Author: RTNM

This is a piece of Flash Fiction.  The topic: A teenage girl gets a letter from George Mason University.

 

Professor Bob

Rosemary bounced through the door, simultaneously kicking off her vans and throwing her backpack against the couch. She didn’t notice that her giant chapstick fell out and rolled under the big, puffy chair her dad used to sit in before he crossed over.

“Rosemary, you have a letter on the table.”

“Mom, geez, you know I hate being called Rosemary! Gah…why do you have to call me that?  Everybody calls me Rosie, you know that.”

“Rosemary is your name.  It was good enough for your grandma, and it is good enough for you.”

“Well…duh!”

Mom put down the parsley she was chopping up to garnish the evening meal and walked over to the dining room table.

“I noticed it was from a university, but I didn’t pay much attention. Which one is it now?  What school is trying to steal my little girl.”

Rosie tried to remain calm; this was bad, really bad. “George Mason mom, well actually it is not officially called George Mason mom, it is just George Mason. I think I’ll go upstairs and research this school. Do I have a little time before dinner?”

“Yes, but first, exactly where is this George Mason University located?  I don’t know how many times I have told you that you aren’t going to school far away from home.”

Rosie pursed her lips as her back muscles tightened.  “It is in Fairfax, Virginia, all right ` It is just outside of D.C.  Guh, do I have some time before we eat or not?”

“A little time is all.  Dinner will be ready in a few minutes.”

Rosie ran upstairs to the computer room – buttons pressed, switches flipped, levers pulled…and (most importantly) the door locked. Rosie concentrated her gaze at the correct spot as her right eye was scanned.  She then reached between two bookcases and touched the wall in the specified pattern to open the portal. The cylindrical staging platform opened up and began glowing steady neon green.  Rosie put on her helmet, adjusted her goggles, took a deep breath, and headed in.

Total silence.  That was always the thing that bothered her most.  It was eerie.  She stood for a few seconds as wispy particles appeared seemingly out of nowhere to form the bust of a figure, a very familiar one.

“Rosemary, good…you got the letter. I wasn’t sure the teleportation had worked properly.”

“Well…duh.  Of course, I got the letter. If I hadn’t gotten it, we wouldn’t be talking now, would we?  What is going on?”

The conjured figure, a sage-like older man (you would never believe how old!) winced as he told her that perdition was upon them. “Rosie, they got out, they escaped. My last experiment went very, very wrong. You and I both know where they are going. I sent out a communique to all the others, they have all checked in and are on their way. You understand exactly what I am saying, right?”

“Uh-huh.  And I also know this must be really bad if you couldn’t just send a message directly to me.  I don’t want to ask, but why did you have to zap a letter into the mailbox?  What’s that all about?”

The old man saw the look in her eyes. He didn’t want to tell her that they were totally compromised, that a data hack and a simple case of blackmail had exposed nearly everything.   “Now listen Rosie, stay right where you are. You are not to leave your house, and even if they show up on your front porch, you are not to engage them. Do you understand me? That is an order. If they come, you are to initiate a complete lockdown of the premises.  If they somehow get through, you are to get your mom and immediately come to the portal, OK?” He looked at her and knew it had been a mistake to warn her, he should have just sent someone to collect her.  Had he been thinking clearly, there are a lot of things he would have done differently.

Rosie stood at the portal, her hands on her hips.  She leaned slightly to the left and shook her head slowly back and forth.

“Rosie, please listen, there isn’t much time…” Rosie cut him off and skipped away from the portal. She was about to get her battery packs and ammunition when her mom’s voice came through on the communication panel.  “Rosie, there is a group of people on the porch asking for you. What is going on, are these new friends of yours?  I certainly didn’t make enough food to feed all those people.”

Rosie looked out a window and saw the group milling around on the front porch and driveway.  “Hey mom, can you come up here for a minute?”

“What is going on?  Is something wrong?”

“Of course not, I just need your opinion on an outfit.  I am trying to impress one of the boys out there.  He is a new kid, and all the girls like him.”

As mom walked through the door, Rosie quickly wrapped her arms around her, lifted her up, and pushed her into the mechanism.  Risking psychosis by getting into the portal unprepared was better than staying and facing the mob.  Easy choice.  As soon as mom was locked in and protected, Rosie did one of those teenage girl waves and then went to the closet for her duffle bag full of ammo.  She threw it on the table and then quickly moved across the room to open a hidden compartment to reveal a silver case, one full of weapons.  She got all the arms locked and loaded, gave her mom a quick glance, and initiated the transport sequence.  You’ll make it.  No worries.  As her mom phased out of existence, she holstered her weapons and headed downstairs.

Author’s note: If you do a little research, you will find that there is a famous professor at George Mason University who is trying to create life in the laboratory.  Sister, you don’t know the half of it.

 

 

 

 

My Favorite President: A Note on The Pythagorean Theorem

Many years ago I had a buddy named Mariah, she was a high school senior working part-time at a restaurant I used to hang out at.  One day she approached me to tell me about a paper she needed to write for both her Government and English classes.  The topic was a President of the United States, to be assigned at random by the teachers.  I immediately said, “Yes!” and started a discussion with anyone who happened by the bar where I was sitting.  The question of the day became: Who is your all-time favorite President?  The surprising thing is that most everyone had an immediate answer.    

As I recall, Mariah was hoping to get assigned George Washington or Theodore Roosevelt, but she ended up with JFK.  That turned out OK because I think she learned a few things, and she completed the assignment on time and to the satisfaction of both of her teachers.  Sadly, I didn’t get to help her as much as I wanted because she had no internet and I couldn’t get my hands on any drafts.  Believe me, I tried, but every time I saw her, she didn’t have her backpack with her, so no draft of her paper was to be found.

So, do you have a favorite President?  I sure do, and isn’t it curious that I can write about him within the context of The Pythagorean Theorem.  It certainly is mysterious unless you know that one of our former presidents actually came up with a smart and original proof of The Pythagorean Theorem.  You had no idea, did you?  Trust me, most people don’t.

Before we get to a discussion of James Garfield, I need to take a detour, a pretty big one.  I am going to tell you a little about how I write these essays and, in particular, what happened when I started to write this one.

There are many different ways that individual writers approach their craft.  I have heard from many sources that Kurt Vonnegut, my favorite novelist, would write three pages of perfect prose in the morning and call it a day.  No editing or revision of any kind required.  If I tried to do that, I would still be on page one of essay one.  I simply could not do it, I am not capable of writing that way.

