Monroe and Brown Counties

Hello. I’m Pleepleus.

From time to time, I hack into The Shaw Show and write stuff about my travels. Even though it is a total hack, the creator of The Shaw Show either

  1. is so dense that he is oblivious to my posts, or
  2. simply doesn’t care that I hack in and write as a guest.

I’m pretty sure that it is 1.

Next week I will travel to Michigan and visit several places along I94. Before I go and report back, I want to write about my travels over the Halloween weekend in 2017.

Here is a map on which I’ve drawn some circles and written the numbers 1 through 4.  I will probably refer to the map at least 4 times.

TripMap
Thanks Google! The surrounding area of Bloomington, IN.

Sometime a little before 1am on the morning of Saturday, October 28 (which still very much felt like Friday evening), I arrived at Paynetown State Recreational Area surprised to still find an attendant on duty to provide me with a site on which to place a tent and sleep.  Paynetown is in the area #1 circled above.

In the morning, I made some pour overs on the camp stove using some coffee I had just roasted a few days prior.

Pro Tip: If you don’t own a camper that needs electricity, you can find just about any campsite that you want at the end of October.  I chose one close to the showers and restrooms, which provided me with the electricity I needed to charge my phone.

Prior to any hike, one must have a decent breakfast. If you don’t bring your own food to make at the campsite, then I have some really good suggestions for places to break your fast each morning.

Runcible Spoon on 412 E. Sixth St. in Bloomington is a unique and wonderful experience.  They have a wonderful breakfast menu, decent Bloody Marys (and Mimosas if those are more your style), and Zombie Dust in the bottle.  I really did not care how early it was, a Zombie Dust was a perfect side car to my Bloody.

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This is my happy place.

Pro Tip: Bring a group of 4 so that pitchers of Bloody Marys or Mimosas can be ordered at a more efficient cost.

The daily dose of nature was at the Hickory Ridge Fire Tower, which is the pin in the circled area #2 in the map above.  This was definitely a highlight of the trip.

Pro Tip: Although climbing to the top of Hickory Ridge Fire Tower is worth it just about any time of year, it is most worth it when there are Fall colors to gaze upon.

Hiking up and around the fire tower made me thirsty and hungry.  I recommend Function Brewing for drinks and food and The Wood Shop for some tasty digestifs.

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Later that evening, I enjoyed a Sirloin Steak cooked to perfection along with an amazingly dark, smooth, and decadent Cabernet at The Little Zagreb.

Pro Tip: Make your reservation early at The Little Zagreb, and prepare to spend a lot of money for the amazing experience that is very much worth it.  If you camp like I did, you can save the money that you need to eat there.

On Sunday morning, October 29th, I had a hangover.  Although I wanted to hike in the trails around the campground that morning, it wasn’t happening.  I slept in until it was time to leave for my 11:30 reservations at FARMbloomington. This place had amazing coffee and bacon.  It offers a full bar and a wonderful menu.

I went back to Paynetown State Recreational Area, and hiked the 3.2 miles along the trails that are in circle #1 above.

That worked up an appetite, so I stopped for some appertifs at Oddball Fermentables. Specifically, I tried First Date, Enzy’s Gold, Hopberry Blue, Hopberry Red, and Melody.  These are all cysers, which are mead-cider hybrids that they specialize in.

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Keeping warm by both the fire and the cysers.

Pro Tip: If you visit Oddball Fermentables, it will feel as if you are walking into someone’s house. Embrace the awkwardness and walk right in with confidence.

To combat my sleepiness, I decided that a pour-over was prudent, and visited Hopscotch Coffee. Their pour-over was fantastic, and it was the fuel I needed for an impromptu hike at Leonard Springs Nature Park.  This is in circle #4 on the map above.

Pro Tip: To hike Leonard Springs, I went down the stairs first and did the hike counterclockwise. Convince yourself to pass the staircase and come up it at the end. I believe this hike is better done clockwise.

Although I was hungry enough for dinner before the hike, I wanted to ensure that there would be plenty of hunger when I went to Mother Bear’s Pizza.

If you only go to one place for a meal in Bloomington, get some pizza at Mother Bear’s.

When I asked what they had on tap, the waitress read through a list of beers. She read off “Space Station, and Middle Finger” as if it were two separate beers.  I told her that “Space Station Middle Finger” was just one beer and that I would have a pint.  She looked at me funny and stated that she was pretty sure it was two beers.

