Many of the memory feats I teach require plenty of practice.
Today's mnemonics however, are short and sweet, and so they don't require much practice. As a bonus, they're all based on something you always have handy - your hand!
This first mnemonic is probably familiar to you already. It's a way of using your knuckles to recall the lengths of each month.
Make two closed fists, and put them together so you're looking at the back of your hands, with the sides of your first fingers touching each other, as in the illustration below. As you can see, all the months located on knuckles have 31 days, and all the months located between knuckles are only 30 days long, or shorter in the case of February.
A similar, yet less well-known mnemonic, uses a piano keyboard. Starting on the F key as January, continuing with the F# (F sharp) key as February, and so on, ending with December on the E key, all the months represented by white keys will have 31 days, while all the months represented by black keys have 30 or fewer days.
Next, we move from the knuckles to the pad at the base of your thumb, which can be used, surprisingly, to tell you how a steak feels at various cooking levels. The graphic below explains this simply:
Using hands for mnemonics is hardly a new idea. In fact, many of the imperial measurements, such as inches, feet, yards, and miles, were originally based on measurements of various parts of the body. In fact, knowing the exact measurements of various parts of your own particular body (assuming you're not still growing) can be very helpful in making accurate measurements without a rule. On Quora, Peter Baskerville explains which measurements are the most useful to know.
Back in the days when few people went any further from their home in their lifetime than 7 miles, basing the measurements on one's own body was the quick and simple. As wider travel and communication became possible, this caused some confusion, as in the classic cautionary tale about the Queen's bed:
The last hand mnemonic I'll mention requires a more practice and a better understanding of trigonometry than the others. When calculating special angles in the unit circle, it's possible to use your hands for quick and accurate calculations for sine, cosine, and even tangent:
The video above only covers the 1st quadrant (the upper right quadrant) of the unit circle, but I expanded on this system to cover the full 360° circle in a post 2 years ago.
It's truly amazing how much knowledge you can keep at the tip of your fingers with a little practice, isn't it?
Many of the memory feats I teach require plenty of practice.
There's an old magic trick out there that's been in the public domain for so long, its origins seem to have been lost.
In the classic version of the trick, a card is chosen, and a mysterious person is called. Somehow, this person is able to name the correct card, despite not even being in the same room, or even the same state or country!
Magicians know this trick as The Wizard, as most of them learned the version by that name from the book, Scarne on Card Tricks. You can read that particular trick for free online (page 42, page 43).
As with many tricks, the presentations grow and change over the years. Some magicians also know this same trick as The Phantom or some other equally mystic name. When Scam School taught this routine (YouTube link), their figure of choice was a secret member of a government conspiracy:
If you think about it, any bit of data which can be identified by two simple pieces of information, in a manner similar to grid coordinates, can be coded in a similar fashion. It's quite obvious that playing cards can be broken down into 2 bits of information, their value (Ace through King) and their suit (clubs, hearts, spades, diamonds). What if the data to be coded didn't have 2 such obvious factors? If we could manage that, this routine could be even more deceptive!
Max Maven developed a version called Remote Pager in which a word is chosen from the following old letter
Impossible, but true! A demonstration of intuition, custom tailored for you by Mister Zulu. Cnoose any word in the paragraph of at least four letters. After you choose a word, contact me by phone. Believe it or not, I'll announce the word you are thinking of! Imagine tne surprise ~ but be on guard: I presume my demonstration is going to haunt you...How would you even begin to code the chosen word? Even being familiar with the above methods, the particular coding isn't easy to work out here.
If you can't figure it out for yourself, Word Ways magazine wrote up Remote Pager here, complete with the explanation. As with all of Max Maven's routines, the approach is subtle and ingenious.
Play around with this routine, which is even more portable now thanks to smartphones. If you have any fun stories of performing this, I'd love to hear about them in the comments!
When you're just starting out in mathematics, infinity is little more than a neat concept.
Infinity, at that point, is simply the idea that numbers go on forever, and once you accept that, you seem to be fine with the concept of infinity. As you learn more about math, though, you start running into more and more problems relating to infinity, and the concept starts to get weird.
David Hilbert came up with a wonderful example that helps people begin to grasp unusual concepts concerning infinity. His example is known as Hotel Infinity, and is explained here by Martin Gardner, with some clear and amusing illustrations.
Imagine a hotel with an infinite number of rooms (especially easy if, like me, you live in Las Vegas). Also, imagine that an infinite number of people are staying there, so every room is occupied. What happens when 1 person, a UFO pilot in Martin Gardner's version, wants a room? Everybody can be moved to a room number that's 1 higher than their current room, so the first room is now available for the UFO pilot.