There is another popular approach to writing called the “scattershot method.”  That is exactly how I write.  When utilizing the scattershot method, you write whatever comes to mind, you don’t worry at all about grammar, spelling, or punctuation.  It is all about ideas and concepts.  When I write, I just try to get the thoughts down on paper as quickly as possible.  There will be plenty of time to clean things up later, right?

Well, guess what?  I have in front of me a sheet of printer paper that contains an outline for this chapter.  I got the idea for this essay as I was finishing up one about an interesting algebra problem I stumbled upon.  The mechanics of that problem (which will appear in a future post) reminded me of The Pythagorean Theorem.  That inspired to write something about the most famous theorem in math, which made me think of the paper Mariah had to write.  Get it?  It all came together in a flash, so I grabbed a piece of paper and wrote down an outline so that I wouldn’t forget what I wanted to write about.  Sigh, much more on that coming up.

Here is a word for word account of what I wrote.  I am reading it off the original sheet of paper.

My Favorite President

-Mariah

-Cohen & Newton

-T.R. & Gould

-Garfield & P.T. proof

I include this because I have a huge problem.  I understand exactly what everything on there means except for the “Cohen & Newton” segment.  Good grief, I know who they are; Cohen is I. Bernard Cohen, the founder of the History of Science department at Harvard University, and Newton is, of course, Issac Newton, arguably the greatest scientist who has or ever will live.  The thing is, I have no idea why I wrote their names down.  I do not know what I am supposed to say about them, I haven’t a clue as to how either man relates to this essay.  I find that a little perplexing.

This is simply more evidence of a disturbing trend, one that I realized a few years ago.  One night some time ago, I was reading an article on a topic relating to evolutionary biology.  I came across an interesting discussion concerning the biological concept of species.  As this was research I was unfamiliar with, I made copious notes in the margin of the paper and went to bed.  The next morning I woke up and decided to quiz myself on what I had read the night before.  I knew there were three major themes that the author had written about.  The big problem is that I could not remember anything about the article.  I couldn’t remember who wrote it or what it was about.  Yikes, that is not good.  I have talked to many other people about this, and they all say that things like that happen to them from time to time, so it is nothing to worry about.  I will say this, I don’t find it reassuring that many of my friends might be losing their minds right along with me.  Seriously though, I hope they are right, and these gaps in memory are not a big deal.  One thing is sure, time, in all its undefeated glory, will eventually chime in with the correct answer.

The fourth line of my infamous sheet of paper reads “T.R. & Gould.”  I know exactly what that means, and I can tell you a little bit about that now.  That line is about President Theodore Roosevelt and Stephen Jay Gould, the greatest science essayist who has ever lived.  Chapter 14 in Gould’s “Bully for Brontosaurus: Reflections in Natural History” is entitled “Red Wings in the Sunset.”  It is a neat little essay about Roosevelt’s accomplishments, not as a politician, but as a scientist.  Bet you didn’t know that Roosevelt published an important scientific article, did you?  Well, he most certainly did.

After he left office, Roosevelt wrote a paper on coloration in birds and mammals.  It was published in the Bulletin of the American Museum of Natural History in 1911.  The technical details of the article are not important, the point here is that Roosevelt was, at heart, a scientist – ever curious about the world and all its inhabitants.  Can you imagine?  A former president sat down and wrote over 100 pages about how and why certain animals are colored the way they are.  Such a thing could have easily gotten Roosevelt my vote as my favorite president, but as it stands now, he is a close second.  Who knows, he may move up to the top spot one day.  I am sure his decedents will be sending me fruit baskets with the intention of influencing me.

Now we can get to my favorite president, James Garfield.  Garfield was college-educated, he got his degree in mathematics from Williams College.  As you are about to see, he put the degree to good use, and that is what makes him my favorite President.

A few years before he became President Garfield, Congressman Garfield came up with a unique proof of The Pythagorean Theorem.  The theorem, the most famous in mathematics, states that for any right triangle, the square of side “a” plus the square of side “b” will equal the square of the hypotenuse, known as side “c.”  That special relationship has been known for centuries, but Garfield was inspired to find his own proof for why the relationship holds.  He did it in a way no one had ever thought of.  He proved that a² + b² = c² by taking an inspired look at the area of a trapezoid.  

Here is the trapezoid President Garfield used for his proof.  It is broken into three right triangles.  I love this proof because it is algebraic, you won’t find any geometry in his argument.  That makes for an intelligent, and unusual, little proof.

The first thing Garfield did was find the area of the trapezoid.  That is easily done in the following way.

 

 

Next Garfield, in an ingenious leap, decided to find the areas of the three right triangles.  The equations are as follows.

He then took the sum of the areas of the individual triangles as follows.

Since the two areas have to be the same, Garfield simply set the equations equal to each other.

Multiplying each side by 2 gives:

Subtract 2ab from each side:

Simply astonishing.  I have to admit that I have a secret ambition (not so secret now) relating to this topic.  I want to come up with the worst, most clumsy proof of the theorem that could possibly exist.  I want it to be Rube Goldberg-esque, the foulest string of mathematics imaginable, yet I want the proof to be correct.  This is the only reasonable hope I have as it appears all the good ones are taken.  There are hundreds of beautiful and thoughtful proofs that can easily be found online.  President Garfield certainly did his part in that regard.  I very much admire his simple, elegant proof.

 

 

 

 

  

 

                       

 

 

             

Bonus Eruptus!

Let me begin by letting everyone know that I love The Simpsons.  The show is now in year 31, and I still look forward to each week’s episode.  I will admit that a few years in the middle of the run were pretty lean, but the show is experiencing a renaissance.  The Simpsons are back on solid footing.

Some of you may remember when Dr. Nick introduced us to Bonus Eruptus.  It was episode 21 of season 7.  The episode is entitled 22 Short Stories about Springfield, and that is exactly what transpires, 22 vignettes about the characters populating Homer’s hometown.  I think it is very clever and I have always wanted them to do more episodes like that one.  This particular episode, one of my favorites, first aired on 4/14/96.  Wow, the show has been around a long time, hasn’t it?  I will gladly take another 30 years.

During that stellar episode, Dr. Nick defined Bonus Eruptus as “a terrible condition where the skeleton tries to leap out of the mouth and escape the body.”  Apparently, Grandpa Simpson had this condition, at least that was the diagnosis of the esteemed Dr. Nick Riviera.  I want to take a closer look at the mathematics behind this ostensibly severe condition.  Why?  I think that we might be able to learn a thing or two about probability theory and the inherent problems that come along with mass medical testing.

