# Omelet Perfection

Ever since starting the Primal Blueprint/Paleo type diet and lifestyle over 2 years ago, I’ve been making an omelet almost every morning.  I estimate this goes into my belly 4.5-5 days a week on average.

In 10,000 Hours of Purposeful Practice, I brought up the theory that one can become an expert in about anything by giving it 10000 hours of deliberate, purposeful practice.  Making omelets has taken up a lot of my time (maybe not 10000 hours yet), and I feel like I’m becoming an expert at making this satiating, and healthy breakfast.

It starts with putting a small frying pan on the stove on medium to medium-high heat, and throwing in a little less than a tablespoon of butter from a pasture raised cow.  While this is heating up and melting the butter, I gather the necessary utensils and ingredients.

• Plate, bowl, and fork.
• Frozen veggies (usually Mirepoix or Broccoli cuts)
• Guacamole (fresh made, or in the single serving size pictured next)
• Two large or three medium to small eggs
• Salsa or Organic Sriracha

Once the butter has melted and begins to bubble, I throw in enough frozen veggies to cover the bottom of my small frying pan.  While the veggies warm and cook, I crack the eggs in a bowl and whip those bad boys up.

Once those veggies are warm and begin to brown only slightly, I turn the burner down to medium and dump in the eggs, using a spatula to craftily mold my amazing omelet into a nice little pancake shape.

When it begins to set, I loosen the omelet to be flipped, turn the burner off, and then flip it over on the other side, setting the pan off the burner.  This took many, many months to perfect.  So many times, the flip was a failure, and it became difficult to discern whether I was making scrambled eggs or an omelet.

The single serving guacamole is opened and spread over the omelet.  Then, it is slid off the pan and folded neatly in half.  It is then covered in salsa or some organic Sriracha.

This whole process takes me about 10 minutes and it is part of my daily routine.  If you checked out my schedule that I posted last week in An Example of Productivity, you’ll understand why I schedule 30 minutes for Breakfast/Coffee.  Making a pour-over coffee each morning from freshly roasted and ground beans is another story.

Since I have such a short memory when it comes to food, I end up having the best cup of coffee and the most fantastic omelet of my life every morning.  It never gets old.

This keeps me satiated until lunch without the need to snack. There sometimes is a desire to snack, but after a glass of water, I’m fine until lunchtime. By skipping toast (and any other type of grain), my body begins burning fat for fuel and providing me energy and mental focus to stay on task all morning.

The featured photo was my omelet on Saturday.  That version involved Sriracha and the individual guacamole cup.  The version below required that I mash my own avocado mixed with spices for the delightful spread, along with red jalapeno salsa on top.  This was last Tuesday morning’s meal:

If you find yourself hanging out with me one morning and you remember, ask for one of my masterpieces, and I’ll happily oblige.

Fun fact: this ode to the omelet is my 200th blog post, and today is Erin and my 8th anniversary!

# 10,000 Hours of Purposeful Practice

In Malcolm Gladwell’s book Outliers, he discusses the theory of 10,000 hours.  By examining several individuals that were considered the elite in their field of expertise, he was able to find that all of them had something in common. Over the course of each of these individual’s lives, they were able to devote a lot of their time, around 10,000 hours in fact, to their craft.

It isn’t just any kind of practice. In Peak: Secrets of the New Science of Expertise, by Anders Ericsson and Robert Pool, they devote an entire chapter to The Power of Purposeful Practice. It is not just the 10,000 hours of practice, it is the 10,000 hours of purposeful practice that is important.

This is a LOT.

If you practice something 1 hour every day, it will take you 27.4 years to get to 10,000 hours.  That lowers to 18.3 years if practicing 1.5 hours every day, and down to 13.7 years if you practice 2 hours every day.  You need to practice about 2 hours and 45 minutes a day at something to reach 10000 hours in 10 years.

The average adult American watches television for 35.5 hours per week.  At this rate, it only takes us 5.4 years of purposeful watching to become an expert at watching television.

This is a huge reason why most conversations are dominated by talking about what is on TV or the recent movie.  We all seem to be experts.  For some, it means talking extensively about sports, and specifically, their favorite sports team.  For others, it means dissecting each episode of Game of Thrones.

If you could go back and trade just a small portion of all that TV watching for the purposeful practice of something would you? If yes, where would you be now?

