When I posted my Day for any Date Online Toolbox, a commenter named Jay notified about a method of getting the year key that was new to me, known by the rather bizarre name of "Odd + 11".

I recently ran across it again in the Scientific American article, What Day Is Doomsday? How to Mentally Calculate the Day of the Week for Any Date, so I figured it was time to give it a closer look.

The standard mathematical formula for finding the year key is as follows:



In my Day for any Date Online Toolbox post, I give a more detail explanation of this formula. While doing mod 7 (also explained at the same link under "Modular Arithmetic") isn't difficult, the rest of the formula isn't that easy to work through in your head.

This is where the Odd + 11 method shines. It give the same year key as the above formula, but is far easier to calculate mentally. So what exactly is this method? Here are the steps:

1. Let T be the year's last two digits.
2. If T is odd, add 11.
3. Let T = T/2.
4. If T is odd, add 11.
5. T = (T mod 7).
6. T = 7 - T.

Adding by 11 is easy, especially if you add 10 first, then add 1. For example, 74 + 11 = 74 + 10 + 1 = 84 + 1 = 85. Even 99 + 11 is easy: 99 + 11 = 99 + 10 + 1 = 109 + 1 = 110. Notice that this approach eliminates the need for any carrying!

Let's try this approach, and say the last two digits of the year are 76:

1. T = 76.
2. T isn't odd, so T = 76.
3. T = 76 / 2 = 38.
4. T isn't odd, so T = 38.
5. T = (38 mod 7) = (38 - 35) = 3.
7. T = 7 - 3 = 4.

If you scroll down to the table of years on this page, you will note that 2076 does indeed have a year key of 4.

Let's try the formula again, but with 63, so that we can see the effects of adding 11. when needed, to the total:

1. T = 63.
2. T is odd, so T = 63 + 11 = 74.
3. T = 74 / 2 = 37.
4. T is odd, so T = 37 + 11 = 48.
5. T = (48 mod 7) = 48 - 42 = 6.
6. T = 7 - 6 = 1.

Sure enough, when checking the year table, the year key for 63 is a 1! Notice, too, that the 63 example is a worst-case scenario, in which you had to add 11 both times. Even Odd + 11's worst case scenario isn't that hard to work through!

As I mention on that same page, taking the time and trouble to memorize the year keys for all years 00 through 99 will get you the key number in the quickest manner. However, if you don't want to put in that time and effort, the Odd + 11 method is easy and can get you the year key very quickly.

Also in the Day for any Date Online Toolbox post, I discuss the 28-year pattern. Within the same century (for our purposes, this means in the same set of years from 00 through 99), any two years that are exactly 28, 56, or 84 years apart will have the same key number. Knowing this, you can make your Odd + 11 calculations even easier!

Let's try taking advantage of this when given a year ending in 82. Before going through the standard process, we realize that we can reduce the 82 itself by subtracting 56 (it's too small to subtract 84, of course). So we figure 82 - 56 = 82 - 50 - 6 = 32 - 6 = 26. So, we'll start with T = 26 instead:

1. T = 26.
2. T isn't odd, so T = 26.
3. T = 26 / 2 = 13.
4. T is odd, so T = 13 + 11 = 24.
5. T = (24 mod 7) = (24 - 21) = 3.
6. T = 7 - 3 = 4.

Check the tables again, and you see that 82 does have a key year of 4, as does 26! Now you see how the 28-year pattern allows you to work with smaller numbers, yet achieve exactly the same results!

Even though I didn't run across it until earlier this year, I like it because of the ease it brings to the mental calculation of what is usually the most difficult part for most people.

Update (Oct. 31, 2011): As near as I can tell, the Odd + 11 method came from a paper by Chamberlain Fong and Michael K. Walters, titled Methods for Accelerating Conway's Doomsday Algorithm (part 2) (PDF).

(NOTE: Check out the other posts in The Secrets of Nim series.)

In the very first Secrets of Nim post, I discussed a variation of the game where you start at zero, and add objects up to certain total, such as 21. What happens if, instead of adding, you were to multiply?