Please indulge me for a moment.  Let’s all pretend that we live in Springfield USA and that Bonus Eruptus is a legitimate concern.  I know I wouldn’t want my skeleton to try to take its leave of me.

Imagine that Mayor Quimby, in a transparent attempt to get reelected, offers free, yet mandatory, testing to all the inhabitants of Springfield.  Since I have no idea how many people live there, let’s say that 10,000,000 people are living in the greater Springfield area.  I know that is more of a Capital City number but just play along, OK?  Of those, let’s say that 50,000 of them have the dreaded Bonus Eruptus.

Now let’s imagine that Bonus Eruptus is caused by a virus, one easily detectable by a simple test.  Like all tests, though, it is not perfect.  Some people who have the virus will test negative, and a certain percentage of the people who are negative will, in fact, test positive.  Imagine that the false-positive result rate is 2%.  Also, the poor people who have the virus will test positive only 95% of the time. So, the simple question is:  If someone actually tests positive, e.g., Bumblebee Man or Jeff Albertson (extra points if you know who that is), what is the probability that they actually have the terrible disease?  Think about that a while before you go on.  As you might already have guessed, the answer is not nearly as straightforward as you might think.  After all, why else would I be writing about it?

So, here we go.  Of the 50,000 people who have the virus, only 47,500 of them will actually test positive.

50,000 x .95 = 47,500

We know that 9,950,000 total people do not have it.

10,000,000 – 50,000 = 9,950,000

Of the people who do not have the virus, there will be 199,000 who will test positive anyway (the false positives).

9,950,000 x .02 = 199,000

So now, we can do some simple addition and see we come up with a total of 246,500 people who will test positive for Bonus Eruptus.

47,500 + 199,000 = 246,500

Of those, we know that only 47,500 will actually have it.  So if you test positive for the virus, there is only a 19.3% chance that you actually have Bonus Eruptus!. D’oh!

47,500 / 246,500 = 19.3%

Isn’t that interesting?  Without walking through the math, there is no way that a 19.3% chance would even seem to be a possibility.

It is time for me to go.  I have to prepare for this week’s show.  I hear that Homer is going to do something stupid, and Marge is going to get upset.  I am about to burst with excitement.

Where is my Dollar?

This is an essay about counting. Simple counting, you know, like the kind that the puppet vampire on Sesame Street does. You remember “The Count,” right? One, Two, Three, Four, blah, blah, blah, blah.  I just did a little research, and I discovered his full name is Count Von Count.  Who knew?

I have sat through a lot of lectures in my life, so many that I couldn’t begin to count (get it?).  This may sound surprising, but one of the best lectures I ever attended was about the proper way to count.  This was in a graduate statistics class.  This seemingly straightforward process is a lot more complicated than you might imagine.

To set the stage, I must tell you that in what feels like a previous lifetime, I was an archaeologist.  I worked in The Bahamas at the Island of first contact, the one made famous by Christopher Columbus and his crew.  For those of you who do not remember, the name of the Island is San Salvador, and it is beautiful.

There are numerous ways we could begin our journey into the strange world of simple counting. We could quickly start with a discussion of how easy it seems to be to count, say, artifacts excavated from an archaeological site. If you do not “dig” that we could talk about how scientists count the number of genes in the human genome.  Instead, and by explicit design, we will start with a fictitious handyman.  This man works at a non-existent motel located in a town that you will not find on any map. His story is much more fascinating and informative than anything else I might dream up.

The story begins like this: there was a man named Ichabod who worked at a campground as a general laborer. Ichabod spent most of his time doing odd jobs, usually something different every day.  On a random Friday, the 13^th three men came in needing a cabin. The manager took their information, charged them $60, and sent them on their way. A few hours after they left, the manager realized he made an error and called Ichabod into his office. He told Ichabod that he had mistakenly overcharged the three guys at campsite number 234 by $5 and gave Ichabod the money to hand over to them.

Ichabod worked his way down a winding dirt road to cabin number 234 with 5 one-dollar bills in his pocket. As he approached their camp, he realized that he had no way to divide the $5 evenly by three. He thought that there would be no harm in just giving each of the three guys one dollar and pocketing the extra two dollars for himself. That is precisely what he did.

A few hours later, Ichabod was bored, his work for the evening was done, and there were no pressing emergencies. He decided that he wanted to figure out why the three guys got a discount. He quickly found out that they had a AAA coupon for $5 off for one night’s stay. Something didn’t seem quite right, so Ichabod got the manager’s calculator and multiplied $19 by three. After Ichabod gave them each a dollar back, that is what each of them paid. He came up with $57. He then looked in his pocket and found the two dollars he decided to keep for himself. He multiplied 19 by three again and once again came up with 57. He added the two in his pocket to get $59. Wait a minute, he thought, they initially paid $60.

Ichabod mulled this over for a few more minutes. Let’s see, each guy ended up paying $19 for the campsite. The three of them together paid $57. I kept two. Ichabod had a perplexed and slightly angry look on his face when he looked up at the stars and yelled: “Where is my dollar?”

Do not bother racking your brain just yet. The answer will be as curious at the end of this essay as it is right now. The point to consider at this juncture is simply the slippery nature of straightforward counting. And yes, the Twilight Zone feel of the end of the last paragraph was put there on purpose.

One of the little known aspects of science is the serious thought that scientists have to put into the seemingly artless process of counting. Counting should be straightforward, shouldn’t it?  It should be easy regardless of whether you have kids learning sums or if you have archaeologists counting projectile points. Unfortunately, this is not the case. Simple counting is anything but.

Consider the dilemma of the archaeologist studying regional settlement patterns.  This form of archaeology had become more commonplace in recent decades.  When studying regional settlement patterns, archaeologists do not look at a single site, they consider the relationship between all the sites in a given area. Typically, the scientist would be interested in the differences in the artifacts found at the various locations. Different artifacts found at a particular site would imply different uses for that site, and this is what studying non-site level archaeology is all about. How exactly to do this, though? Simple counts are virtually useless because Site A may have twice as many projectile points as Site B, but maybe the site itself is 10 times as large. The naive investigator might think that Site A was an area where the use of projectile points was much more important than it actually was.