Soon she brought my pint of Space Station Middle Finger and apologized for her blunder. How dare she question a stuffed monkey beer connoisseur!

Pro Tip: Be really hungry when you go to Mother Bear’s, and if you can swing it, have a big group. That way, you can order all the different kinds of amazing pizza they have available.  You need to have room in your stomach for at least 4 slices.  I ordered 10 Hot Garlic Wings, a 10 inch Spinnocoli, and a 10 inch Dante’s Inferno.

There were a mere 4 slices left when I was done being a glutton.

I had a fire that night at the campground, slept hard, and got up for some coffee at the campsite. Breakfast: 4 slices of pizza.

My journey on Monday, October 30 would take me to Brown County State Park and Hesitation Point.  This is in the area circled by #4 on the map above.

Brown County State Park is stunning in late October. I hiked Trails 8 and 7 starting at Hesitation Point. This took me around Ogle Lake.

Pro Tip: Visit Brown County State Park and hike anywhere within during the Fall.  It is gorgeous and will capture a piece of your soul.

When one is already this close to Nashville, IN, you might as well stop in and visit this quaint little artsy town.  I went to Big Woods Brewery Company (not to be confused with Big Woods Pizza Company which is very close by) and ordered a Hare Trigger IPA. When I found out that was in the bottle, I purchased some to take home.

There was another cozy fire that night, and in the morning, there were ice crystals on the tent.  For my last breakfast in Bloomington, I ate at The Village Deli. This is a staple breakfast place, and is where you go when you want to get going and you’re not in the mood for a Bloody Mary.

Mark Schultz vs. Ed Banach in 1982

On Saturday morning, January 6, 2018, Erin and I participated in the Topeka to Auburn Practice Run with the Sunflower Striders Running Club.  I did not want the full 13.1 miles yet, as the race is in two weeks, but I did want to get 10 miles in. Erin got in a little over 9.

During the run, I listened to Mark Schultz be interviewed by Ryan Ford, the host of The Grappling Center Podcast.  Mark Schultz was played by Channing Tatum in the movie Foxcatcher, which tells the story that led up to the tragic murder of Mark’s brother David.

The podcast is very interesting, and worth listening to, but for a super fascinating highlight use the link above to listen from 37:00 to around 41:45 for Mark’s description of the match with Ed Banach.  Listen all the way to 44:10 for post analysis of the match and the encouragement to watch the match itself, which I’ve put below.

Although you could watch the match without listening to Mark’s description, you won’t get as much out of it, so listen first!  I hope you enjoy the experience as much as I did.

 

The 2017 Review and 2018 Plan

Failures of 2017

There are many resolutions of 2017 that I did not accomplish or stick with.  I may have been very ambitious with my Resolutions of Twenty Seventeen.

Failure 1. The first personal resolution was to maintain a weight between 163-169. My plan was to have two weigh-ins per week with an allowance of 4-8 weigh-ins above 169.  I failed in having two weigh-ins per week.  By strategically weighing in fewer times, I was able to keep that allowance… but we all know that is cheating.  And cheaters are failures.

Having the resolution in place did keep me in check, however.  Whenever a weigh-in resulted in something above 169, I stuck with the primal diet for the week following so that I could fall back into the range.  Observed a few days ago after the holidays had ended: my highest recorded weight in 2017: 171.6 lbs.

My resolution for 2018 is to drop to 160 at some point during the year, and then let that slowly creep to my maintenance region (163-167).

Failure 2.  Remain off of social media.  At the end of 2016, I had quit Facebook, Twitter, Instagram, and Untappd.  This had a lot to do with the influence from Cal Newport’s Deep Work, a fantastic read that I recommend to everyone.

While I remained off of Twitter, Instagram, and Untappd, I failed at remaining off of Facebook.  Although I am back, I’m trying my best to post very infrequently, and am trying to make sure that each post is adding value to my life (by perhaps adding value to others).

My resolution for 2018 is to spend less time on Facebook (measured by my Quality Time app) and post less often.

Failures 3, 4, and 5. I failed at making my own limoncello, orangecello, and ginger beer. I failed at kayaking the rivers and lakes around my area, and in the amount that I wanted to.  I failed at completing a triathlon.

Insert a sad face here.

Failure 6. After the election of 2016, and during the first part of 2017, I was really gung-ho about starting my own chapter of Represent.US in Topeka.  I wrote a letter that was included in the League of Women Voters’ Newsletter in Topeka.  I gave a presentation in front of the Sunrise Optimists Club of Topeka. I talked about the organization in front of Washburn’s Student Government.

