes, and is referred to as Rational Tangles, or just Tangles, for short.

Imagine 4 people standing as if they were at the 4 points of a square. In the diagram below, each colored dot represents a distinct person:

Next, 2 ropes are handed out. One of the ropes is held by Blue at one end, and Red at the other end. The other rope is held by Green at one end, and Black on the other end, like this:

In reality, both ropes would be the same color. 2 different color ropes are only used here to make things clearer. Later on, I'll be referring to the above illustration as the starting position. Now that we have the basic set-up, it's time to introduce the rules of the puzzle.

From this point, there are only two moves allowed. The first is a simple 90° clockwise rotation, as seen from above, of all players. This move is referred to as a rotation. From the starting position (above), a clockwise rotation would end with the players and ropes in this position:

The only other legal move, known as a tangle, is performed by having the players in the upper right and lower right (again, as seen from above), switch places. As this happens, the player in the upper right lifts up their rope, and the player in the lower right goes underneath. Not surprisingly, this results in a crossing of the ropes.

Going back to our original starting position, with Red in the upper right and Black in the lower right, a call for a twist would start with Red holding its end up while heading to the lower right, and Black going underneath Red's rope, while heading for the upper right. At completion, the result would appear like this (again, remember this is starting from the original starting position, NOT the previous illustration):

The idea is that you have the 4 people involved, and/or anyone else who is watching to call out a long combination of tangles and rotations, in order to ensure that the 2 ropes are well intertwined. The challenge is, with the tangled section of the rope hidden from view, to untangle the ropes once again.

You might think the solution would simply be to use a memory system to memorize all the calls made, and then call them out in reverse order, making sure you untwist and unrotate at the appropriate steps. The problem here is that untwists and unrotations aren't legal moves.

The only moves you can use, remember, are a 90° clockwise rotation, and a switching of the two rightmost people, such that the rope of the upper right person goes over the rope held by the lower right person as they switch.

Using only those two moves, is it always possible to return to an untangled state? Surprisingly, the answer is yes. The question of course, is how do you do it?

The answer involves keeping track of the tangles mathematically. As there are only two types of moves, this is easier than it might sound. However, the mathematics do involves working with fractions. Should you need a refresher course on fractions, I recommend starting with Math Dude's series of podcasts on fractions (starting with the Nov. 16, 2012 episode). WhyU's video series on fractions, mainly from episode 12 to episode 17, are also very helpful. You'll also want to make sure you're comfortable adding and subtracting negative fractions with a common denominator.

Yes, I know the mentions of fractions, which many math students consider the real F-word, makes this sound scary, but once you get used to the process, it's not as bad as it may seem at first.

Getting back to Conway's Tangles, James Tanton's Rational Tangles PDF is the easiest introduction to the mathematics behind this puzzle.

If you're viewing this on a device that supports the Flash plug-in, NRich.maths.org has some wonderful tools to help you understand Conway's Rational Tangles, as well.

Not surprisingly, you're going to get the best understanding of Rational Tangles from the inventor himself. I'll wind this post up with John Conway's full, 74-minute Tangles, Bangles, and Knots lecture, courtesy of UCTV Prime:


Making The Most Of What You Know

Published on Thursday, March 07, 2013 in , ,

Ianmacm's number pictureI once met a man who didn't get too far in school. He explained that he had to leave school to help support his family, so the most advanced math he ever learned was multiplying and dividing by 2. He said he never got to understand more advanced things like fractions.

Like me, you might expect that this guy wasn't too bright when it came to math, but he understood how to make the most of what he did know.

I chose two random numbers, not so high as to embarrass him, and asked, “So, if I asked you to multiply, say, 38 by 29, you couldn't do it?” He explained that he could, but he just had to simplify the problem in his own way.

He asked me to write down the problem on a piece of paper:

He reminded me that he had no problem multiplying and dividing by 2, so he explained that he was going to simplify the problem. He calculated, “38 divided by 2 is 19, so put 19 below the 38. 29 times 2 is 58, so put 58 below the 29."

Now, I've seem multiplication simplified this way before, but since he'd mentioned he didn't care for fractions, I was wondering how he could simplify the problem any further.

