One of my favorite mental challenges, as many regular Grey Matters readers know, is the Knight's Tour. The challenge is, using only the chess knight's L-shaped move, to land on each of the 64 squares once.

Mentalist Richard Paddon recently released a download resource titled The Knight's Tour: A Scenic Journey. In this post, I'll take a close look at this new take on a classic feat.

We'll start with a quick peek at Richard Paddon himself performing the Knight's Tour, via the teaser ad:



The Knight's Tour: A Scenic Journey comes as a set of 3 files: A 45-page PDF of the same title, a 16-minute MPEG file of Paddon's performance, and the Knight's Tour Windows application used by Paddon, and programmed by Dave Everett.

In the PDF, right away, the author emphasizes the importance of developing drama in the Knight's Tour presentation. The first parts of the actual instruction, however, focus on developing the path through the board. Much of this part of the book may be familiar to readers of the “Knight's Tour” section of Paul Brook's Chrysalis Of A Polymath. However, Richard Paddon does add some new and helpful notes, such as the section on what he has dubbed “delta values”, which are familiar to those who have programmed a Knight's Tour, but little discussed in the use of performances.

In the next half of the book, Paddon discusses the presentational details. He starts with the benefits of the Knight's Tour, including its uncommon nature, and its huge potential on an emotional and theatrical scale. The thoughts behind the presentations are well laid out. Even if you disagree with any aspect of the presentation as written, you at least have a good starting point of why particular choices were made.

One of the more interesting choices is ending on a selected square, as seen in the above video. As the board empties, the chosen square becomes a more and more important focus, and becomes a natural point of building tension. The PDF winds up with a detailed description of how to use the program.

There are very few weaknesses in this product overall. One of the one that stands out to me as both a programmer and a blogger of mental feats was the choice of the Comic Sans font for the numbering of the board. If you're taking as much care as this author does to make an impact on the audience, there's probably better ways to label your board than a font designed specifically to have a comic-book appearance. On an equally minor note, the lightning in many of the shots of the performance video could be better. The importance of the video is for a more complete understanding of the presentation, so this isn't a huge drawback.

Overall, this is an excellent value for anyone seriously interested in performing the Knight's Tour. The basics of working through the path may be easily accessed in multiple sources, but the depth of knowledge that is presented, as well as the use of multiple media to demonstrate this make this the most complete lessons about all aspects of the Knight's Tour and its proper performance.

It's available for only $9.95 over at Lybrary.com and is a remarkably great value for that money. If the Knight's Tour interests you, Richard Paddon's The Knight's Tour: A Scenic Journey is a must-read.

People are often confused as to why the dates of Easter moves around so much from year to year. It moves so much much because Easter is the first Sunday after the first full moon after the first day of spring.

If this sounds confusing on its own, consider that the Roman Catholic and Eastern Orthodox churches use different calendars, which can yield different dates as a result!

Thanks to the work of John Conway, though, it is possible to work out the date of both Roman Catholic and Orthodox Easters in your head!

Some basic understanding and practice are all that's really needed to be able to calculate the Easter date in your head for any year from 1900 to 2099. In order to help make everything clearer, I've posted my new Easter Date For A Given Year tutorial over in the Mental Gym. To make it easier to learn, the tutorial is broken up into several steps:

• The introduction explains the rules for Easter calculation in detail, as well as what you need to know to get started.

• The next section explains how to calculate the date of the traditional Roman Catholic Easter. After learning how to work out the date of the Paschal full moon (the first full moon after the first day of spring in a given year), you then learn how to work out the date of Easter for that same year.

• If you want to impress others by performing this feat, there's an entire section of presentation tips that can help make this feat entertaining.

• The method for calculating the date of Orthodox Easter is covered another section, as well. Assuming you can work out Roman Catholic Easter, there are surprisingly few changes involved in working out the Orthodox Easter date.

• Finally, there's another section for those adventuresome souls who want to venture on and work out Easter dates in other centuries. Here you can find out what changes need to be made to the original calculations.

Since practice is important, I've also developed a set of interactive Easter date quizzes. Since you work through each section verbally in a step-by-step manner, the quizzes work the same way. In the first quiz, you simply work out the paschal full moon date for Roman Catholic Easter. In the next quiz, you're asked about the paschal full moon and Easter dates. The Orthodox quizzes are similar, and start with the Roman Catholic dates first, since you need that information as a starting point.

If you put in a little understanding, a little practice, and a little time, you may surprise yourself (and others) with an impressive new skill!

One of the more popular Mental Gym tutorials, probably because it's short and simple, is the Squaring 2-Digit Numbers Mentally tutorial.

Playing around with the methods in the math section of that tutorial, and doing a little research, I've run across an interesting pattern that make the calculations even simpler.

I'd noticed a pattern in some of the squares concerning their last two digits, but never really thought about the possibilities until I ran across this page about squares.