What to do then? There are nearly as many strategies employed as there are archaeologists to do the counting. Many people find percentages to be very powerful.  If projectile points are found to be 85% of the total number of artifacts found at Site A, then this is a very important observation, especially if Site B has three times as many projectile points, but they only represent 2% of the total excavated from that site.

Some archaeologists prefer to work with volumes. They weigh the artifacts and then calculate volumes with respect to all other excavated material. You can even consider the weight of individual types of artifacts vis-à-vis the total amount of dirt dug up. This can lead to percentages or even ratios. The point being that simple counts are almost always deceptive and are rarely useful.

Animal bones, including those of humans, present more issues. A simple, yet potent technique, called Minimum Number of Individuals, or MNI, is often used. If you find 15 sheep vertebra at your site, they may have come from 15 sheep. That is not a whole lot of help. But what happens if you are studying mass burials in the U.S. Southwest and you discover 35 human vertebrae, 8 finger bones, 22 foot bones, and 5 right clavicles? The minimum number of individuals required to explain that group of artifacts is 5. Why 5? Because humans have only one right clavicle each. That might prove to be something worth knowing.

The types of issues we have been talking about are fundamental and might even be surprising to someone who thinks only of Indiana Jones whenever archaeology is mentioned. Once the archaeologist decides how they want to count, the problem with counting is just beginning. Then issues of a statistical nature raise their ugly heads.

It might well be the case that some aspects of the size, e.g., length, of the projectile points that have been excavated are diagnostic. In other words, they can tell us something vital if we are smart enough to tease the information out of the material. You might think that it would be a good idea to get an average length of the points so that you could compare this number with a similar amount from other sites. Let’s see if we can illustrate just how tricky that is.

Determining the average length of points could not, on the face of it, be more simple. Just add all the individual lengths up and then divide by the total number of points to get an average. No calculus involved here. The problem is that it might very well be the case that you will end up with an average size that is not representative of anything in existence. What if you found 6 points 1 centimeter in length and 5 that were 4 centimeters in length? You will calculate an average projectile point length of around 2.5 centimeters. This 2.5-centimeter long point does not exist, you did not find even one that is that length. It should be clear that this is a terrible idea.

The solution to the averaging problem is to split the points up into two separate “batches” of numbers that will be considered separately.  It is pretty obvious that the larger points were used for different purposes than the smaller ones. Reviewed independently, two distinct averages can be calculated that will prove very informative when compared to comparable data from other sites.

This essay has been about simple counting. I am sure that you now appreciate how much care goes into adding, subtracting, multiplying, and dividing. Truth be known, we have only scratched the surface of what is an important scientific issue. This topic becomes even more outrageous when advanced statistical techniques are employed. This is true for all scientists, not just archaeologists.

I am sure that there are many clever ways to end an essay of this nature. Ridiculous puns about counting ways to make mistakes come readily to mind. Or I could remind you to count yourself lucky that you do not have to think deeply about such nonsense. You and I both know, though, that I must end with a simple, elegant, yet slightly disturbing question. Tell me, exactly where is Ichabod’s dollar?

 

 

Negative One Twelfth

There is a perfect epigraph for this essay, but I advise you not to use it. It is too on point.  If you think about it, I am sure you can come up with something.
Buford Lister (personal communication)

I have been reminiscing about my time at Harvard University. I admit that for someone who does not like looking back, that is a bit unusual. There is a good reason, though. We will get to that in due time.

As for my experience there, I found that place to be full of the smartest and the hardest working people I have ever known. Nearly all of them had big ideas and the ambition to go along with them. The people I knew cared deeply about making the world a better place. Without question, the time I spent there was the best in my life.

I was “Pahking my cah in Hahvahd Yahd” before the internet took off. The Web was in its infancy; AOL, Prodigy, and CompuServe were all years off. There were only a few sites I could visit with my home built computer (the one with no hard drive and a nine-inch mono screen). There was lots of excitement, though, about how the internet was going to give voice to people who historically did not have one. My colleagues, friends, and associates at Harvard imagined discussions amongst people who would never meet in real life. They envisioned an exchange of ideas that, otherwise, would never have been possible. The hope was that these people from diverse backgrounds would realize they were basically the same, that their similarities outweigh the differences in geography or ideology. There was a kind of magic in the air. There was real hope that democracy would substantially benefit from a thoughtful and nuanced exchange of ideas from people of different backgrounds.

Did you make it through that last paragraph without shaking your head and saying, “Yeah, right!” to yourself?  Did you throw up on your shoes? I almost did while writing it because we all know what ended up happening. Nearly every thread I have ever seen on any web page I have ever visited devolved into a racial hate fest within a couple posts. One insult after another hurled by the cowards who hide behind the anonymity that their user names afford them. It really is disappointing; instead of rational and informed discussion, I see lots of intolerance and bullying.

This is the main reason I do not have a social media presence. I have had my fill of hate and, as you know, there is lots and lots of it out there. Trolls, by nature, are always looking to take a negative stance, regardless if they understand the topic or not. I simply do not need it.

So, why THAT introduction? The title (Negative One Twelfth) implies that this might be a math essay, even though I am the first to admit it is curious that the number is spelled out. The other way (-  ) seems like it would be cleaner, doesn’t it?  Well, this is a math essay. As usual, I am going to bury the lede by taking off on a seemingly irrelevant mathematical tangent. 

Let’s start by doing some straightforward calculations and see if we can eventually make our way back to the introduction. Unfortunately (and I do mean UNFORTUNATELY), by the end of this essay, my reasoning will become clear.

We are going to start here: What is: 1-1+1-1+1-1+1…? We will call this SUM_1.

[That little string of addition and subtraction has its own particular name. In 1703 Guido Grandi, an Italian scholar, did some interesting work on this problem. As the years went on, other mathematicians started to refer to the problem as Grandi’s Series. Lots of hard and thoughtful work has been done on that simple string of 1s.]

So, do you have an answer? Let’s do what any good, curious mathematician would do; let’s play around with the series and see what we get.
SUM₁ = (1-1)+(1-1)+(1-1)+(1-1)… = 0. Fair enough, right?
BUT:
SUM₁ = 1+(-1+1)+(-1+1)+(-1+1) … = 1. Now something interesting is happening.
If we solve the series one way, we get 0. If we move our parentheses around, we get 1. So, if you like, we can split the difference:

Angry yet?  Probably not, my guess is you are more confused than angry.  I can understand why you might not be convinced that SUM₁ does equal (it does, perhaps we should look for some better evidence).