Similarly, if 5 couples show up, everyone can be moved to a room number that's 5 higher than their current room, so five rooms are now available for the new couples.
Let's make this more challenging: What happens if an infinite number of people now want a room? You can't simply have everybody move to a room number that's infinitely higher than their current room, so how would you solve this problem?
The answer is surprisingly simple. Have everybody move to a room number that's twice as big as their current room number! Now, the infinite number of previous guests are all staying in even numbered rooms, and the infinite number of new guests can now move into the odd-numbered rooms! Since there are an infinite number of even numbers and odd numbers, this works.
In the late 19th and early 20th centuries, Georg Cantor started talking about different sizes of infinities in a manner similar to this, and even the great mathematical minds of the time scoffed. Eventually, however, mathematicians did come to accept this idea. How exactly can there be different sizes of infinities? You can learn more about this unusual concept in a basic way via Martin Gardner's Ladder of Alephs article. Videos from TED-Ed and Numberphile examine this concept in more detail.
Even though such discoveries about infinity are relatively new, even the ancient Greeks understood the importance of analyzing infinity. Zeno of Elea developed several paradoxes involving infinity which still challenge mathematicians today. TED-Ed's video below explains the Dichotomy paradox:
A video from Numberphile discusses both the Dichotomy paradox and the Achilles and the Tortoise paradox, and how they relate to infinity:
Not that Hilbert's Hotel Infinity thought experiment even makes this clear. It's even a little startling to realize that it can help you reduce this to a simple algebra problem.
In BetterExplained.com's newest post, An Intuitive Introduction to Limits, these odd ideas about infinity help you understand the concept of limits in calculus. The introduction sums up the challenge perfectly: Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there… ” Ugh. Here’s how I learned to enjoy them: Concrete examples, including a buffering soccer video, make even this odd concept clear.
If you grasped limits from that article, you're probably ready for the concept of an infinite series, explained in detail in this 15-minute video from WhyU. It's amazing how a little knowledge of infinity can quickly take you through such advanced concepts.
If you're confused by the infinite series video, take some time and go back through the earlier concepts of infinity to make sure you understand them. Start by reading the first half of this post, followed by the next quarter of this post, then the next eighth of the post, then the next sixteenth...
Werner Miller has certainly been keeping busy!
Not long after the release of sub rosa 3 and 4 comes his newest book, EZ-Sqaure 5!
E-Z Square 5 is available as an ebook from Lybrary.com, available in English and in German.
As with previous books in the series, this one features a particular routine concerning magic squares. The major difference here being that these magic squares are created using playing cards, similar to Richard Wiseman's The Grid and Chris Wasshuber's Ultimate Magic Square, both of which are acknowledged in E-Z Square 5.
Werner Miller explores the possibilities through 3 main routines, and a bonus routine. The first routine is the simplest, in which the spectator generates a total by selecting 4 cards out of 16, and you quickly deal a 4 by 4 square with 16 different cards whose rows columns and diagonal give the same total. The second routine, which is my personal favorite, has the spectator cut off about half the deck, and you as the performer are able to create a 4 by 4 grid whose rows, columns, and diagonals are equal to the number of cut-off cards.
In the 3rd routine, the spectator cuts off a group of cards, and deals them into 2 piles, while the performer uses the remainder of the deck to create a 5 by 5 grid of cards. When the magic total is revealed, it proves to be the same as a number created from the top 2 values on the spectator's piles!
The bonus routine may be familiar if you've purchased Werner Miller's da capo 3, as it is Squaring the Cards. In this 4 by 4 magic square routine, the magic square's total is equal to the total of the remaining cards not used in the routine!
If you're nervous about handling the various arrangements and calculations required in normal magic square routines, EZ-Square 5 is an excellent choice, as the routining and use of playing cards takes care of much of the work automatically. As any Werner Miller fan already knows, not much more than basic card knowledge is required in his routines. I recommend E-Z Square 5 highly!
Is it time for April's snippets so soon? It only seems soon because March's were so late.
This month, the focus is on resources which help you remember more effectively!
• Just today, Forbes.com posted a wonderful article titled 6 Easy Ways to Remember Someone's Name. In addition to the standard advice, I especially like the tip of asking them a question, so you can take some time to mentally link their name with their face.
If you want to examine this in more detail, I've prepared a YouTube playlist focusing on memorizing names and faces. There's also an excellent book titled How to Remember Names and Faces: How to Develop a Good Memory (originally published in 1943, but the advice is still very sound!). I've also covered various mobile apps that help you practice these techniques.