I know little about politics and political activism, but I learned one very important lesson through all of this.  I would rather sign up for a 50 mile run across a desert packing my own water than try and motivate anyone around a political cause.  Although there may be some initial interest on the surface, nobody (it feels like) has any desire to give time for such a thing.

This deflated me, as I was faced with this harsh reality.

Successes of 2017

Success 1. I rode my bicycle over 2017 miles.  Using the Cottonwood 200, BAK, Ragbrai, and the Buffalo Bill Century Ride, among several other personal rides, I was able to bike over 2017 miles last year.  The weather in 2017 allowed me to ride my bike on January 2 last year!

This will be a new bike year.  In 2018, I resolve to explore more with my new bike, which will be able to ride on peat gravel and gravel trails.  Although 2018 miles will be in the back of my mind, this goal will not make the cut this year.

Success 2. Watch less TV and fewer Movies than I did is 2016. With my post Time Spent Watching TV, I analyzed the data I had collected on my TV and movie watching in 2016.  Here is a summary.

  • I watched TV and movies for approximately 15500 minutes in 2016.
  • This worked out to be about 42 minutes and 21 seconds per day, accounting for 4.41% of my day.
  • Counting time awake as a day (~16 hours, or 960 minutes), I watched TV and movies for a little over 16 days of my life in 2016.

Here is the summary of 2017.

  • I watched TV and movies for approximately 7500 minutes in 2017.
  • This worked out to be about 20 minutes and 33 seconds per day, accounting for 2.14% of my day.
  • I watched TV and movies for almost 8 days of my life in 2017.

In 2018, I resolve to watch less than or equal to the amount I watched in 2017. This year, however, I will also track what I watch on YouTube.

Success 3. Although not part of my resolutions, I picked up some daily habits in the latter part of 2017 that will continue into 2018 and beyond.  In my post Homebrew Daily, I wrote about creating a daily habit of learning something related to homebrewing.

Not long after that post, I began a daily habit of studying Norwegian using Duolingo.

These habits are now a part of me.  Yes, there have been a few days since I have started where I missed a day, but the habit is a part of me now.  So, going to bed having missed a day feels as if I didn’t brush my teeth.

In 2018, I resolve to continue learning Norwegian (even after we travel over there and back in May).  I also will continue learning something about homebrewing each day.

Success 4. Again, this was not part of my resolutions, but I feel like experimenting with something new can lead to a success.

Last night was my 4th class in Brazilian Jiu-Jitsu.  I’m working out and am a student under Cody Criqui, who owns and operates the Criqui Academy in Topeka. After two classes, I purchased my own Gi (pronounced with a hard “g”, they are essentially Japanese pajamas to outsiders), compression shorts, rash guard, and mouthpiece.

This will take priority in my “health” category in 2018.  As long as I can stay uninjured, I resolve to get a few stripes on my white belt by the end of 2018.

Aliquot Divisors

When the idea came to write about this post, I had never came across the term “aliquot” to my recollection.  Let’s quickly deconstruct that sentence and assign some probabilities.

  • I’ve never came across the term “aliquot”: probability = 0.01
  • I don’t recall having come across the term “aliquot”: probability = 0.99

From this, I hope you’ve come to understand a few important things.

  • Probabilities must sum to 1
  • All probabilities must be non-negative and less than (or equal to) 1
  • I most likely don’t have a great short term memory.
  • I am most likely a pretty funny dude.

Sum of Divisors Function

After searching the term “Sum of Divisors function” a page with a wiki on the Online Encyclopedia of Integer Sequences came up.  On this, they define a function for the sum of the divisors for a number.

Let’s do a quick example before introducing notation and look at the number 40 (that’s how old I am for another month and a half).  What numbers divide 40 evenly?  That is, if you push 40, and then \div, what numbers can you push next so that the result is an integer (not a decimal).

Here are the divisors of 40: 1, 2, 4, 5, 8, 10, 20, 40.  A function that computes the sum of the divisors of a number should give an output of 1+2+4+5+8+10+20+40 = 90 whenever you input the value 40.

Now for the notation.  The sum of the divisors function is given as \sigma(n).  To apply the above example, this means that \sigma(40)=90. In other words, you input the value 40 into the sum of divisors function \sigma(\cdot), and the output is 90.

Let’s try a few others. What if we input 10?  That is, what is the value of \sigma(10)?