Let’s tackle the problem using a different methodology. Consider the following series:

Any guesses where that sequence is going?  One way we can solve this problem is by looking at partial sums. We can do that like this:

Do you see what is happening? We are taking the first term, then the sum of the first two terms, then the sum of the first three terms, etc. Our partial solutions are approaching 2, which is the correct answer to the sequence. There is another way (among many) we can solve this problem, and we need to take a look at it.

We can treat the partial sums we just found as a new series, i.e., 1, 1.5, 1.75, 1.875, 1.9375… Now we can take the average of the partial sums in the following way:

We get: 1, 1.25, 1.42, 1.53, 1.61,…

As before, the sequence is moving toward 2. This method, the averaging of the partial solutions, is simply one other way of getting to the correct answer. Got it? Don’t worry if it takes a bit of time to let it sink in. It is a somewhat unusual concept.

Why did I bother bringing this up? Because we can use the same logic to solve SUM₁.

SUM₁ = 1-1+1-1+1-1+1…

First we get the partial sums: 1, 1-1=0, 1-1+1=1, 1-1+1-1=0, 1-1+1-1+1=1, 1-1+1-1+1=0…

We end up with: 1,0, 1, 0, 1, 0… This does not help much, all we are getting is alternating 1s and 0s. But, as you are about to see, something interesting happens when we average the partial sums. Take a look at this:

Using this method, we find that the sequence is approaching . The further we go out, the closer we will get to . It has taken a bit of time, but most of you should be convinced that SUM₁ =.  If you are still skeptical, I suggest a Google search. The Wiki page on Grandi’s Series is full of useful information.

Here is our next problem, we will now tackle SUM_2.

SUM₂ = 1-2+3-4+5-6…

This next step is totally legitimate. We are not breaking any mathematical rules. We are simply going to add SUM₂ to itself.

     

That sequence should be familiar, it is SUM₁.  Now we can easily solve SUM₂.

I am the first to admit that you should be fairly astonished at this point. I know I am.

Now I am going to let you in on a little secret, one that mathematicians and physicists (and me) have known about for a long time. It is a bit of information that makes people very angry. Here we go:

I know, trust me, I know. I came across this little nugget some time ago while taking a course on Number Theory. It didn’t make sense then, and it does not appear to make any sense now, but it is important to note that this result pops up in physics quite a bit. You can find it in String Theory and something called The Casimir Effect (I know a young man who got his Ph.D. in physics from MIT, his dissertation: The Casimir Effect). So yes, if you keep adding numbers forever, you get a negative fraction. Here is the proof:

Easy enough.

Even easier.

 

So, we now know that 1+2+3+4+5+6+7+8… = Negative One Twelfth. I will admit, the result is slightly counterintuitive. Maybe a little more than “slightly.” Bizarre might be a more appropriate word. I wish everyone reading this the best of luck. It might be helpful to keep in mind that physicists have provided experimental evidence that this solution is correct. Simply astonishing.

I always write these essays intending to use the most straightforward methods possible to introduce the readers to things that I find unusual and fascinating. This Negative One-Twelfth nonsense is undoubtedly a worthy topic. As always, I hope that anyone coming across this essay will become interested enough to do further research. There are lots of blanks to fill in, and a computer search will provide any and all answers you might want. This topic, as you might have guessed, has a fascinating history. Take a look around the internet, you will learn about (among other things) convergent and divergent series, Cesaro Summation, and Abel’s quote that Buford Lister thought was too obvious. If you are genuinely inspired, you will become familiar with the concept of analytic continuation. If you can’t sleep at night because this result hurts your brain, you will be introduced to Leonhard Euler and Bernhard Riemann (true geniuses) and their Zeta Function, a methodology that will also get you to Negative One Twelfth.

I guess it is time to close out this essay. I still owe you an explanation for the introduction. I started the piece the way I did because there are numerous threads out there on this very topic. In general, math and science discussion boards are respectful and informative places where intelligent people discuss the issues of the day. Three Cheers and a Tiger from and for all those Harvard people from so many years ago. Well, you can guess where I am going, right? Discussions concerning this topic get people so riled up that they do little more than fling insults at everyone, including the professors who are trying to explain why Negative One Twelfth is a perfectly respectable (and reasonable) answer to SUM₃.

Words such as “stupid” and “moron” are hurled at Ph.D.s by people who have no idea what they are talking about. This is discouraging and totally unacceptable. This problem is elusive; there are instances where Negative One Twelfth is the correct answer, and there might be other instances where it is not appropriate at all. As usual, nature (with all her subtly and nuance) will chime in and let us know when and where to use it.

My final point is that people interested in a problem such as this one are precisely the people who should know better than to head straight to the sewer when they are presented with something they do not understand. Lashing out at people who know more than you about a math or science issue serves no purpose (HACK, HACK, CLIMATE CHANGE, COUGH, COUGH). The internet certainly has given voice to people who otherwise would not be heard, and I am sometimes left wondering just who is benefiting.

 

 

1 = .999…

Q. How many mathematicians does it take to screw in a light bulb?
A. .999…

One of my hobbies is Number Theory. I have a bunch of DVD courses on the topic as well as quite a few books. Consequently, I have always had a secret ambition to write an entire essay consisting of nothing but equations. Mathematics can be so much more beautiful and concise than the words we all struggle to string together.  As the language of nature, mathematics has an elegance that far surpasses anything I can try to tap out on my keyboard. So, what do you say; let’s find out if 1 does indeed = .999…

You know what “.999…” means, right? It is a repeating decimal, that means that the 9s just keep going and going; they never, ever stop. So what am I talking about when I say that 1 = .999…?  Well, why don’t we take a look at some simple math?  I can easily prove to you that 1 and .999…are the same number.

 Could the math be any simpler? Interesting, isn’t it? Take a look at this:

So, are you becoming a believer?  The thing about mathematical proofs is that they do as advertised, they prove things.  Finally, this proof also gets us there but in a slightly different way.

Phil, from Phil’s Slight Oversight fame, read this when he was a college student.  He brought it to one of his professors to see if I was doing some sort of magician’s trick just to fool him.  His professor told Phil that it is true, 1 does, in fact, equal .999… Phil then asked him if that is so, can he write .999… instead of 1 on a test when it comes up?  “Of course,” he replied, “but I wouldn’t do it in any class other than this one.”  