Maybe you would be an expert in coding and software development.

Maybe you would be an expert small start-up investor.

Maybe you would be fluent in 2 or 3 languages.

Maybe you would be an expert piano, guitar, or drum player.

At this point in my life, I probably won’t reach 10000 hours in my language learning, guitar practice, or jiu-jitsu that I’ve recently taken on in my life.  However, there definitely won’t come a time when I wished I had traded all the hours that I will inevitably put into these new activities for some more TV or movie watching.

# Paths to Work

My house is close to the corner of 15th and High in Topeka, KS.  My office is close to Boswell (5 blocks east of High) and 17th (2 blocks south).  How many paths can I take if I choose only to travel south or east (and not north or west, which would be counterproductive)?

This is a small enough problem that you could literally count up the number of paths pretty easily. In fact, why don’t you try that out and see if you get the correct answer!

Now, what if you were faced with counting up the paths for this situation?

Would you want to count up the number of the paths to work in the same way? Instead, let’s find a pattern that we can work with and understand, and then apply that to any grid of any width and length.

Let’s go back to the original 5×2 grid and analyze why there are 21 total paths.  It’s easier to start at the end and work backward. Notice that in the 5×2 grid that there are 18 different “corners” that we could potentially find ourselves.  To go from home to work, we must go east 5 times and south twice in some order.

What if we find ourselves 5 blocks east and one block south, or 4 blocks east and 2 blocks south? According to the diagram below, we only have 1 option in each of those cases.  We must travel the final south path in the first case, or the final east path in the second.

Now, if we look at the corner diagonal from work (this would be 4 blocks east and 1 block south), you’ll notice that we have an option of going south or east from here. If we choose east there is only 1 option, and if we choose south, there is only one option. Adding the two together, we get 2 paths from this corner.

Now, let’s expand out a single block.

If we find ourselves 5 blocks east, then we have only one path choice, and that is to travel south two blocks to work.  If we find ourselves 2 blocks south and 3 blocks east, again, we have only one path option, and that is to travel the two more blocks east to work.

Now, the fun part. When we look at the corner that is 4 blocks east, we have the option of traveling south or east.  South will give us 2 paths to choose from, and east will give us only 1 path to choose, for a total of 3 paths from that corner.

If we are at the corner that is 3 blocks east and 1 block south, we again have the option of traveling south to a corner that has only 1 choice and east to a corner with 2 choices for a total of 3 paths from that corner.

Finally, at the corner that is 3 blocks east, we can travel south to a corner that has 3 paths to choose from, or east to a corner with 3 paths to choose from, giving us a total of 6 paths to follow.

Let’s fill out our original map, to detect any patterns.

If you placed “Work” at the top of a triangle it might begin to look like the triangle I talked about in The Yanghui Triangle, Part I or  The Yanghui Triangle, Part II. Indeed, we can find the number of paths using combinations.

Previously, I described how to get from home to work we must pass 7 corners.  Each of the 7 corners must have a decision to go south or east, but there must be exactly 5 corners in which we choose to go east and 2 corners in which we decide to go south.

With 7 corners, how many ways can we choose 2 of them in which to go east (and therefore south on the other 5)?  The answer is $\binom{7}{2} = 21=\binom{7}{5}$.  This can be found using the ${}_nC_r$ button on your calculator. First, push 7, then use the ${}_nC_r$ button, then push 2 (or 5), and then press enter.

We now have the tools to answer the bigger 6×10 grid puzzle.  If our work is 6 blocks south and 10 blocks east, we will need to visit 16 corners on our path to work.  Of these 16 corners, we will need to decide which of the 6 we will travel south on (and, therefore, which 10 we travel east on).  This is $\binom{16}{6}$ or ${}_{16}C_{6}$, which is 8008.  This would have taken a little longer if we counted the paths directly.

# Rates of Change

Last week, I was busy studying for (and then taking) the MFE Exam through the Society of Actuaries.  I hope I passed.  That was the reason I did not post.

Next week, I plan on posting about my experiences truly taking advantage of a break when there is no travel planned.

Until then, I solve problems.

Consider the following two problems.