• Speaking of apps, there's a new free iOS app called Brain Athlete (iTunes link). This focuses on memory-competition feats, including memorizing numbers, word lists, and playing cards. If you've read Joshua Foer's Moonwalking With Einstein and/or read my PAO system post, you should have a good understanding of the basics.
If you get stuck finding a certain person for your PAO system, here are links to lists totaling 10,000 famous people to help. No actions are objects are included, as these need to be developed based on how you imagine each of these famous people.
• Every so often, I run across free memory web apps that I find useful, such as these. The newest one I've found is the Major System Database. It's very simple and direct. You can find words for a given number, the numeric equivalent of a given word, or even break up numbers into small groups and give you mnemonics for each group!
• For Windows users, there's a new free program available, simply titled Memorization Software. It's designed to help you remember various types of texts, such as lyrics, poems, and speeches word for word. The tutorial video below (no audio) gives you an idea of the various approaches used here.
If you like this approach, but don't have a Windows machine (or even if you do!), my web app Verbatim 2 (Video Tutorial link) is also free, works in a similar manner, and runs in any modern browser.
Scam School's newest episode is right down the alley of many regular Grey Matters readers.
Their 265th episode teaches how to memorize a list of 20 items very quickly. This is a classic feat and a classic technique, but it's a rare treat to see it actually performed.
You can find the episode on Scam School's own site, and on YouTube, as well. Let's get right to the memory feat and the explanation:
As you see it performed in the video above, it's a pretty bare bones technique. That's a great way to learn it, but there are other handy tips that can take it to another level.
First, as the items are called out, make sure to specifically ask for objects you can encounter in everyday life. That way, you don't get hard-to-picture images such as sickle cell anemia, as in the video. Hopefully, you don't encounter maggot-infested tacos, either, but at least it's easier to picture.
Also, ask for more details. If someone calls out a car, ask for a specific model of car, or even the color. This additional level of detail makes the feat seem more difficult, but actually makes the image more vivid, and thus easier to remember.
In the video above, you always see them calling out numbers, and having the item given in return. As long as you've formed your images effectively, there's no reason you can't have them call out the items and give the number in return, as well.
Once you've memorize the list, and they're starting to call out numbers or items, each time you recall the image, imagine your mnemonic frozen in a block of ice. If someone calls out 2, and you recall that's a unicorn, imagine the unicorn with a shoe for a horn frozen in a block of ice. Have them call out numbers or items until they've covered about 60% of the list or so. After that, you can recall the items and numbers that were never called by mentally going through the list from 1 to 20, and recall which images weren't frozen! This is a great finish!
You don't have to use rhyming pegs, of course. On my Memory Basics page, you can also learn shape-based pegs, or even the Major System, which allows you to turn any number into a vivid image.
If you've already learned pegs from something else, such as the pegs I teach for 1 through 27 in my Day One calendar feat, you can quickly adapt those, too.
To learn more about various peg systems, I have one YouTube playlist focusing on simple peg systems, and another focusing specifically on the phonetic peg system.
Yes, this feat requires a little more work than most Scam School feats, but it's worth it not only for the results, but also for practical everyday uses!
Last Sunday, I posted about wordplay in the context of memory challenges.
Wordplay is a far richer field than you may expect, so how about a look at wordplay just for the sake of wordplay?
Let's warm your brain up with a few word puzzles, shall we? Puzzles.com has several great word puzzles, and many are easy if you think of them in the proper way. The challenge, of course, is getting to that way.
If you think about it, the history of languages is wordplay in and of itself. Words get used, misused, understood, and misunderstood as they travel from culture to culture, eventually leading to their modern meanings. TED-Ed.com's recent series of videos, Mysteries of Vernacular, each of which focus on a single word, teach about some fascinating word origins. My favorite, at this writing, is the one about the origin of the word CLUE:
If you're not already familiar with the myth, watch Jim Henson's Storyteller: Greek Myths - Theseus and the Minotaur here.
Probably the best regular resource for wordplay would have to be Word Ways, a quarterly journal that's been in publication since 1968, started after Martin Gardner first suggested the idea. PDFs of every article are available via the Butler University's digital comons, although accessing the most recent 8 issues require an account. You can access text versions of the most recent issues via thefreelibrary.com, but being text versions, illustrations are naturally missing.
One of the great things about Word Ways is that, even though it's focus is words, the variety of wordplay is astounding. Regular Grey Matters regular will enjoy a wide variety of articles on words mixed with math, mnemonics, and magic. This is one of those cases where you should simply spend time exploring, and let yourself get lost in articles that grab your attention.
What are some of your favorite types of wordplay and wordplay resources? Let me hear about them in the comments!
My regular readers might be looking at the title and picture and wondering whether they're on the right blog. I normally post about doing math and memory in your head, not on a calculator.
