Saturday, November 30, 2013

How YOU Can Memorize 2000 Digits of a Number

Since it is two days after Thanksgiving, and many of us are probably eating leftover pie, I thought it would be appropriate to do a post somewhat relevant to the numerical pi. And I could do something very mathematical, but I just finished my first trimester at Phillips Academy Andover last week, and I needed a break after my nearly impossible MATH-380 final exam. As a result, I thought it could be fun to talk about memorizing numbers, pi and tau in particular.

When people hear about my Tau 2000 event, they often ask me if I have what they call a "photographic memory." This is not at all true. I don't even think the types of photographic memories advertised in pop culture really exist (I'm not an expert on neurology, so for more on that, I'd recommend reading this Scientific American article). The way I memorized 2012 digits of tau was all learned and practiced techniques, similar to my mental math presentations. I was not born with some gift or natural talent, it was just learning the methodology and practicing until I could do it quickly. Just like anyone can do mental math, anyone can be a memory expert as well, ranging from being able to remember 57890 digits of pi to being able to remember your car keys as you leave for work. There are techniques for it all.

First, let me introduce you to the Major System. This is a phonetic code that enables you to turn numbers into words. You store them as words, and later retrieve them as numbers. Basically, each digit is associated with a specific consonant sound.

1 is the t or d sound. It can also be either of the th sounds (see note below).
2 is the n sound.
3 is the m sound.
4 is the r sound.
5 is the l sound.
6 is the j, ch, sh, or zh sound.
7 is the k or g sound.
8 is the f or v sound.
9 is the p or b sound.
0 is the z or s sound.
Note: th (both the th in "that" and the th in "thing") is normally paired with 1, but there are other variations on the system that will put it with 8 or not include it.

This looks hard to memorize on its own, but it is actually not that hard. Here are some mnemonics that can help you.
  1. A t or d has 1 downstroke.
  2. n has 2 downstrokes.
  3. A m has 3 downstrokes.
  4. The number 4 ends in the letter r.
  5. If you hold up your hand with 4 fingers up and your thumb at a 90° angle, you will see 5 fingers shaped like an L.
  6. A J looks somewhat like a backwards 6.
  7. A K can be drawn with two 7s back to back.
  8. A lowercase f in cursive looks like an 8.
  9. The number 9 is a backwards p or an upside-down b.
  10. The word zero begins with the letter z.
You will also notice that the consonants that were paired together sound very similar. Your lip movement and tongue placement are the same in any of the consonant sounds chosen for a number (except for the th sounds, hence the inconsistency of its use).

You might be wondering why there are no vowel sounds on the list. There is also no h, w, or y sound. This is because you can insert these wherever you want between consonants and they mean nothing. With all of this in mind, you can begin turning numbers into words. Let's take the number 15. What words can this become?

Well, one is the t or d sound. Five is the l sound. Insert vowels, and you can get doll. Or tile. Or tail. You can also insert vowels at the beginning or end of the word and make deli, or Adele. You can also insert hs, ws, and ys to get hotel, towel, or yodel. Here are a list of the 66 words that can be made out of  the number 15 (I put the ones that I might use in a mnemonic image in bold print):

Addle, daily, dale, dally, deal, delay, dell, dial, dole, doll, dual, duel, dull, duly, dwell, ethyl, hastily, hostile, hotel, hotly, huddle, ideal, ideally, idle, idly, idol, it'll, italy, oddly, othello, outlaw, outlay, saddle, sadly, seattle, settle, societal, stale, stall, steal, steel, still, stole, stool, style, subtle, subtly, suicidal, sweetly, tail, tale, tall, tally, teal, tel, tell, they'll, tile, till, toil, toll, tool, towel, waddle, widely, yodel
Note that some of the words start with s. Since s is zero, this is referring to the number 015, which is normally still 15. These words do not work if 15 is part of a string of other digits such as in pi or tau.