1. A vehicle travels 30 mph for 1 hour, and then 60 mph for an hour. What is the vehicle’s average speed?
2. While traveling an unknown distance, a car travels half the distance at 30 mph and the other half at 60 mph. What is the vehicles average speed?

While the first problem is pretty easy to solve, the second one can be a bit more challenging.

First, recognize that mph stands for miles per hour, or in mathematical speak, mi/hr.

We do this so that we can see that when we multiply the speed by the time, we get the distance:

$30 \text{mi}/\text{hr} \cdot 1 \text{hr} = 30 \text{mi}$

For problem 1, we traveled 30 miles in the first hour, and 60 miles in the second hour for a total of 90 miles in 2 hours.  This means our average speed was $90/2=45$ mph.  Like I said, that one was easy.

When we’re working with distances and speed, to get the time you need to take distance divided by speed.  To see this, remember dividing by a fraction is the same as multiplying by the reciprocal:

$\frac{\text{mi}}{\text{mi}/\text{hr}} = \text{mi}\cdot \frac{\text{hr}}{\text{mi}} = \text{hr}$

For problem 2, we don’t know the distance. But that is OK.  We don’t need to. It turns out, we could pick any distance we wanted to solve the problem.  To make it easier, we will do that.  Let’s pick 120 miles.

If the vehicle travels 30 mph for 60 miles (half of 120 miles), then it will take the vehicle $60/30=2$ hours to do so. If the vehicle travels 60 mph for the second 60 miles, it will only take the vehicle $60/60=1$ hour to do so. Thus, the total travel time is 3 hours.

If the vehicle traveled 120 miles in 3 hours, then this means the average speed is $120/3=40$ mph.

Now that you know the answer, see if you can follow along by exploring any distance, we’ll call it d.

If the vehicle travels 30 mph for d/2 miles, then it will take the vehicle $(d/2)/30=d/60$ hours to do so. If the vehicle travels 60 mph for the second d/2 miles, it will only take the vehicle $(d/2)/60=d/120$ hours to do so. Thus, the total travel time is $d/60+d/120$ hours.  Getting a common denominator to add these fractions, we get a total travel time of

$\frac{d}{60}+\frac{d}{120}=\frac{2d}{120}+\frac{d}{120}=\frac{3d}{120}=\frac{d}{40}.$

If the vehicle traveled d miles in d/40 hours, then this means the average speed is $d/(d/40)=d\cdot \frac{40}{d} = 40$ mph.

See, the distance doesn’t matter.

### Riddler Express on March 2, 2018

The problem as stated on FiveThirtyEight was a rate problem:

Andrea and Barry both exercise every day on their lunch hour on a path that runs alongside a parkway. Andrea walks north on the path at a steady 3 mph, while Barry bikes south on the path at a consistent 15 mph, and each travels in their original direction the whole time — they never turn around and go back the other way. The speed limit on the parkway is the same in both directions and vehicle traffic flows smoothly in both directions exactly at the speed limit.

In order to pass the time while they exercise, both Andrea and Barry count the number of cars that go past them in both directions and keep daily statistics. After several months of keeping such stats, they compare notes.

Andrea says: “The ratio of the number of cars that passed me driving south on the parkway to the number of cars that passed me driving north was 35-to-19.”

Barry retorts: “I think you’re way off. The ratio for me was 1-to-1 — the number of cars that passed me going south was the same as the number that passed me going north.”

Assuming Andrea and Barry are both very good at stats, what is the speed limit on the parkway?

Although this problem may be too difficult to solve for many of you, some of you may be able to follow how it is set up.

We eventually want to find the speed limit on the parkway. I’ll call that v, for velocity.

There are a few other unknowns in this problem.  We are not given the flow of traffic northbound and southbound.  I’ll call these values $R_N$ and $R_S$.  An example value might be 30 cars per hour.  In this case, you would know that a car is passing a stationary object once every 1/30th of an hour (or 2 minutes).

First, let’s focus on the southbound traffic.  From a stationary point, the flow is $R_S$ as stated above.  However, if you travel south at a velocity of v (the same speed as the cars), the flow at which you see cars pass you that are going southbound reduces to 0.  This is a linear relationship.  If y is the perceived flow, and x is the velocity at which you travel, then

$\displaystyle y= -\frac{R_S}{v}x+R_S$

is the equation that describes this relationship. For a stationary object, x=0, and $y=R_S$, which is what we expect.  If we travel south at speed v, we perceive traffic flowing by us at $y=-\frac{R_S}{v}(v)+R_S = -R_S+R_S=0$ cars per hour.

