Happy End Problem

As this is the last post of the year, I thought it would be appropriate to end with a post on the "Happy End Problem." I will explain the problem, which is pretty cool in itself, and then talk a little bit about the history behind it.

The initial question was if any five points were placed on a plane with no three of them in a straight line, will four of those points always form a convex quadrilateral? For example, in the following image:

The following convex quadrilateral can be formed:

Will this always work? As usual, I encourage you to grab a scrap piece of paper and try out a few examples. Have fun with it. Get creative! You will end up finding that no matter how you position the five points, you cannot get a combination without a convex quadrilateral.

Why is this true? In fact, there is a very easy way to prove it. Let's analyze three cases.

The first case is the top left one in the red, where the five points form a convex pentagon. In this instance, connecting any four of the points will form a convex quadrilateral by nature.

The second case is the top right one in the blue, where one point is located in between the four outside points. The illustration shows the inside point being included in the quadrilateral, but it could have just as easily been made as just the four outside points. This will continue to work for any combination of this nature by logic.

The third case is the bottom one in the yellow, where two points are enclosed in a triangle. When you draw a line between the two center points, two of the outside points will end up on one side and one will be on the other. Using the two outside points as your third and fourth vertices will form a quadrilateral without flaw.

Now, you might be wondering if one can prove a similar case with a convex pentagon. Could it be done with six? Seven? Eight? Turns out, nine points are required for it to work every time. As you can see below, eight points is just one too few.

What about convex hexagons? Or heptagons? Or octagons? Or chiliagons (1000-sided polygons)? Well, what many mathematicians will do from here is look for a formula to figure out how many points are required for a given n-gon. We know that for a triangle (n = 3), just 3 points are needed (all triangles are convex). For a quadrilateral (n = 4), we proved that 5 points are needed. For a pentagon (n = 5), I mentioned that 9 points are needed. Do you see the pattern?

3, 5, 9, ...

It is not easy to spot at first, but what if I subtract one from each of those terms:

2, 4, 8, ...

They are all now powers of two! This pattern seems to fit the formula A_n = 2^n-2 + 1. Plugging six in for n would give:

A₆ = 2^6-2 + 1
A₆ = 16 + 1
A₆ = 17

This formula predicts that seventeen points would be required for a hexagon. Mathematicians would then work to try to prove that this is the case. Further, they would try to prove that for any value of n, the A_n formula holds true.

George Szekeres (1911-2005; a Hungarian-Austrailian mathematician and analytical chemist) and Esther Klein (1910-2005, another Hungarian-Austrailian mathematician) worked together to prove that all values of n will have a finite A_n output (there will be a number of points that creates the ability for a convex n-gon to be formed), but they could not get this bound down to the formula above. Soon after this proof was published, Szekeres and Klein married each other, which inspired the name "Happy End Problem."

Paul Erdös (1913-1996; a Hungarian mathematician), possibly one of the most influential of the twentieth century), was able to prove successfully that with 71 points, a hexagon can always be drawn.

Sixty years later, Ronald Graham (born 1935; a Californian mathematician) and his wife, Fan Chung, decided to take a swing at the problem. While on a plane ride to a math conference in New Zealand, they were able to lower Erdös's bound to 70 points, which doesn't sound like much, but it brought the problem back into the minds of mathematicians. It was also ironic that another achievement pertaining to the Happy End Problem was from a couple.

Daniel Kleitman (born 1934; an applied mathematician at MIT) and Lior Pachter (born 1972; an Israeli mathematician and molecular biologist at Berkeley College) worked together to lower the upper bound to 65 points. The number was then lowered to 37 points, and has yet to be lowered further.

Although lots of progress has been made on this problem, the overarching proof still has not yet been found. There has been no counterexample to the A_n formula, and there has certainly been no guaranteed formula to generate the future values. People often wonder what a mathematician actually does for his/her job. A big part of it is trying to figure out the answers to these unsolved problems, which can often be understood by the average person. Try playing around with it and you might make a discovery too.

Figure Out The Number Of Digits In Gigantic Numbers

In science and math, you often run across numbers that are too big to be written in standard form. They are usually written in scientific notation, but they are also sometimes written as a number to a certain power. For instance, one might say that there are 2²⁰ outcomes of the flipping of twenty coins rather than saying 1.049x10⁶ ways.

By using that power, your information is likely more accurate. However, this power does not tell you much about the number. Most of us would have no idea if 2²⁰ is in the thousands, millions, billions, etc. at the first glance.