However, using your brain in conjunction with a simple 4-function calculator, you can get much more out of them than you may have ever thought possible.
Since you only see functions for addition, subtraction, division, and multiplication (and sometimes a square root function) on a 4-function calculator, most people limit their use to just those few functions. However, even a simple pocket calculator has a few hidden features that, when combined with an understanding of varous aspects of math, allows you get much more out of it.
Before you try these out, make sure you're using an actual 4-function calculator, as more complicated calculators act differently. Many calculator apps on mobile devices appears to be 4-function calculators in one orientation, and scientific calculators in another orientation. Unless you discover for yourself otherwise, these calculator apps are generally always working as a scientific calculator, even when it appears otherwise.
Over at Ted's Math World, there's a very complete course in using a 4-function calculator, which includes these sections:
1: Introduction to Programming a Four-Function Calculator
2: Integer Powers
3: Integer Roots
5: Compound Interest
7: Extra Decimals for Square Roots
8: Some Arithmetic Shortcuts
Even if you don't go through every section, at least go through the introduction section, as you may learn about some hidden features of your pocket calculator. Ted's Math World also features a very simple continued fraction approach to square roots, and in the Integer Roots section, you can learn how to enter this into your calculator.
Eddie's Math and Calculator Blog also has a course on calculator usage called Calculator Tricks. Surprisingly, there is very little crossover with the above course, and this one gets as far as dealing with 2 by 2 matrices! Eddie's course is available at these links: Part 1, Part 2, Part 3, Part 4, Part 5.
Back in 1974, when 4-function calculators were just starting to become affordable and popular, Popular Science wrote up an excellent guide, including many common real-word uses, such as photography, cooking, and shopping. True, you might have apps on your mobile device that handle similar functions today, but it's still good to know how to handle them yourself. The article, titled New Tricks For Pocket Calculators, can be found in the December 1974 issue of Popular Science, on page 96, page 97, page 98, page 118, and page 119.
Go through these resources, and you'll start to get a good idea of just how much more powerful your 4-function calculator can truly be!
Don't forget to keep an eye out for the occasional individual tips, as well. For example, here's a quick way to find any root on a 4-function calculator, as long as you have a square root button available:
One kind of math that doesn't get much coverage on calculators is modular arithmetic. If you're not familiar with modular arithmetic, BetterExplained.com and Martin Gardner (page 1, page 2) have excellent introductions.
Surprisingly, even many models of scientific calculators don't have basic modulo functions. In the few places I have seen methods for working out the modulus on a calculator, the methods were similar to the ones taught in this xkcd.com forum thread.
That method is certainly useful, but I never cared for the back-and-forth nature of it. I developed another method (other people must have come across this, but I've never found a reference to it) which takes you straight to the answer. Let's say you're trying to figure out what 83 mod 13 equals. Simply enter 83 - 13 = on the calculator, and you'll see 70. Hit equals again, and you see it drop down again to 57. Keep hitting the equals button until you come to a positive number that is less than 13, and that's your answer! In this case, the answer is 5.
For any number x mod y, just start with by entering x - y =, and then keep hitting the equals button until you wind up with a non-negative number that's less (LESS - not LESS THAN OR EQUAL TO) than y, and that's the final answer.
This answer works well when the numbers are relatively close, or at least have the same number of digits. What happens, though, if you have to work out something like 96,528 mod 17?!? In this case, we use powers of 10 to help. What number starts with 17, ends in 1 or more 0s, and is less than 96,528? It's easy to see that 17,000 fits the bill, so we start with 96,528 - 17,000 =, and keep hitting the equals button until we get a non-negative number that is less than 17,000. After this, we wind up with 11,528. Now, drop a zero from 17,000 to get 1,700, and repeat the process starting with 11,528 - 1,700 =, resulting in 1,328. Repeating this with 170, we work our way down to 138. Finally, we go through this process one last time with 17, and we come to our final answer, which is 2.
So, when working through any modular problem, you can not only take the number itself out, but add an appropriate number of zeroes to the end, and them out by the hundreds, thousands, millions, or whatever scale is needed! This approach may take longer, but it goes to the answer directly, and helps you understand the process of modular arithmetic.
You can also use a similar process with addition in order to find congruent numbers. What numbers are congruent to 2 modulo 6? Start with 2 + 6 =, and you'll get 8. Hit equals again, and you should get 14, then 20, and so on. Each of the displayed results are numbers that are congruent to 2 modulo 6: 2, 8, 14, 20, 26, 32, etc.!
Give a little thought, a little fun, and a little effort to your simple 4-function calculator, and you just may be surprised by what you can do with it!