The ones that I bolded are all nouns that you can create a mental image of in your head. As the Scientific American article that I linked to states, people naturally have a better memory for visuals (the reason why you might remember someone's face, but not be able to place the name). So, you might not be able to remember the number 15, but you can probably picture a doll, or a hotel, or a yodel (for this, I would think of the chocolate pastry, not the verb). If you are trying to remember that it is someone's address or apartment number, picture a relationship between the object and the person. Maybe the person is standing up on a stool shouting to a crowd of confused, awestricken people, or they are on the couch stuffing their face with yodels. The sillier your image, the easier it is to remember.

There are lots of memory experts who will create a list of "peg words," which are essentially 100 words that they will refer to when they are trying to remember a number between 1 and 100. It is certainly not a necessity, but it can often help if you are trying to come up with a word on the fly. Every person has a different list of words that works for them, so this is something that I would encourage you to make on your own. The website works very well to help create this list.

To memorize shorter strings of digits (something like memorizing 100 digits of pi), the best approach in my opinion is to create sentences out of your words. For instance, take the first five digits of pi: 31415. The only word that can be formed out of this is moderately, which isn't a great start to a sentence. However, it could be turned into "my turtle" or "Madrid law" or "Mother Yodel." The first 24 digits of pi create the sentence:

My turtle Pancho will, my love, pick up my new mover, Ginger.

Say this a few times and you will sadly have it memorized. And since you now know the code, you now have the first 24 digits of pi memorized. If you want to keep going, the next 17 digits are:

My movie monkey plays in a favorite bucket.

The next 19 are:

Ship my puppy Michael to Sullivan's backrubber.

If you want to take it to 100 digits, you can use:

A really open music video cheers Jenny F. Jones.

And my personal favorite:

Have a baby fish knife so Marvin will marinate the goosechick.

This method works great for condensing large quantities of numbers into a small amount of silly, memorable sentences. However, once you get up towards 300, 400, 500 digits, it is really tough to remember the exact prepositions and linking verbs you used, which contribute to the digits. Because of this, the method I used for memorizing 2012 digits of tau is a different variation. Rather than just memorizing plain sentences, I used a technique called the memory palace.

A memory palace is essentially a place that you can mentally visualize that you put the images that you create in. For instance, your drive from your house to work might be a memory palace. Your elementary school campus could be your memory palace. You can even create an imaginary place to be your memory palace. Let's pretend your memory palace is inside of your house. The first ten loci (places to put the images) might be:
  1. Your bed (in your bedroom)
  2. Your closet
  3. Bathroom
  4. Hallway
  5. Other Bedroom
  6. Stairs
  7. Living Room
  8. Dining Room
  9. Kitchen
  10. Front Porch
And you might have a grocery list with the following items:
  • Grapes
  • Carrots
  • Corn on the Cob
  • Yogurt
  • Cheddar Cheese
  • Marshmallows
  • Cheetos
  • Salt
  • Pepper
  • Ice
All you need to do is mentally "put" each of these items into the corresponding locus in your memory palace. For instance, the first item is grapes. You would put the grapes on your bed. But you wouldn't just put them there, you must do something to make the image stand out. First of all, you must embrace the image. Not only do you see grapes, but you smell the grapes, you taste the grapes. The more of your senses that you alert, the easier the image is to remember. The image also needs to be less dull than just a few grapes sitting on your blanket. Maybe have grapevines growing out of the back of your bed. Maybe visualize the grapes to have legs, and jumping on the bed. As long as it is a silly image that stands out in your mind, you will be able to remember it.

The next item on the list is carrots. The corresponding locus is your closet. Carrots grow out of the ground, so maybe you picture all of the mud that your sneakers have tracked into the closet has carrots growing in it. As long as you pull a carrot out of the mud, you will remember it is carrots. Or maybe there is a snowman inside with a carrot nose, or a carrot shoe-horn. The actual carrot aspect of the image can absolutely be subtle, as long as you can remember the image and this image triggers the thought of carrots in your mind.