First, lets ask a question. What is the common logarithm of a number, or what can you gather from it? Well, the common logarithm is the power that ten has to be raised to to obtain that number. For instance:

log(100) = 2

log(5000) = 3.69897

log(6283185) = 6.79818

What do you notice about these numbers? It's not clear at first, but count the number of digits in each of the inputs. You will find that the common log is always just a little bit below that number. In fact, to figure out the number of digits in a number, all you have to do is take the common log and round up to the nearest integer.

How can this be used to find the number of digits in a power? Interestingly enough, there is a logarithmic identity stating that the log of a number raised to the power is equal to the power times the log of the number. For example,

log(27) = 3log(3)

ln(32) = 5ln(2)

log(2²⁰) = 20log(2)

Look at the last example there. We just simplified the gigantic 2²⁰ to a reasonable looking 20log(2), which is the formula to figure out the number of digits it has. In other words, the number of digits in 2²⁰ is just 20log(2) rounded to the nearest integer. Plugging this into a calculator tells you that the log is 6.0206, meaning that there are seven digits in the number. If you multiply it out, you will find that 2²⁰ = 1048576, which does indeed have seven digits. 

So whenever a type of problem pops up with a power of this sort, try to determine how many digits it is. Chances are you will gain a much better understanding of the statistic when you perform this quick calculation.

Math in the News: The Influences of Politics

Though mathematics is normally a pretty concrete subject, people's intuition for it is not. Probability and statistics in particular is a very difficult area for us to grasp, as you've seen with the Monty Hall Problem I talked about a few weeks ago.

Here is another example of mathematical aptitude being influenced by an outside source, but this time, it is not just a matter of lack of skill or desire to be correct. It is also influenced sometimes by political views, as Kevin Drum shows in this news article. Check it out!

http://m.motherjones.com/kevin-drum/2013/09/politics-destroys-math-ability

Isosceles Triangle Theorem

The field of geometry is comprised of many different types of questions. Some are construction based, such as “what is the area of this circle?” On the other hand, many are proof based, like “why are these two triangles congruent?” This is slightly different from the proofs I normally discuss; proofs I normally post are very generalized theorems while these questions are more similar to a specific algebra or arithmetic problem.

When writing a geometric proof, many different theorems come into play. One must use the generalized theorems, properties, and postulates to arrive at a conclusion. Let’s try a simple proof just to demonstrate the nature of these theorems. Let’s prove that triangle ABE is congruent to triangle CDE.

First, we would say that it is given that segment AC is parallel to segment BD and that segment AB is parallel to segment CD. We would then say that ABCD is a parallelogram by the definition of a parallelogram (which requires two sets of parallel sides). The definition of a parallelogram also requires AB and CD to be congruent. The vertical angle theorem can be used to say that angle AEB is congruent to angle CED. The alternate interior angle theorem says that angle ABE is congruent to angle DCE. The angle-angle-side triangle congruence postulate then concludes that triangle ABE is congruent to triangle CDE.

As you can see, there are many steps in this proof, and each one uses a different definition, theorem, or postulate. In high school geometry classes, students are told these theorems and postulates, and expected to memorize them for future examples. This makes geometry boring and pointless, when it can be quite fascinating. A way to easily spice up geometric proofs is to actually prove the theorems before they are used in class. If Euclid could do it, then we can do it.

A fun one to prove is the Isosceles Triangle Theorem. This theorem states that when a triangle has two congruent sides, it also has two congruent angles. This can be proven in a similar way as the congruent triangle question I posed earlier. Take an isosceles triangle:

If you were to bisect that top angle, it would create two new triangles. Since the original triangle is isosceles, it is given that the top left segment is congruent to the top right segment. The definition of a bisection (cutting an angle in half) states that the left part of the top angle is congruent to the right part of the top angle. The reflexive property of congruence states that the middle segment is congruent to itself. By the side-angle-side triangle congruence postulate, the left triangle is congruent to the right triangle. And finally, by CPCTC (common parts of congruent triangles are congruent), the bottom left angle is congruent to the bottom right angle.

This sort of geometric proof language sounds extremely long and boring. However, finding uses for it such as proving the Isosceles Triangle Theorem can make it a little more fun. Among many things, I think that schools should teach the reasons behind these theorems to make it more logical and fun to apply them to class.

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. A 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 www.phoneticmnemonic.com 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:

6.28318530717958647692528676655900598...

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.