I've reproduced a slightly modified version of the number arrangement from that page below, but with the last two digits of each square highlighted. Although it's not shown here, the 2-digit pattern does continue on forever.

x	x2	x	x2
1	01	49	2401
2	04	48	2304
3	09	47	2209
4	16	46	2116
5	25	45	2025
6	36	44	1936
7	49	43	1849
8	64	42	1764
9	81	41	1681
10	100	40	1600
11	121	39	1521
12	144	38	1444
13	169	37	1369
14	196	36	1296
15	225	35	1225
16	256	34	1156
17	289	33	1089
18	324	32	1024
19	361	31	961
20	400	30	900
21	441	29	841
22	484	28	784
23	529	27	729
24	576	26	676
		25	625

Presh Talwalkar's Mind Your Decisions blog, has many interesting posts, some of which I've covered before.

I was thrilled recently when he began to make a series of videos about various mental math techniques!

It seemed to start about 2 months ago, when he posted the following video about squaring numbers ending in 5:



At the same time, he posted a companion video explaining why this trick worked:



Apparently, they've proved popular, as he's released a small series of them so far! Just like the above pair, each lesson provides a video teaching the method and an explanation behind the method. This is a good approach, as some may just want to impress their friends and family with a technique, without worrying about why it works, can just learn and go, while those who are more curious can take the extra step in learning about it.

Among the more unusual methods taught is the following method for squaring 34, 334, 3334, etc. and 67, 667, 6667 (explanation video):



A more standard technique is taught in this video about multiplying any 2 numbers between 11 and 19 (explanation video):



Presh Talwalkar has also made several short mental math videos that each teach simple techniques and includes simple explanations along the way. The first of these was a video on the rule of 72, a way of calculating how long it will take an investment to double:



Another one was this handy lesson in calculating percentages in your head:



My favorite of this group, however, is the following video on converting decimal numbers to binary in your head, as I've managed to impress even well-posted computer programmers with this feat:



If you enjoyed these videos, take a look around the rest of the Mind Your Decisions blog. Presh posts math puzzles every Monday, game theory posts every Tuesday, and other posts as thing grab his interest.

Pictures of the Mandelbrot set, such as the one to the left, are pretty to look at, and fascinating when you zoom in to see more detail.

But the origin of the design often isn't made clear. In this post, I'm going to slowly step through each important detail in way that helps you understand it.

Recently, Numberphile posted a video about the Mandelbrot set, as explained by Dr. Holly Krieger of MIT:



Even this video tends to gloss over some points, so my explanation will be somewhat slower, and build up differently.

THE BASICS

Let's start with something very familiar, the number line you probably grew up seeing in your elementary classroom every day.


We're also going to start with a very simple formula: z² + c. The variable c will act as a chosen starting point, and we'll be putting different numbers in the formula, except that c will remain constant (c is for constant).

What about that z², though? That, of course, represents z multiplied by itself, and we'll be changing the value of z. Where do we start? As in the video, the starting point for z will always be 0. The next value for z, though, is whatever we get out of the equation from the previous calculation.

One of the simplest starting point is 1, so let's demonstrate with that. Since we chose 1, the formula becomes z² + 1. Using 0 for our initial value of z, we have 0² + 1 = 0 + 1 = 1. If we take this 1 and use it as our new value of z, and keep using the previous result for our new value of z, we get the following values:

• 1² + 1 = 1 + 1 = 2
• 2² + 1 = 4 + 1 = 5
• 5² + 1 = 25 + 1 = 26
• 26² + 1 = 676 + 1 = 677

You can, of course, continue doing this as long as you like.

Each of these repeated calculations is referred to as an iteration of the formula. Perhaps not surprisingly, these numbers get bigger quickly after only a few iterations. We can have Wolfram|Alpha run through the z² + 1 formula for us. Click on the More button in the Values pod will show that after only 9 iterations, the number is already up to about 1.94727 × 10⁴⁵ (roughly 2 followed by 45 zeroes)!

NOT EVERY NUMBER GETS BIGGER

What's the big deal? Doesn't every number get bigger? No, not all of them do. One of the more interesting numbers that doesn't get bigger is -2. If we run z² + (-2) through a few iterations, it quickly reaches a result of +2 and never gets any bigger!

If we use a number that's even slightly smaller than -2, such as -2.000001, we can see that such numbers will continue to grow. That's why the 2 cases in the video are defined as Case One: Blows Up and Case Two: Stays Small (<= 2), and why 2 is considered a special value in the Mandelbrot set.

Now, in the video, Dr. Krieger mentions that ¹⁄₄ is on the boundary, so what happens with that number itself? Not only does ¹⁄₄ NOT surpass 2, it never even reaches 0.5!

So, what we have so far is just a line from 2 to about ¹⁄₄. Why about? Well, ¹⁄₄ isn't as solid a border as -2. For example, we can move a tiny bit to the right on the number line, say to ¹⁰⁰⁰⁰¹⁄₄₀₀₀₀₀, and we find that point never exceeds 2. That point is too small to accurate mark on the number line, as compared to the next handy fraction, such as ¹⁄₃, which eventually does grow past 2.