Years ago, a high school student named Tayler also read this.  She brought it to the attention of her high school math teacher, who immediately told her that I didn’t know what I was talking about.  She told Tayler that the idea was nonsense after she spent a few minutes consulting her calculator. Such is life in Iroquois County. 

This short post is a prelude to a longer one that is coming next.  I will be building on these same principles to reach a bizarre conclusion to a simple math problem about counting.  If you are unsettled by this post, I recommend a Google search.  What I have done here is just a sample; .999… has its own Wiki page, visit it, and you will find lots of different and varied proofs.   

 

A Pitcher in Zipper Boots

I had an uncle named Dallas.  This post is about him and a couple things he did during his life, random and unusual things I will never forget.

Dallas used to wear a leather band on his watch, an extraordinarily thick one.  The watch was always worn with the dial under his wrist, not above it.  Why?  I have no idea.  I asked him about it once, and he rotated his left wrist up and to the left and said: “If I want to know what time it is, I do this.”  Fair enough.

Dallas was a big wrestling fan.  Andre the Giant was the strongest, baddest man on the planet, he wouldn’t let anyone dispute that.  I remember him booing The Shiek, an apparent and blatant cheater.  How is the ref not seeing that?  He cheered Bruno Sammartino and got riled up when Gorilla Monsoon got up to his usual shenanigans.  That world was black and white, no one was wearing a grey hat, and Dallas loved it. 

He had a great sense of humor, totally out of proportion to everything else about him.  And, believe me,  that was an excellent thing.  He used to hand me a hammer as he held a piece of rebar vertically with the end touching the ground.  He would say, “OK…when I nod my head hit it.”  He thought that was hilarious, but not nearly as funny as his favorite comedian, the remarkably unfunny Raymond  J. Johnson, Jr.

As far as I can tell, Johnson had one bit.  People would address him by his name, and he would say to them that there were numerous other names they could call him by; they didn’t have to use the particular one that they just used.  Here is a typical Ray Jay Johnson inspired encounter between my uncle and me:

Hi, Dallas.

Dallas, you doesn’t hasta call me Dallas.  You can call me Dal, or you can call me Donley, or you can call me Sonny, or you can call me Junior…but you doesn’t hasta call me Dallas.

Now, this went on whenever I mentioned his name.  He loved that bit.  Ray Jay Johnson has clips on the internet, some of them are national beer commercials.  Such was the fame of Ray Jay Johnson. 

We used to go bowling when I was a kid.  One Saturday night, we went to a local bowling alley for what they called Razzle Dazzle.  There were colored pins mixed in with the regular white ones.  If a head pin came up a particular color, you got money if you threw a strike.  Different color pins were worth different amounts of money.  The red pin in the headpin position was worth $25, a nice chunk of change back in the 1970s.

You guessed it, it was Dallas’ turn, and there it was front and center,  the red headpin.  He waited for the person at the desk to acknowledge the situation.  It only took a few seconds for the loudspeaker to engage.  “Red pin on lane 16.”  The other bowlers stopped.  The $25 shot was the big one.  Some nights went by without it ever coming up.  This was a big deal.

Dallas took his ball and cradled it as he dried his hand.  He might have said a silent prayer, I really don’t know.  With all eyes upon him, Dallas went through his regular routine.  He raised the ball up in the air with both hands as he took a giant step to his right.  As the ball dropped, he slowly started his approach.  He reared back and released, as the ball left his hand it drifted right, completely missing the headpin.  Dallas turned and started walking back toward the seats, totally dejected.  Then something happened, something I had never seen before or since.  The pins started dropping from the back forward, slowly one after the other as if they were in slow motion.  Dallas saw me point down the lane, he turned just as the headpin, the last pin standing, started to slowly wobble and then fall.  The place exploded in cheers as Dallas jogged up to the counter to get his money.  Simply remarkable.

Dallas was not just a bowler, he, along with my friends and I, used to play a lot of softball when we were younger.  I had a long list of names and numbers by the telephone at my parent’s house.  I would call someone to try to get a game together, they would make some calls, and then those people would make some calls, and when everyone was done, we could usually get a bunch of people to play.  We would have neighborhood games, and then we would often challenge people from other schools to play against us.  We played a lot of those types of games.

One day we played a bunch of kids from the local Catholic school.  The games could get pretty competitive.  Most of the kids who showed up played high school baseball, along with lots of other sports.  In fact, those kids from the Catholic school would go on to win a state championship in baseball a few years later.  And yes, among their ranks was a young man who would grow up to become a famous football coach.  I can’t quite remember if Urban Meyer was there on the day my story takes place, he certainly could have been.  Lots of his teammates used to show up for these games.

We were all warming up when Dallas appeared just as the game was about to begin.  I asked him if he was ready, and if he wanted to pitch.  He said sure and took the mound with no warm-up.  The first three batters were mowed down in quick succession.  Then something unusual happened.  He did it the next inning and the next and the next.

Around the fourth or fifth inning, I realized Dallas hadn’t allowed a run.  He was pitching a shutout, in a softball game, against a group of young men who were to become state champions in a few years.  When I mentioned to one of the other players that Dallas hadn’t given up a run yet, he nodded and said: “Yeah, I know.  I can’t believe it.”

Keep in mind that many of these games finished with scores in the 20s or maybe even the 30s.  Pitchers did not fare well in our games, offense dominated…except for that day.  Dallas pitched a shutout.  He hadn’t realized what he had done, and no one made that big of a deal of it.  We all got on with our day after the last out was recorded.  And here I am, over four decades later, checking the time on my upside-down watch.  Every time I look at it, I am reminded of the day my Uncle Dallas pitched a shutout against a team of future state champions while wearing zipper boots.         

The Notorious MFT: The Checkered Story of the GOAT of GOATS

On April 3, 1995, the human race lost the GOAT of GOATS, a stone-cold assassin, a man whom Michael Jordan and Wayne Gretzky can only admire from down the road.  As for Wolfgang Mozart and Isaac Newton, let’s just say that the three of them can argue over which appetizer to order with dinner.  The subject of this essay belongs at that table; those three would have a lot to talk about.  And yes, five days after his death, The New York Times gave him a healthy two-column obituary.

The Notorious MFT was a bad man, how bad?  Humans posed such a weak challenge that in 1958, he gave up his world title out of boredom.  For the next 50 years or so, he would become World Champion whenever he wanted.  Everyone knew he was the GOAT, so he didn’t need to rub elbows with us mere mortals.  He could do whatever he wanted, his legacy was secure.  If he felt like playing, he did; if not, he stayed home.