Confidently, he continued, “Next, we do the same thing again. 19 divided by 2 is 9, so...” I interrupted, “19 divided by 2 is 9½, not 9.” He repeated that he never could deal with fractions, so he just told me to forget the fraction, and just put down 9.

I had to snicker a little, as I couldn't see how this would work. I then did as I was told, put the 9 down, and then put down the double of 58 he'd calculated, 116.

He continued in this manner, always getting the doubling right, but putting down things like 4 has half of 9.

I was about to explain that, due to the lack of fractions, the problem didn't come down to 1 times 928, so 928 wasn't the correct answer, when he stopped me. He explained he wasn't finished yet, and said that he was no going to run through the left column and idenetify which numbers as odd or even.

At this point, I was intrigued, as I couldn't see how he was going to get any kind of answer to the problem.

He explained that he was going to mark a little 0 next to the even numbers, and a little 1 next to the odd numbers. He quickly went down the column, muttering, “19 odd, 9 odd, 4 even...” and marking them accordingly:

"Now what?” I wondered out loud. He replied, “Now, I add up ONLY the numbers in the right column whose numbers in the left column are marked with a 1! According to what we have on the paper, that's 58 + 116 + 928."

Instead of writing the answer and carrying all the values, he did the addition problem in his head by working from left to right, starting in the hundreds column! He said, “100...plus 900 is 1000...plus 50 is 1050...plus 10 is 1060...plus 20 is 1080...plus 8 is 1088...plus 6 is 1094...plus 8 is 1102! That's it!"

"That's what?” I asked as he wrote the total proudly below the other numbers. He explained, “The answer to your multiplication problem, 38 times 29!"

I was astounded. Double checking, I found that not only did 58 + 116 + 928 equal 1102, but that 38 times 29 was 1102, as well!” His way of working through the problem was strange, but I had to admit it worked!

"There's more!” he announced, while I was still recovering from the shock. He pointed out that we hadn't identified one of the numbers on the left as odd or even, and quickly marked 38 with a 0, denoting it was even.

He then asked if I would read off the 1s and 0s column in order, reading from the bottom up. I verbalized, “100110.” This man, with a gleam in his eye, inquired, “You do realize that's 38 in binary, don't you?” Sure enough, Wolfram|Alpha backed him up on this.

"How could you possibly know about binary? I wondered. He joked, “Originally, I figured computers were beyond me, but once it was explained that they dealt mainly in powers of 2, and they had difficulty with fractions, too, I could instantly relate."

"Since school, of course, I learned about a few more things about math here and there, such as factorials. You know, 4 factorial is 4 × 3 × 2 × 1, or 24. 5 factorial is 5 × 4 × 3 × 2 × 1, or 120, and so on,” he recalled.

After I stated that I was familiar with them, he pointed out that when you work with larger numbers like that, you tend to get more and more multiples of 5, and of course many more even numbers, so factorials of numbers larger than 5 would always ended in 0. The factorial of larger numbers, of course, ended in many zeroes.

He then explained that calculations like this helped him work out exactly how many zeros would be at the end of a number's factorial. I was confused, but I no longer doubted him. He added, “We've been ignoring the 29 over there, so let's work out how many zeroes are at the end of 29 factorial. I'll start with 58, and erase the 1 rightmost digit from it, leaving 5 there. With the number below it, I'll erase the 2 rightmost digits. In the numbers below, I'll erase the 3 rightmost digits, and so on.” When he'd finished, the paper looked like this:

Using the remaining 5 and 1, he simply added them up to get 6, and claimed that 29 factorial had exactly 6 zeroes at the end of it (known as trailing zeroes). I had Wolfram|Alpha work out 29 factorial, and sure enough, while there were 7 zeroes in the entire number, exactly 6 of them were trailing zeroes. Embarrassingly, he pointed out that I could have asked Wolfram|Alpha about the zeroes directly, and save myself the trouble of counting them.

Like I wrote earlier, this guy may not have known much, but he could definitely make the most of what he did know.


OK, it's time to admit I lied about this guy. He's completely made up for the purposes of storytelling.

The math, however, is 100% real. The process of doubling and halving numbers, and adding only particular numbers is commonly referred to as “Russian Peasant Multiplication.” I discuss both this and the halving method to convert to binary last year, in October's Powers of 2 post. I first discussed this approach to the trailing zeroes phase last December.