Continue through the list, and you will have ten images in your head that will in fact be stuck there until you use other techniques to remove them (yes, there are techniques people use to forget things). Try this out a few times, and I'm sure you will find it very useful. If you have a list of things to do at work, you need to remember when to pick up your kids and bring them to their activities (you may even use the major system for translating times into words - if you need to bring your son to baseball practice at 4:15, you may just picture your son swinging his bat at a "hurdle" (r=4, d=1, l=5) in the appropriate locus), or anything else, the memory palace is a great way to go.

How does this help one memorize the digits of a number, like tau? Well, what the major system does is turns numbers into words, which can then be turned into images. The memory palace then acts as a place holder for those images. For instance, take the digits of tau:


The first two digits are 62. What words can this form? You can say chain, gin, maybe you know someone named Jane or John. I ended up choosing the word ocean.

The next three digits are 831. This forms the word vomit. Yes, it is disgusting, but it is a word that will create a memorable image.

The next two digits are 85. From this, we can create the word waffle. So the first image will be "an ocean vomiting a waffle." It sounds very silly, but it will be memorable. The smell of the saltwater, the taste of the waffles, the sound of the ocean waves crashing. This all will go into your first locus. My memory palace for tau was my middle school campus, so I remembered this image in the back parking lot of the school.

The next image is comprised of the digits 30717958. This can be turned into "a mask tugging on a bailiff." This was put inside of a staff room that the back parking lot has a door to. It is a very weird image, but still memorable. Picture the bailiff really struggling to get away from this mask, while still fearfully reciting his lines: do you solemnly swear to tell the truth, the whole truth, and nothing but the truth. Make yourself feel scared of this moving mask, and sympathize with the bailiff. The more you relate to and embrace the image, the more memorable it will be. Especially when you are memorizing 2012 digits of tau (which took me 272 images), you need each image to be extremely vivid.

To retrieve the numbers from this memory palace, all you do is go back to the image, find the subject, root verb, and object of it, and translate the consonants back to numbers with the major system. With practice, this becomes easier and easier to do. I strongly recommend practicing at least memorizing grocery lists and to-do lists with the memory palace, and if you want to take it further, learn to convert numbers to words with the major system for more advanced lists and situations. Maybe even memorize your family and friends' phone numbers with the major system and memory palace. These are all great exercises for your mind, and will definitely give you a better memory.

Saturday, November 23, 2013

The History of the Monty Hall Problem

One of my very first blog posts was about the Monty Hall Problem. This is an extremely classic example of a probability paradox. Let me quickly describe the problem:
Pretend you are on a game show, and the host gives you three doors to select from. One of these doors has a car behind it, while the other two have goats. Let's say you select door number one. Then, the host (who knows where the car is) opens another door to reveal a goat. Let's say he opens door number three. You are then given the option to either stick to door one or switch to door two. Does either strategy have an advantage?

The common answer would be that it is 50-50, and there is no advantage either way. However, the correct answer is that there is only a 1/3 chance of winning by staying put, and a 2/3 chance of winning by switching. Click here to learn why.

This problem was first posed by Steve Selvin, but it was popularized by Marilyn vos Savant in 1990. Vos Savant is famous for once having the highest IQ in the world, as well as her "Ask Marilyn" column in Parade Magazine.

One week, her column was about the Monty Hall Problem. She posed the question, and then explained her reasoning as to why there is a 2:1 advantage for switching. This created a pandemonium of angry readers who insisted that she was incorrect, and furthermore, accused her of adding to the problem of innumeracy and lack of mathematical intuition in America. Some of these complaints came from a statistician at the National Institutes of Health, the deputy director of the Center for Defense Information, and professors at George Mason University, University of Florida, University of Michigan, Millikin University, Georgetown University, Dickinson State University, Western State College, and more. Even the legendary Paul Erdős couldn't wrap his brain around the paradox.

This problem has continued to baffle everyone it encounters, from average people to accomplished mathematicians. In 2010, Walter Herbranson and Julia Schroeder of Whitman College performed an experiment to see if playing the game multiple times could end up refining the player's strategy. The human test subjects failed to revert to the optimal strategy and switch doors in the experiment. However, when the test was performed on pigeons, with mixed grain as the prize, they were able to pick up on the fact that switching doors gave them the best chance of success. The fact that a pigeon can do better than a human in this situation is fascinating to me.