Before I get to this man and his remarkable story, I have a few thoughts on the word goat and the now ubiquitous acronym GOAT.  When I was younger, the word goat had a negative context when it came up in sports.  The goat was the person responsible for their team losing.  Poor Bill Buckner, who booted a ground ball for the Boston Red Sox in the 1986 World Series, is a famous example.  I still remember where I was when I saw the ball roll by him as much of New England screamed in dread.

Not too long ago, something interesting happened, the word goat became the acronym GOAT, and everything changed.  Now when I hear anyone mention The GOAT, they are always speaking of The Greatest Of All Time.  Many people attribute this change in meaning to the great Muhammed Ali, a man who was much more than one of the greatest boxers who ever lived.  Ali’s wife, Lonnie, incorporated G.O.A.T. Inc. in 1992.  That was the beginning of the metamorphosis.

Ali fought in and outside the ring.  As much as he was a fantastic fighter, he was undoubtedly a historic crusader for social justice.  Our man, while not nearly as famous, also dedicated his life to improving the circumstances of those born less fortunate.  Few would argue that Ali was one of the greatest boxers who ever lived.  No one would ever say that The Noriorious MFT was anything other than the greatest in his field.  It’s not even up for debate.   

Now we get to the man for whom this essay is named.  The Notorious MFT, also known as Marion Franklin Tinsley, was the greatest checkers player who ever lived.  He never lost a match, you read that right, he never lost a single match!  He only lost four games his entire life when he sat down to play checkers against a fellow human being.  Can you imagine?  I don’t know what to say in the face of such a ridiculous record.  All I can do is stand in awe of the genius that was Marion Tinsley.

Tinsley was a mathematician and a lay preacher.  He got his Ph.D. in mathematics from The Ohio State University.  The records at that university will show that Tinsley concentrated in combinatorial analysis, those same archives will not document all the time he spent studying checkers while there.  The man never married, he lived checkers, the game was his one true love.

Tinsley left his position teaching math at Florida State University for a similar post at Florida A&M, a public university historically attended by African-Americans.  Instead of becoming a missionary to Africa, Tinsley decided to stay closer to home to spread the word about mathematics and his faith.  I have been searching, and I can not find anyone who has gone on record with a single bad word to say about him.  By all accounts, he was a kind and gentle soul, except when seated to play checkers.  Then he transformed into an aggressive menace. 

Tinsley’s full story can not be told without considering the development of computers and the software that runs them.  Many people took it upon themselves to try to defeat Tinsley.  When the humans failed and failed again, a man in Canada decided to write some code.  He and his team made it their mission to create a computer program that could defeat the most magnificent checkers player of all time.    

Jonathan Schaeffer of the University of Alberta in Canada led a team that dared to believe that they could beat Tinsley.  How did they fare?  Do you have any guesses?  What do you think happened?

They failed.  Schaeffer’s program, named Chinook, defeated Tinsley twice in individual games but lost the match.  It is essential to note that no human ever beat Tinsley twice in a game of checkers.  His four losses were to four different people.  

After Tinsley’s death, the team did something extraordinary.  They designed a software program that solved checkers.  What does this mean?  The best any opponent can hope for is a tie.  The software can not be defeated.  It plays perfectly every time it is engaged. 

It took the team decades to dial the software in.  Obviously, this is a significant accomplishment, one worthy of world acclaim.  Think about this: there are 5 x 10²⁰ possible moves on a standard checkerboard.  Imagine the computing power, as well as the brainpower, required to attack that problem.

Schaeffer and his team were motivated and inspired by the greatness of Tinsley.  They were driven to build a program that could defeat him.  They were on the cusp of perfecting the program when Tinsley succumbed to pancreatic cancer at the age of 68.     

Tinsley’s grave is in Columbus, Ohio.  Schaeffer hopes to make a pilgrimage to the site one day.  Chinook lives on, you can find it with a Google search.  You can even challenge the program to a game if you dare.  If you play, prepare yourself to lose, you certainly will.  Same as with the people who came up against The Notorious MFT, the undeniable GOAT of GOATS.            

Phil’s Slight Oversight

Phil was pacing back and forth in his sparse, tiny cell. His mind was racing: man, man, man, man, man…what has happened to me? How did I let it come to this? I don’t belong here, this isn’t right at all.

“Guard, guard!  You have to let me out of here.  This is a big mistake.  I don’t belong here!  Do you hear me?”

Jebediah, known to most of the inmates as Big Jeb, approached the cell.  He slowly lifted up the index finger of his right hand, pressed it to his lips, and softly said, “shh.”

Phil retreated to his bunk and sat down.  He took one of his tennis racquets and slowly pointed it in turn at the pictures meticulously taped on the wall. One photo each for 15 federal indictments. The images were of him and each of his co-conspirators.  Shown, of course, in happier times.

So, how does a fine young man like Phil, the stone-cold pride of his hometown, Iroquois City, end up behind bars?  My, my, my…what a saga. As with many Shakespearean tales of tragedy, Phil’s story begins long before he was born. This tale of woe starts sometime in 1881 (not really sure of the month) with a man few people have ever heard of, a Harvard trained astronomer and mathematician named Simon Newcomb.

In Newcomb’s day, if anyone needed to know the logarithm of a number, they had to go to a log table book. Thick books composed of nothing but logarithms were commonplace in the days before calculators. I hate to say it, but I can remember consulting them when I was in high school. Anyway, Newcomb noticed that any log table book he looked at had soiled pages at the front while the back pages remained relatively pristine.  What this meant is that people were looking up numbers whose first digit was a 1 or a 2 far more than they were searching for numbers that started with an 8 or a 9. Finding this very curious, he wrote up a paper that was instantly (and unceremoniously) ignored by everyone. The astronomer had no idea that about 150 years later, a man named Phil would get arrested because of what Newcomb had discovered all those years ago. I have no further comment; circle of life or some such…at least I guess.

Now we skip ahead to 1938. The plot thickens as a physicist in the employ of General Electric rediscovers what Newcomb noticed all those years ago. Let me be clear about this: What Newcomb discovered is that numbers people were interested in, numbers that came up in their work or studies, started with a 1 or a 2 much more than any of the other numbers. Geez, numbers are numbers, aren’t they? Shouldn’t a number with a leading digit of 1 (e.g., 1,452,325 or 125) appear about 11% of the time when all the possible numbers are considered? I mean, the digits 1 thru 9 should each be represented about 11% of the time, shouldn’t they? That makes nothing other than perfect sense.