INTO TWO DIMENSIONS

Let's finally get off the number line and jump into 2 dimensions. We could attempt to use standard graph paper-style Cartesian coordinates of the form (a, b), but the mathematics required to handle 2 separate coordinates quickly complicates the formula.

This is why the imaginary number plane and complex numbers come into the picture. Read BetterExplained.com's A Visual, Intuitive Guide to Imaginary Numbers to get a good grasp of how these work. If those interest you, you may want to read Intuitive Arithmetic With Complex Numbers and Understanding Why Complex Multiplication Works at the same website.

For c, let's try the complex set of coordinates 0.5 + 0.5i. Running those coordinates through Wolfram|Alpha we see that after only 14 iterations, the results become roughly (-6×10²⁸⁷) + (-4×10²⁸⁷i)!

Thanks to the effects of i, the results can obviously get into negative numbers quickly, and the answers get confusing. What we need to do, then, is translate each result into polar coordinates. We're not so much worried about θ (the angle), but rather r (the radius), which is measured as the distance from the origin point. The radius is always a positive number, and we can simply judge whether this is bigger than 2.

Converting (-6×10²⁸⁷) + (-4×10²⁸⁷i) to polar coordinates, we get a radius of roughly 7.2111×10²⁸⁷ units from the center, so these coordinate definitely escape 2!

To cover two dimensions, we need to focus on a particular range. In the video, Dr. Krieger talks about using a radius of 2 units in all directions, but most of the Mandelbrot pictures you see tend to use a range of -2 to +1 on the real number line, and i to -i on the imaginary plane.

Besides a range, we also need to choose a “resolution.” To keep things simple, we might choose a resolution such as 70×30. So, we'd break up the space from -2 to +1 into 70 equally-spaced coordinates horizontally, and we'd break up the space from i to -i into 30 equally-spaced coordinates vertically.

Then, for each of those 2100 (70 × 30) coordinate pairs, we'd put them in complex coordinate form (a + bi), and assign that to c. We'd run through z² + c for that set of coordinates and see if the radius ever became larger than 2 (in other words, whether there's a limit to the iterations). If so, we would assign a white square to it. If not, we would assign a black square to it (or an asterisk, if we're working in ascii).

Mapping those results gives you an image that appears as follows (click the image to see a larger version):


This picture is the result of 2100 calculations, with each calculation being iterated to varying degrees. Is it any wonder than the Mandelbrot set couldn't have been discovered until computers became commonplace?

Now things are starting to look much more familiar! For a better picture, just as you would with any display, you increase the resolution. Given 300×220 as the resolution, you would get an even clearer picture (click the image to see a larger version):



PLAYING WITH THE RESULTS

Those simple black and white versions only show which sets of complex coordinates, when iterated through z² + c, fall into which of the 2 cases. The pictures you usually see are much more colorful. What is the deal with those pictures?

Instead of asking just a yes-or-no question (Does the radius of the iteration ever exceed 2?), we change the question to “How long before the radius of the iteration exceeds 2?” If the radius of the iteration never exceeds 2, we use black. If the radius of the iteration exceeds 2 after 1 iteration, we'll use a color we'll call color 1, after 2 iterations we use a 2nd color, after 3 iterations we use a 3rd color, and so on.

That approach, known as the escape time algorithm, gives us the standard Mandelbrot set images with which you are likely familiar (click the image to see a larger version):


Notice that you even get better contrast with this version, as you can see “branches” that aren't easily seen in the black-and-white versions.

As an aside, that name, escape time algorithm, brings up a good point. In the Numberphile video above, they make it sound as if distance is important. Instead, it's actually the amount of iterations it takes a number to escape the value of 2. The Mandelbrot isn't a measurement of distance, but rather time.

The colors chosen are usually a matter of personal preference, which is why you see so many different variations. If you don't like the banding effect of sudden color changes, there are even ways to smooth that out.

Remember how it's impossible to tell when moving the tiniest amount to the right of ¹⁄₄ on the number line would result in a set of iterations whose radius would ever exceed 2? Thanks to computers, we can choose a smaller range, an appropriate resolution, and zoom in to study the complexity of that particular area (known to Mandelbrot set fans as “Elephant Valley”):



Consider that all these results have been coming from iterating the same equation, z² + c. What happens if we try taking z to another power, such as z³ + c or z⁴ + c? Here's what happens when changing those powers from 2 to 5:


Once you understand the basics, there's plenty of ways to explore the amazing world of fractals such as these. I've written on fractals several times on Grey Matters, including Iteration, Feedback, and Change: Fractals, R.I.P. Benoît Mandelbrot (November 20, 1924 - October 14, 2010), and A Closer Look, so you can explore further if you wish.