The Monty Hall Paradox is something that reminds us of how humans are not wired to understand probability and statistics. This is why people can be fooled by mathematical scams and why casinos are packed full of gamblers. If our math curricula put an equal focus on probability and statistics as it did on algebra and calculus, then our world would have much better math minds and critical thinkers in general.

Saturday, November 16, 2013

The Mathematics of Ghost

A couple months ago, we were talking about various mathematical games where player two can very easily force a win, such as Chomp and Anti Tic-Tac-Toe. Those games were pretty easy to spot the unfairness, but this is one that is actually commonly played by everyday people. You might even play it yourself. The game is called Ghost.

Ghost is definitely not a scary game, despite the name. In this game, player one starts by saying a letter. Player two must respond with another letter, and they keep taking turns saying letters and forming a word. The goal is to not create a real word. However, you can't call out letters that don't allow a word to be made either. For instance:

Player 1: T
Player 2: O
Player 1: U
Player 2: C
Player 1: A
Player 2: Umm... L
Player 1: I challenge.
Player 2: I didn't have a word in mind.
Player 1 = Winner

If your opponent challenges you, you need to say a word that can be made out of the letters to win. If you can't think of a word, then you lose the round. A better strategy for player 2 in this game might be:

Player 1: T
Player 2: W
Player 1: E
Player 2: A
Player 1: K
Player 2 = Winner

This game seems pretty fair, right? If you play this game with a friend, you will find that the wins are pretty even (unless one of you has a much better vocabulary). However, it is possible for player 2 to force a win.

When all of the words in Scrabble Player's Dictionary are in play, every letter of the alphabet can be followed with another letter that forces an odd lettered word (assuming that player 2 continues perfect play). However, this dictionary has a lot of obscure words in it that might get challenged, and the average player will not know or use.

By keeping it to words that are known to the average player, player 1 does have a chance. If they play H, J, M, or Z, they can force a win if player 2 does not have the dictionary memorized. Here are the words to remember for player 1:

H: Hazard, Haze, Hazily, Hazy, Heterosexual, Hiatus, Hock, Huckster, Hybrid
J: Jazz, Jest, Jilt, Jowl, Just
M: Maverick, Meow, Mizzen, Mnemonic, Mozzarella, Muzzle, Muzzling, Myth
Z: Zaniness, Zany, Zenith, Zigzag, Zombie, Zucchini, Zwieback, Zygote

For any other letter, player 2 can still force a win. Once you know the correct followup letter to use, it is a pretty simple game. Here is the strategy:

Letter Followup Possible Words
A O Aorta
B L Black, Blemish, Blimp, Bloat, Blubber
C R Craft, Crepe, Crept, Crick, Crozier, Crucial, Cry
D W Dwarf, Dwarves, Dweeb, Dwindle, Dwindling
E W Ewe
F J Fjord
G H Ghastliness, Ghastly, Gherkin, Ghost
I L Ilk, Ill
K H Khaki
L L Llama
N Y Nylon, Nymph
O Z Ozone
P N Pneumonia
Q U Quaff, Quest, Quibble, Quibbling, Quondam
R Y Rye
S Q Squeamish, Squeeze, Squeezing, Squelch
T W Twang, Tweak, Twice, Two
U V Uvula
V U Vulva
W H Whack, Where, Whiff, Who, Why
X Y Xylem
Y I Yield, Yip

By memorizing this list, you can become a perfect Ghost player, and win every time, despite the seemingly fair nature of this game. The other two games were both fun, but with this one, you can really start to impress people with your ability to win.

Saturday, November 9, 2013

Math in the News: Teacher Salaries

One of the biggest issues in math education is teacher quality. I have discussed this in both of my TEDx talks about this topic.

We explained it the best we could in our Capstone Research Paper (link is at the top of the page), but I think this New York Times article that just came out describes it fantastically. So, I couldn't resist posting it here.