First, let’s find the divisors of 10: 1, 2, 5, 10. Next (and finally), sum them together: 1+2+5+10 = 18.  Thus, \sigma(10) = 18.

How about \sigma(11)?  The only divisors of 11 are 1 and 11 (which, as a side note, means that 11 is a prime number).  This means \sigma(11)=12.  If you inspect this example thoroughly, you should be able to deduce that the sum of the divisors of any prime number will be one more than the number.

Since 5 is a prime number, \sigma(5)= 6 (the sum of 1 and 5).  Likewise, since 7 is a prime number, \sigma(7)=8.

Now, try a few on your own.  Find \sigma(15) and \sigma(24).  The answers are below, but you can also check them by typing “sigma(15)” or “sigma(24)” into WolframAlpha. Technically, the correct code is DivisorSigma[1,15] and DivisorSigma[1,24], but you need not concern yourself with that.

So What are Aliquot Divisors?

The aliquot divisors of a number are what I have always known to be proper divisors of a number, which are all the divisors except the number itself.

So, the aliquot divisors of 40 are 1, 2, 4, 5, 8, 10, and 20 (we leave off 40).  And the sum of the aliquot divisors function is denoted by s(\cdot).  For our example, then, s(40)=1+2+4+5+8+10+20 =50.

With this sum of aliquot divisor function, s(n), we can now explore some fascinating things about different numbers.

Prime Numbers

For every prime number (a number only divisible by 1 and itself), the only aliquot divisor is 1.  So, by plugging any prime number into the function s(\cdot), the result will be 1.

Deficient Numbers

The numbers 4 and 10 are deficient numbers. Why? When you plug them into the sum of aliquot divisors function, s(\cdot), you will get a value smaller than the number.

The aliquot divisors of 4 are 1 and 2.  So, s(4)=3.  The aliquot divisors of 10 are 1, 2, and 5. So, s(10) = 8. Notice that each of the output values are smaller than the input values.  That means the numbers are deficient. 

Let’s recall what you learned about prime numbers in the previous section.  If p is a prime number, than s(p)=1.  From this we can deduce that all prime numbers are deficient.

Furthermore, since we have the above stated examples of 4 and 10 as deficient numbers, and we know they are not prime since they are divisible by something other than themselves and 1, then we can also deduce that not all deficient numbers are prime.

Abundant Numbers

The numbers 12 and 40 are abundant numbers? Why? When you plug them into the sum of aliquot divisors function, s(\cdot), you will get a value larger than the number.

We have already seen that s(40)=50. The aliquot divisors of 12 are 1, 2, 3, 4, and 6.  Thus, s(12) = 1+2+3+4+6=16.  Each output value is larger than the input value. That means the numbers are abundant. 

Perfect Numbers

The numbers 6 and 28 are perfect numbers? Why? When you plug them into the sum of aliquot divisors function s(\cdot), you will get the same value!

The aliquot divisors of 6 are 1, 2 and 3.  Thus s(6)=1+2+3=6.  The aliquot divisors of 28 are 1, 2, 4, 7, and 14.  Thus, s(28) = 1+2+4+7+14 = 28.  Each output value is the same as the input value!  That means the number is perfect.

Amicable Chains

Recently, I was introduced to the concept of amicable chains.  What if we continue to apply the sum of aliquot divisor function s(\cdot) over and over again?  For example, let’s try it with 12.  As we saw in the Abundant Numbers section, s(12)=16.

Now, what if we plug in 16?  Aliquot divisors of 16: 1, 2, 4, and 8.  Thus, s(16)=15.

Again? What if we plug in 15? Aliquot divisors of 15?  1, 3, and 5.  Thus, s(15)=9.

Again? Aliquot divisors of 9? 1 and 3.  Thus s(9)=4. The aliquot divisors of 4 are 1 and 2, so s(4)=3.  Since 3 is prime s(3)=1 and now we’re trapped at 1 forever.

This created the sequence

\displaystyle 12\rightarrow 16\rightarrow 15\rightarrow 9\rightarrow 4\rightarrow 3\rightarrow 1

If you do a lot of exploration, a lot of numbers will produce a sequence that ends at 1 and becomes trapped. Some don’t. Take the perfect numbers 6 and 28 for example.

Since s(6)=6, then 6 has a chain of length 1 since it goes directly back to itself.  So does 28.