Just as with previous versions, we need to start by laying down a few ground rules.

First, what's the goal? Since we're going to be multiplying, and numbers tend to get large quick, we'll need a number that's large enough to keep the game interesting, but small enough to keep from dragging the game on for too long. Those who remember Scam School's first Pi Day Magic Trick will understand why 1 million makes a good goal when multiplying numbers.

Since you can't really use objects, as in many other versions of Nim, the best thing to do is use a calculator. Obviously, multiplying by 0 is out, since you can't get out of that. Also, multiplying by 1 can't be allowed, for the same reasons you can't add 0 in other versions of Nim. So, the only numbers you're allowed to multiply by in this game are 2 through 9.

We'll play this as a standard Nim version, in which the first person whose multiplication results in a product of 1 million or more. As with all versions, two people will take turns.

Yes, because this is Nim, there's a definite winning strategy. So the major questions are: 1) What is the winning strategy? 2) Should you go first or second? And 3) How do you even go about determining a strategy?

Based on what you know about previous versions of Nim, see if you can work out a strategy. Once you think you've got an answer, check out Presh Talwalkar's post, Monday Puzzle: Alice and Bob race to 1 million. Pay special attention to the logic behind finding the winnable and losable ranges.

Once you understand which number ranges are safe and which ones are not, you'll realize that there's a problem. It would be nice if there were a simple pattern to the ranges, such as being 2-9, 20-90, 200-900, and so on, but that's not the case.

Because of this, you'll have to memorize the numbers. To simplify the task before starting, only memorize the winnable ranges. I recommend using the Peg/Major System to memorize the needed numbers. You can learn it via the videos on this playlist, as well as more detailed help here under Major system (“Peg” System). You can also find great tools to help generate mnemonics for the needed numbers here.

For example, when trying to remember the range 343 to 3086, you might use Rememberg to get a mnemonic for 343 such as "homeroom" or even "Sam Raimi". You then need a mnemonic for 3086. Rememberg came up with novelist Sam Savage, and pinfruit came up with "mass fudge". Of course, they both offer many more choices of mnemonics, so find something that suits you.

Once you've got your mnemonics, simply link them together to identify the range. For example, you might picture Sam Raimi diving into "mass fudge" (a large amount of fudge?). Then move on to the other ranges.

When you know the ranges by heart, you're ready to play. If you want to try practcing on your own, you could simulate another person choosing numbers with help from random.org. Naturally, it won't be an intelligent opponent, but this means you'll be better prepared for unexpected plays in the real world.

Halloween is a little over a week away. To add to the fun, I have a little challenge for you!

Using my free Verbatim 2 app, try and memorize a Halloween poem by the time Halloween itself arrives!

This purpose of this challenge, of course, to help you learn both how to use the app, and to have a fun poem you can recite at your halloween gathering!

If you've never used the Verbatim app before, you'll definitely want to watch the instructional video (at the link above), as well as use a short poem.

What are some good short Halloween poems? Children's poet Kenn Nesbitt always has a good selection of poems at his site. Among the ones I suggest are Boney Mahoney, I'm Not Afraid of the Dark, Melvin the Mummy, or My Brother's Not a Werewolf (this last one is best recited by kids).

If you're up for a longer work, try I Cloned Myself on Friday Night, or Oh My Darling, Frankenstein (the fact that it's sung to Clementine helps). Here's a video of the latter being performed at a school presentation:



You might know the classic Halloween movie The Nightmare Before Christmas, but did you know that it was based on an original poem of the same name by Tim Burton? Christopher Lee gives an excellent reading of this poem here.

Of course, the most classic Halloween poem has to be Edgar Allan Poe's The Raven. That classic is 108 lines long, so it may seem difficult to learn in less than a week. However, when I decided I was going to memorize this poem, it only took me 3 days to learn this, using the Verbatim 2 app.

Another good classic that's a challenge is The Cremation of Sam McGee by Robert Service (click here to listen to the author read his poem). If you've never read this poem, take this opportunity to read or listen to it now.