Benford, just like Newcomb, was intrigued by those curious soiled pages he found in the various log table books he looked at. Unlike Newcomb, Benford took his discovery much more seriously and analyzed around 20,000 data sets to see if numbers whose leading digit was a 1 occurred naturally more than numbers that started with, say, a 7, an 8, or a 9. Benford studied all kinds of different data sets, any naturally occurring group of numbers he encountered became part of his study. Remember, he was only looking at the leading digits of these numbers. By now, you should be finding this very curious indeed.

Phil wanted to pull his hair out. He kept going over it. Damn, leading digits of numbers…ARE YOU SERIOUS? How was I supposed to think anything different? How was I supposed to know all this? He remembered how he meticulously cooked the books for his investment firm, how he had gone through the spreadsheets very carefully. How could it have never occurred to him that there was a natural pattern that these numbers are supposed to follow? How was it that CurlyMatt or Butera, or anyone who knew what was going on, didn’t know this? Like it is with many people who end up in prison, Phil’s incarceration was simply “one of those things.” In this particular case, though, it was a giant thing, a tragic oversight, a life-changing life lesson.

Phil was escorted into a meeting room, his lead attorney, a tall, lanky man in a bolo tie and a stetson hat, shook his hand and got right to it. “So, Phil, you are not going to like this.”

“C’mon Harky, tell me already. I think I am still paying you by the hour, so get to it.”

“OK, I found this on your bookshelf in the library of your Miami mansion. It was sandwiched in among all the Athena stuff. Remember all those long chapters about the chick some guy met at a rock show?  Well, that same guy also wrote this.” Phil took it in his hands and gasped.

“How could I not remember this?”

Harky just shook his head. “Ok, that is all I’ve got. I have to get going. I’ll talk to you soon enough. Just do the best you can, stay safe, and try to remain calm. We are working hard on the appeal.”

Phil shook his head violently. He threw the pages of paper across the room. He yelled so loud that a couple guards rushed to tackle him. After they restrained Phil, this is what one of the guards picked up:

BENFORD’S LAW: A SHORT DISCUSSION

Ryan-Tyler N. Mason

In this brief chapter, I will introduce everyone to a simple yet stunning law of nature. It is called Benford’s Law, and it can be summed up in the following equation: log₁₀ (1 + 1/d). So, what does that mean? Something that I hope all of you find fascinating.

BENFORD’S Law explains a very unusual characteristic found in many, if not most, sets of numbers. The law says that in naturally occurring data sets, e.g., populations of countries, length of rivers, numbers that appear on the first page of any newspaper, numerous untold economic data, etc., the distribution of leading numbers (i.e., the first digit in the number) follows a pattern defined by the equation in the last paragraph. Common sense might suggest to us that the number 1 and the number 9 might show up as leading digits about 11% of the time. Makes perfect sense, right? The following table shows the distribution predicted by Benford’s Law:

1 = 30.1%
2 = 17.6%
3 = 12.5%
4 = 9.7%
5 = 7.9%
6 = 6.7%
7 = 5.8%
8 = 5.1%
9 = 4.6%

Here is the expected distribution of leading digits of numbers in chart form. I include it here so that a sharp distinction can be drawn between natural and unnatural sets, or batches, of numbers.

Nature, in fact, does reflect this type of distribution. There are technical reasons why this is true, but I do not want to bog down this discussion with a lot of math. Suffice it to say that Benford’s Law is legitimate and if you really are interested in finding out why further research on this topic can quickly be done on the internet.

Benford’s Law actually has many practical applications. For example, if someone (say a forensic accountant) is hired to go through the financial records of a large company and finds numbers with leading digits that match the distribution found in Figure 1 then no suspicion will be raised. If, on the other hand, a collection looking something like what is displayed in the following bar chart is found, then the CEO and his cronies better find the best criminal defense attorneys money can buy. The type of distribution found here very strongly implies fudged data.

All this information can be obtained just by looking at the first digit of entries in virtually any naturally occurring data set. Isn’t that simply remarkable? Benford’s Law is one of the more curious rules of nature that I have been lucky enough to come across. My only practical recommendation is that if you are going to work in finance or accounting, you need to memorize this law. Phil, I am talking directly to you.

Nickels

I am a big fan of McDonald’s restaurant.  I really like their hamburgers, I will take theirs over those of lots of different chains.  Currently, a burger is $1, quite the bargain.  You can also get any size beverage for $1, my go-to is the large Diet Coke.  Lots of days, I drive through and get those two items for lunch.

So this is how it plays out; I sit in my truck, trying my best to look cool as they tell me that I owe $2.07, ask me to confirm my order, and then drive around to pay for it.  $2.07?  Right, pop is taxed.  They don’t tax the hamburger, but if you want a Diet Coke, you have to fork over the government’s share.

Virtually every time I order, I struggle to find the correct change on my short journey to the pay window.  I always have lots of quarters, dimes, and pennies.  But, for me, nickles remain rare and elusive.  It is always a pleasant surprise when one appears from the console between the front seats.  The problem is, rarely do I find one.

This got me thinking about change, all kinds of change, the change I get when I go to the store, and the change I am constantly rifling through in an attempt to come up with the exact amount I need.  I ended up making a spreadsheet (big surprise, I know).  I considered all possible amounts I could be charged and then input the change I was most likely to receive.  By that, I mean, if I was owed 78 cents, I assumed I would get three quarters and three pennies, not seven dimes and eight pennies.  I created a simple bar chart of the results, this figure should put my constant search for nickles in perspective.

As you can see, the nickel is by far the rarest of all coins I can expect to get back as change whenever I make a transaction.  Consequently, I spend an undue amount of time searching for them. 

The chart shows that for every nickel I get, I am receiving 5 pennies, 2 dimes, and 3.75 quarters.  The big assumption is that every amount of change I might get, from 1 cent to 99 cents, are equally likely.  For this short essay, I believe that is fair enough.

If you are like me and still pay cash for most everything go ahead and take a look in the change compartment in your vehicle.  My guess is you won’t find a lot of nickels.  The same should be true if you unload the change from your pocket.  As for me, I am going down to the local Circle K to get my breakfast.  The cost?  Of course, it is $2.05.