There are other numbers that do weirder things.  Take 220 and 284 for example (you will definitely need to use WolframAlpha on these if you want to check. However, you’ll need to type in “sigma(220)-220” and “sigma(284)-284”.)  The sum of the aliquot divisors of 220 is 284, and the sum of the aliquot divisors of 284 is 220.  This produces a chain of length 2. There are several chains like this one.

With the aid of a computer program, I’ve found one of length 5 and another of length 28.  Using a similar program, I also stumbled upon interesting numbers like 1230 and 1248.  Although they do not produce chains, they take a while to get back to 1.  Using a very cool trick of emailing a former colleague with a higher level of number theory understanding, he was able to find out that it takes 185 terms to get back to 1 starting at 1230 and 1076 terms to get back to 1 if you start at 1248.

The interesting thing about these sequences are how really, really, big they get.


Answers to Above Posed Questions

\sigma(15) = 24, and \sigma(24) = 60.

The Yanghui Triangle, Part II

Welcome back for round two!  If you missed Part I, you can catch up here.

An Alternative Explanation

In the first part of this post, we explored the following specific consequence of the triangle that we generalized:

\displaystyle \binom{6}{2} = \binom{5}{1}+\binom{5}{2}.

I recognized that I lacked an example of why this might be true from a counting perspective.  Using the triangle below, we see that the equation above is saying that 15=5+10.

YanghuiTriangle
Here is Yanghui’s Triangle for convenience.

So, why is this true from a counting perspective?

Let’s say we want to select 2 of these 6 letters: ABCDEF.  There should be 15 ways of doing this according to the triangle.  Let’s list them below.

AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, ED

Now, let’s try it another way.  Let’s first make a decision on the F.  We can either select it, or not select it.  Let’s explore both cases.

We Select The F. Now, our task is to select 1 more of the 5 letters ABCDE.  We can do this in \binom{5}{1}=5 different ways.  We are left with AF, BF, CF, DF, and EF.

We Do Not Select F. Now, our task is to select 2 of the 5 letters ABCDE.  We can do this in \binom{5}{2} = 10 ways. Here they are: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.

Now, combine these two together for the total number of ways to select 2 of 6 letters.

OK, now onto some more fun properties.

Powers of 2

If we sum each row of Yanghui’s Triangle, you will notice a pattern.

  1. Sum of 0th row: 1
  2. Sum of 1st row: 2
  3. Sum of 2nd row: 4
  4. Sum of 3rd row: 8
  5. Sum of 4th row: 16

And so on.  From the title of this section, you probably have already guessed the pattern.  If we think of the first row as the zeroth row instead, then we simply take 2 to the power of the row to get the sum.

Why does this work? You may have to review the first post to recall that the numbers in each row can be used as the coefficients when expanding expressions such as (x+y)^6.  From the previous post,

\displaystyle (x+y)^6 = x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6.

Let’s use this result, and plug in 1 for both x and y. The left part of the equation says that this is just 2^6.  The right part of the equation says that this is the sum of the numbers in the 6th row of Yanghui’s Triangle.

Fibonacci Sequence

PascalsTriangleFibonacci
By using hexagons to construct the triangle, the lines that create Fibonacci are easy to construct

The picture above says it all.  By creating lines that run parallel to the top left side of the hexagons and summing up the numbers that it passes through, we get Fibonacci’s Sequence.

Why is this?

Let’s explore the line that goes through 1, 4, and 3, which sums to 8.  The 1, 4, and 3 correspond with the following counting principles: choosing 0 of 5 things, 1 of 4 things, and 2 of 3 things.  To see why this must sum up to the 5th Fibonacci number (again, thinking that the first 1 in Fibonacci’s sequence is the null, or zeroth number), review my explanations in Approach 1 and Approach 2 in Jumping Lily Pads. You can also watch the quick video there.

For a brief explanation of why \binom{5}{0}+\binom{4}{1}+\binom{3}{2} sums up to the fifth Fibonacci number, it is like counting the number of ways that a frog can jump from shore to the fifth lily pad if the frog can jump one or two lily pads at a time.

He can make 5 total jumps, 0 of which are skipping any lily pads: \binom{5}{0}=1.

He can make 4 total jumps, 1 of which must be a double: \binom{4}{1} = 4.

He can make 3 total jumps, 2 of which must be doubles: \binom{3}{2} = 3.

As was described in that post, this is equivalent to summing up the number of ways you can jump to lily pad 3 (3) and the number of ways you can jump to lily pad 4 (5).