Because the Verbatim 2 app requires text input, I've tried to include links to the text version of all the poems above. If you're not above doing a little transcribing while listening to a poem, there are also several good poem for Halloween in my Bizarre Poetry post.

Those are my suggestions, but don't let that stop you from exploring on your own. Do you have an favorite Halloween poems, especially ones you've memorized?

Michael Frink, who developed a great blackjack presentation for the card memory feat in the Mental Gym, is back with more ideas of interest to Grey Matters readers!

This time around, he's developed a complete series of videos to help improve your mental math skills!

The entire set of Michael Frink's Mental Mathematics videos is available on this page.

Here's the first video, which starts you off with basic mental addition techniques:



If, like me, you find that the dark blue color on the blackground is hard to see at smaller resolutions, I suggest watching the videos in full screen mode. Compare the above video to the same video in full screen mode.

Currently, there are 15 videos, and I understand Michael has plans to add more as time goes on. If you're familiar with the more basic techniques, you may wish to jump ahead to ones such as the one dubbed My favorite party trick. If you like the trick for multiplying squares ending in 5, you'll like this one, as well.

Do you have any favorite techniques from this series, or even your own favorite mathematical techniques not covered in this series? If so, let's hear about them in the comments!

BIG NEWS: The mobile update to my Train Your Brain and Entertain (TYBE) memory training software is done and tested!

It is now available for sale at Lybrary.com for only $9.99 (a permanent $20 drop in price!). In this post, I'll show you the new look and features of TYBE.

If you previously purchased TYBE from Lybrary.com, the new version is available in your Lybrary.com account already. Simply download the new version.

If you purchased the CD version, contact me with the subject heading "TYBE Update", and in the body, include the first full sentence of the Introduction (The sentence immediately after the "Introduction" and "Benefits of Memory Training" headings), and I'll send it to you at no extra charge. Even if you can't run the old version of TYBE anymore, this can still be found in the Help folder as 001-Introduction.html.

It's important to understand that TYBE is a web app, not a native app. On desktops and laptops, you can place it anywhere and run it in your browser. On mobile devices, it needs to be run from a browser or file app that can store HTML programs, as well as run programs with HTML5 and CSS3 capabilities. More details are available in the ReadMe.txt file included with the program.

When you start it up, you see the main menu, consisting of Support (Manual and Preferences), Basics (basic memory techniques), and Memory Feats. Touching any individual line will take you to the corresponding section:

When you click a technique or a feat, you'll go to a quiz's start page, some of which feature settings (especially under basics).

For example, when you go to numbered lists, you'll see this page:

Starting this particular quiz will show you a numbered list (6 seconds per item), then the list is closed, and you're quizzed on the contents:

Each individual section also has help available in one of two ways. If you're having trouble with the interface, you can select Guided Tour, and an arrow will go through all the features with comments (as shown on the left), or choose the help for the technique itself, and it will jump to a series of pages that will detail the technique (as shown on the right).

More TYBE screenshots can be seen in this slideshow.

Among other features, you can save your own preferences, including Phonetic Alphabet settings (e.g. what letter sounds are represented by the number 8), peg words for the numbers 0 through 99, card peg words, and even a particular card stack. These will remain in the program once set, even if you close your browser and shut off your mobile device.

Since its an HTML program, it should be able to run in any standards-based browser with HTML5 and CSS3 capabilities, preferably with at least 5 MB of browser cache.

For only $9.99, you can buy Train Your Brain and Entertain, and use it to gain a memory that will return that value many times over!

October's snippets are here! This time around, there's lots of goodies for those of you interested in memory techniques!

• First, a big thank you to Lifehacker, for linking to Grey Matters in the post How to Quickly Figure Out the Day of the Week Any Date Falls On. Since that link, more than 20,000 new visitors have come to check out this site! The recent broadcast of National Geographic's Brain Games special no doubt helped interest in the subject matter.

• Since my post on the PAO system, I've discovered a great memory technique forum called Mnemotechnics. Actually, Mnemotechnics is much bigger than that, as it includes a memory technique wiki, a blog, custom user journals, and more! For me, though, the chance to engage in a forum with other memory enthusiasts is the heart of the site.