Relative to Andrea, who is traveling north along with the northbound traffic, the flow of traffic that is going northbound is reduced to $-\frac{R_N}{v}(3)+R_N$.  From the perspective of the southbound traffic, she is traveling at -3 mph!  This increases the flow from which she sees the southbound traffic pass her to $-\frac{R_S}{v}(-3)+R_S = \frac{3R_S}{v}+R_S$.

We are given that the ratio of the second equation to the first is 35:19.  This gives us the complicated mess

$\displaystyle \frac{3R_S/v+R_S}{-3R_N/v+R_N} = \frac{35}{19}$,

which is equivalent to

$\displaystyle 57R_S+19R_Sv = -105R_N+35R_Nv$

after a little algebra (first, cross multiplying, and then, multiplying by v on both sides).

Using a similar argument for Barry, the flow of the traffic going southbound reduces to $-\frac{R_S}{v}(15)+R_S$ and the flow of the traffic going northbound increases to $\frac{R_N}{v}(15)+R_N$.  Since he witnesses a 1:1 ratio, this gives us the complicated mess

$\displaystyle \frac{-15R_S/v+R_S}{15R_N/v+R_N} = 1$,

which simplifies to

$\displaystyle -15R_S+R_Sv = 15R_N+R_Nv$

after a little algebra (cross multiplying while thinking 1=1/1, and then multiplying both sides by v).

We’re in a dilemma right now as we have two equations and 3 unknowns.  In order to be able to solve for a number of different variables, we need to have the same number of equations as we do unknowns.

We can get around this issue since we do not really care about the values of $R_S$ and $R_N$.  What we will do, is divide both the equations above by $R_N$, to create one unknown $R = \frac{R_S}{R_N}$.

Now, the two equations become

$\displaystyle 57R+19Rv = -105+35v$

$\displaystyle -15R+Rv = 15+v .$

This seems like a headache to solve, so let’s use our friend WolframAlpha.  This computational engine really likes x’s and y’s, so let’s plug

$57x+19xy=-105+35y, -15x+xy=15+y$

into that website and see what we get.

It provides us with the two solutions $x = -\frac{21}{19}, y = \frac34$, and $x=\frac53, y = 60$.  The second solution is the only one that makes sense, which suggests that the cars are traveling at 60 miles per hour.

Less important, but we might as well address it, are the flow rates.  We’re only able to determine the ratio of the flow rate of the southbound traffic to the northbound traffic.  If the southbound traffic has about 5000 cars per hour, we know the northbound traffic has about 3000 cars per hour since $x=5/3$.

# A Unique University Experience

Have you ever heard of professors taking classes with students?

This semester, I’m teaching Stochastic Processes on Mondays and Wednesdays at 11am.  However, at 10am on those same days, I’m taking an Investments course through the School of Business.  Four of the students in my Stochastic Processes class are in the Investments course.  A few of us walk from one class (that I’m taking with them) to the other class (that I’m teaching to them).

This provides quite a unique university experience.

### From the Professor’s Perspective

My students are watching me.  I better be the model student that I want them to be.  Hmmm.  No skipping class I guess.  Unless that is what I will find OK if they decide to skip mine.

What if I don’t do so well? I better study and work hard enough to ensure I do well. That’s what I want out of my students, so that is what I better do.

Wow… that’s how other professors use that statistical notation?!?  I need to work that into my stat lectures from now on, and warn them of the relaxed notation they’ll encounter as they move from course to course.

Excel does THAT!!  I thought I was a master of Excel by now.  Nope.

### Perspective from a Student

I think I know that answer to that basic statistical or mathematical question, but my statistics professor is taking the course with me and sitting right over there.  It would be embarrassing if I got it wrong.  I better not answer.

It is really cool how what I learned in statistics (from that guy over there) is being applied to this Investments course.

Why in the hell is he taking this class? He’s got to be bored out of his mind.  There goes the curve!

Wonder if he’ll work with me on my homework assignment?  Never mind, I don’t need it.  It was easy.  Maybe that has to do with the fact that I took stats from him? Hmmm.