Amazingly the process above, once understood, works for multiplying any two whole numbers. Unlike many mathematical feats taught on Grey Matters, the focus is not on speed or mental ability, but rather the sheer variety, power, and unexpected simplicity that can be derived from just a single math problem.

Try it out, have fun, and explore this unusual approach to math.


The Weird World of Continued Fractions

Published on Sunday, March 03, 2013 in , , , ,

Pau's Golden Spiral illustrationIn my previous Quick Feats post, I briefly made use of continued fractions.

As a concept, they aren't well known, yet they are well worth exploring. When you start learning about continued fractions, there are many seemingly-endless surprises!

Let's start with a simple division problem, such as 47 ÷ 17. If we run that problem through Wolfram|Alpha, its' pretty much what we expect. 4717 simplifies to 21317, the decimal goes on forever, and that the first 16 digits after the decimal point repeat.

Let's learn something new by changing our point of view. Go through this desmos.com demo, which uses a 47 by 17 rectangle. At each stage, the rectangle is divided up into the largest squares possible. The same is then done with any remaining area, until the entire area is divided up into squares of various sizes.

Assuming you've gone through the process visually and geometrically, we're now going to repeat the process arithmetically. We're going to be dealing with fractional division and reciprocals, so here's a video refresher course, should you need it.

Instead of starting with a 47 by 17 rectangle, we'll just start with the problem 4717. We've already seen that this simplifies to 21317, and isn't hard to see how this relates to dividing up our rectangle into 2 perfect squares, with a 13 by 17 rectangle left over.

The next step was dividing up the 17 by 13 rectangle into a one 13 by 13 square, leaving a 4 by 13 rectangle. We need to keep our fraction as the same, but somehow redefine it in terms of 13s at this point. Starting from the fact that a fraction multiplied by its reciprocal equals one, we can work out the following:

Yes, it's a rather strange-looking result, but at least we have the 13 on the bottom, where we need it, and the value of the fraction remains the same. Putting the 2 back into the equation, 21317 becomes:

Remember how we took that 13 by 17 rectangle and divided it up into a single 13 by 13 square with a 4 by 13 rectangle left over? Simplifying 1713 into 1 + 413 is the same thing. Not surprisingly, we can repeat this process of flipping and simplifying the fractions until we get down to our 1 by 1 squares:

Due to the way in which we flipped the fractions, it's not hard to understand why all the numerators (top numbers of the fractions) are 1. In fact, this is the standard way in which continued fractions are written, with all the numerators as 1 (there are exceptions, of course).

Ignoring the numerators for the moment, look at the sequence of the other numbers - 2, 1, 3, 4. If you walked through the desmos.com demo I linked above, you'll recognize this right away! The geometric process resulted in TWO 17 by 17 squares, ONE 13 by 13 square, THREE 4 by 4 squares, and FOUR 1 by 1 squares, just as our continued fraction resulted in 2, 1, 3, and 4!

That's basically what continued fractions do. They show you how to break up a number so as to better understand its structure, and can often help you discover useful patterns in the process.

To get a better understanding of continued fractions in a very clear manner, there's a wonderful series of father-and-son video series called MathForKids that explains them to any beginner very well. The following is their first continued fraction video:

Towards the end of that video, there's another surprise; the continued fractions help solve quadratics equations with far less difficulty than you probably remember from your days in school!

The second video in the series starts with the simplest continued fraction (all 1s), and yet another surprise develops from this simple pattern. The third video in the series shows you a wonderful shortcut for evaluating continued fractions that automatically generates approximate fractions for any number! The fourth video focuses on working out the square root of 2, and the final video focuses on generating Pi approximations.

For a detailed understanding of the amazing power of continued fractions, R. Knott's course, complete with homework assignments, is tough to beat. It even begins with a similar rectangular division explanation with which you're already familiar.

Plus magazine's Chaos In Numberland article goes on to show you some of the amazing uses to which continued fractions have contributed.

As I mentioned the beginning, the surprises you get as you understand more and more about continue fractions are a consistent treat. Take the time to explore them, and the treasures you'll discover will be well worth the time.