• The Great Courses (originally known as The Teaching Company) has posted a free sample video lecture online, including a download link. It's Dr. Arthur Benjamin giving a 33-minute lecture on How To Memorize Numbers, from the series Secrets of Mental Math. If you haven't already learned the Major/Peg System, this is a great way to start. I've also preserved it over at YouTube, but will take it down when they ask me to do so.

• In my PAO System post, I featured a video of a computer-generated walk-through of a building as a mnemonic, mentioning that it was built using Google Sketchup. If you like this idea, you'll be thrilled to know that it's not hard to use. I've put together a YouTube playlist of example and Sketchup tutorial videos, so you can learn to do this yourself.

Quick, name any city, state, or country! Before you look up the population of that geographic area, what would you guess is the leftmost digit of that number?

Similarly, what would you guess the odds are of that leftmost digit being, say, a 1?

Many people guess that the odds are no different from it being any other number, so they say 10%. While there are 10 digits, 0 can't be the leftmost digit of a population. So, there are actually only 9 choices.

“OK, so the odds are 1 in 9, or a probability of about 11.1%”, you might say. Surprisingly, the odds of the first digit of a randomly-chosen population, or a number representing just about any other naturally occurring quantity, being a 1 is more than 30%!

This very counter-intuitive observation is known as Benford's Law. It was first discovered in 1881 by Simon Newcomb, who noticed that the earlier pages of books of logarithm tables, which contained logarithms that began with the number 1, were far more worn than later pages. If you have, say, a dirty keyboard, you may have noticed a similar phenomenon.

In 1938, physicist Frank Benford rediscovered the principle, and researched it with a wider variety of data sets. This is why it's referred to as Benford's Law and not Newcomb's Law.

It's certainly an interesting observation, but what good does it do? Fortunately, James Grime is here to help us answer that question:



James Grime does go through some very complex explanations very quickly in the above video, so if you prefer to go through that information at your own pace, you may want to check out the article Looking out for number one, by Jon Walthoe.

Probably the most important thing to remember about Benford's Law, besides the distribution of the digits themselves, is when the Law applied. As has already been mentioned, when the data is too random, such as randomly chosen numbers, or too constrained, such as people's heights or ages (which only happen in a certain range), then the law doesn't apply.

Besides the population example with which I opened this post, another good random set of data would be something like Twitter. As a matter of fact, Christian S. Perone programmed his computer to lift numbers out of Twitter's public timeline, take a look at the first digit, and graph them. The video on his site shows the amazing results, with Benford's Law predictions in blue, and the twitter results in green. Note how they conform closer to the prediction as more tweets are read.

Fans of mathemusician Vi Hart will be glad to know that she gave her unique take on Benford's Law during a visit to Khan Academy. In part one of the video, she and Salman Khan discuss the mysteries of it. In part two of the video, they delve more into the whys and wherefores.

Of course, even when you understand the idea behind Benford's Law, it still seems surprising. For those who aren't familiar with it, the concept still seems mind-boggling. So, we've got a tool that works naturally in certain circumstances, yet produces surprising results when demonstrated. What do we do with such a tool? Correct, we use it to win money, as Brian Brushwood shows us in this week's Scam School:



Even with everything you may have learned at the earlier links, it's fascinating to watch the psychology behind the Scam School version of Benford's Law. One side is made to seem more attractive in the bet by being loaded with more numbers. Also, 4 is taken out of the equation entirely, so as to actually worsen the odd for the team that chooses the larger numbers (they're forced to use 5 through 9 instead of 4 through 9). About the only legitimate part remaining in the presentation is the lack of concern for the choices made, as long as they're not from too constrained or too random numbers.

The fact that you can present this in a bar without carrying an almanac (which some scammers used to do in the days before smartphones) makes this much more pleasant, as well.

Just out of curiosity, what was the population of your chosen geographic area at the beginning? Look it up now, and see if it would fall on the 1-3 side, the 5-9 side, or the no-man's land of 4.

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.