May 2016

The Pigeonhole Principle

After looking at the potential topics for this paper I was initially drawn to “Mathematics and Music” because my wife is very musical and likes the topic but as I read about it, it didn’t really sing to me. As I read the rest, I thought the pigeonhole principle seemed very interesting and I decided to write about that. In chapter three in the book “To Infinity and Beyond” the pigeonhole principle states: “Suppose we have four pigeons but only three pigeonholes. No matter how we assign a hole to each pigeon, at least two pigeons will have to share the same hole.” We can apply this principle to conclude some surprising results with very little work.
Before I give some other examples this principle is based on certainty and not chance. So each example will happen but the specifics may differ from case to case. One example I found outside of the book was “if you pick five cards from a standard deck of 52 cards, then at least two will be of the same suit.” (16 fun applications of the pigeonhole principle) The reason that I liked this example is that I could easily test it and see for myself at home. While this principle made logical sense, it was good to be able to test it and see for myself that it worked. I then scaled this principle up. I found that when I had nine cards, three of them were of the same suit. While many of the conclusions of the principle are difficult to confirm I found this to be fun to scale and confirm. [Calculation 9 = 4 x 2 +1]
You might wonder to yourself at this point about how this is a nice trick but what use is this principle to mathematics or science? As I was reading I learned the pigeonhole principle is called an “Existence theorem” because it confirms that a solution exists but doesn’t give a method for finding it. So when working on a difficult problem we can use an Existence theorem like the pigeonhole principle, to confirm ahead of time that a solution exists before working on it. This would be preferable to wasting time on a problem where a solution doesn’t exist and for which that could have been confirmed previously.
Another principle spoken of in the chapter is “Uniqueness of a solution.” The Uniqueness theorem is essentially if there is a solution, it is the only solution. While the pigeonhole principle itself is not a unique solution with two pigeons that can be random among the group, this theorem is an extension of that principle. This theorem can be applied to many things in life where there is a presumption of one solution. The book uses an example of finding the one true church. An example popular in the world would be finding your one true love. After you found that person continuing to search would be illogical as you have already found the person.
The pigeonhole principle was fascinating to learn about and test myself with its predictive ability. While I’m not the sort of hard mathematics person to use this in my working life, I think it will be fun to share this sort of thing with friends as a conversation piece they can see and test in person with a few examples.

The Pigeonhole Principle Read Post »

If the 2016 Presidential election were held today…

Clinton-Johnson-Trump-750x400If the 2016 Presidential election were held today with Hillary Clinton (D), Donald Trump (R) and Gary Johnson (L) as the candidates, I believe the electoral votes would come out far more skewed than than what the most recent national polls suggest. When looking at what States are safe for candidates and which ones candidates have a strong lead I would place Clinton as currently receiving 285 electoral votes, Trump receiving 189 and 64 electoral votes up for grabs. So even if Trump was able to grab all the states that are too close to call, he would still lose. The only way I really see this election getting interesting is if Gary Johnson can pull enough votes to get his home state of New Mexico and Trump gets all the states I’ve marked as toss ups along with North Carolina, where Clinton has a smaller lead. Johnson is currently polling at 10% or higher so he has the chance to make the election interesting if he gets enough support to win a state. If he doesn’t get that level of support, he is unlikely to be a spoiler for either Trump or Clinton as he draws roughly equal support from disenfranchised members of each their parties. So unless Trump kicks it up a notch and Johnson gains enough momentum to win his home state, we will have a President Clinton. If Johnson and Trump are successful then the vote would go to the House of Representatives who would choose from the top three finishers. So this election will either be a boring repeat of 2012 or very interesting as it’s fueled by the dislike of two very polarizing candidates.


Click the map to create your own at 270toWin.com

If the 2016 Presidential election were held today… Read Post »

Joseph and reading

batman

Joseph continually surprises us with his reading ability today head to me “Batman: Reptile Rampage.” Which is level 2 in the “I Can Read!” series but level 3 in the ATOS book level. He also read to me “Henry and Mudge and Mrs. Hopper’s House.” Which is also level 2 in the “I Can Read!” series but level 2.8 in the ATOS book level. I prefer to use AR level as a guide as I find it a better guide to compare against different books. This is pretty amazing to us because Joseph won’t enter Pre-K until next year.

Some of the things that I think have helped Joseph with his reading are:

  1. We read at least two books to him daily before bed time. These books come from his collection at home (that we mostly got for free) and from library books we trade out every few weeks on Monday night. We also read him a few scriptures every night that also gives him some more advanced reading exposure.
  2.  PBS shows like Word World, Word Girl and Super Why.  Sure gluing his face to the TV wouldn’t be good but when he does watch a show I feel programming like this does very well at exposing children to reading and new vocabulary.
  3. Video Games. Your probably saying to yourself wait, “video games”? Yes, video games have helped encourage him to read and sound out new words. Sure some games don’t require reading but RPG’s like Twilight Princess pretty much require it and he really wanted to read all the text boxes from the villagers. I just modeled it a bit for him (played) and he wanted to be just like dad reading and playing.

At any rate, I thought I’d just share with others what worked with my family for encouraging my son to read. I hope this is helpful to someone else.

Joseph and reading Read Post »