Saturday, November 2, 2013

Logarithmic Proofs and Identities

I have talked about logarithms quite a bit on this blog, but they were always being applied to something else, whether it be Benford's Law, the Law of 72, or some other practical use. This week, I would like to show that logarithms are very much a part of pure mathematics as well. Of course they are in mathematical equations just as much as exponents and radicals, but they have some pretty cool features of their own.

First off, let me review what a logarithm is. I have explained it before, but once you understand the notation, you shouldn't need to have done Precalculus to understand this post. They are very easy to understand.

For instance, we know that 102 is 100.

102 = 100

If we take the logarithm of both sides, we are essentially bringing the two out of the exponent. Rather than doing an operation on the 10, we do an operation on the 100 to determine what the exponent is.

log(102) = log(100)
2 = log(100)

In most situations, it is clear what type of logarithm you are using, especially in this one because ten is a common logarithm to use. However, many people will write a subscript to clarify. For instance:

log10(100) = 2

Logarithms become extremely useful when you need to solve an algebraic equation where the variable is in the exponent. For example:

2x = 64

As you know, algebra is about doing the inverse operation. If there is addition going on, you subtract. If there is multiplication going on, you divide. Similarly, if there is exponentiation going on, you use a logarithm. In this instance, it would be taking the log2 of both sides.

c2x) = log2(64)
x = log2(64)
x = 6

In practice, there are three bases that are extremely popular to use in a logarithm. We just used two of them: log10 and log2, which are also known as the common logarithm and the binary logarithm. The third one is loge (using the number e that is described here), which is called the natural logarithm, or the natural log. This one is found on most calculators, usually next to the common logarithm.

Though logarithms are not a part of most people's day-to-day life, they do have lots of practical applications. The common logarithm is the basis of the pH system which describes the acidity of water. The natural log is a huge aspect of finance and compound interest (as we saw with the Law of 72). And of course, they are all over nature.

Let's look at an identity of logarithms. Take the following problem:

log6(24) + log6(9) =

If you just used a calculator to do this, you would get:

log6(24) + log6(9)
1.773705614 + 1.226294386

That's odd. Two random numbers happened to have logarithms that summed to three. Let's look closer at this and see if we can figure out why. What number could you find the log6 of and get 3?

log6(n) = 3

First of all, let me point out that we just asked an algebraic question. We suddenly got curious about why something happened, so we asked a "what" question, which calls for an unknown quantity, which later becomes an algebraic variable. So when algebra seems like a drag, remember that it is all techniques for answering that "what" question. And "what" is a question asked in all branches of mathematics, science, and engineering.

Anyways, for this equation, we would want to do the inverse operation. We turn both sides into the exponent, and create a base of 6. This gives:

6log6(n) = 63

The left hand side cancels, leaving just n. The right hand side is six cubed, which is 216. So, we end up with:

n = 216

So, this means that the log6 of 216 makes you end up with 3, or the sum of log6(24) and log6(9).

log6(24) + log6(9) = log6(216)

What is the relationship between these three numbers? Well, it shouldn't take to long to determine that 24 x 9 = 216, or:

log6(24) + log6(9) = log6(24 • 9)

In other words, the sum of the logarithms is the logarithm of the product. Wow! That's pretty cool! Is that always the case? Well, let's try to prove that it is for all logarithms.

Let's set a few terms equal to each other and see what happens. Since there are logarithms, we will need a lot of variables.

x = loga(p)
y = loga(q)

In other words:

p = ax
q = ay

Let's multiply those two equations together. Since they are both equal, multiplying the terms on each side by each other won't make a difference.

p • q = a• ay
pq = ax+y

The right hand side was simplified using the Law of Exponents, which is explained very well here.

Now, we must take the logarithm of both sides, or specifically, the logof both sides.

loga(pq) = loga(ax+y)
loga(pq) = x + y

But what were x and y? We defined them in terms of a, p, and q earlier. So, let's substitute those values in and see what we get.

loga(pq) = loga(p) + loga(q)

And this creates the identity that we were trying to prove: the sum of the logarithms is the logarithm of the product, and thus, completes our proof. There are other logarithmic identities like this one, but I will save that for another post.