Day 17, MAT115

• A key
• A small curve (add 3.5 points to your score)

• The Bridges of Konigsberg problem: "is it possible to set off and walk around Konigsberg crossing each bridge exactly one (without crossing any bridge twice)?"
  • 

    As you can see, the Konigsberg bridge graph has 4 vertices of odd degree.

  • There is only an Euler path if a graph has either 0 or 2 vertices of odd degree. If there are two odds, then you will start the path at one, and end it at another; if there are no odd vertices, then you will start and end at the same vertex (and create what is called an Euler cycle).

• Definition: Planar graphs are graphs that can be drawn such that no two edges intersect. Which of our complete graphs are planar?

  • Exercise: $K_4$ is planar: it can be drawn without false intersections.
  • Exercise: $K_5$ is not planar:
  • The three utilities and three houses graph is not planar:

  • Kuratowski's theorem: Any non-planar graph has a copy of one of these two graphs in it somewhere, as a subgraph:
    

1. On problem 1, I expected to see three trees, and then see the number written in three different ways: as a product of primes, as a sum of powers of 2, and as a tally.

   These I call the "great factorizations" of mathematics. There will be one more, and it's the coolest of them all....

2. Read the instructions: some of you attempted all six parts of problem 2, whereas you were allowed to skip two. The most amusing of these was the "Vi Hart" problem -- a lot of folks seem to think that Vi's a guy, which, from the video, she's clearly not! So do your readings, and visit the websites I ask you to visit.

3. Problem 3: the student does a good job of explaining one of those factorizations: every counting number is either a power of two, or can be written as a sum of distinct powers of 2, in exactly one way.
4. Problem 4: I knew that students were concerned about sensitivity and specificity, so I just went ahead and used the data (including the fact that 15% of American males are smokers) to build the tree of natural frequencies.

   At that point, the calculation of conditional probabilities is easier (according to Strogatz, at any rate!).

   1. When we say "given that a male is a smoker", we mean that you need only look at the smokers (there are 81+144=225 of them). Then calculate the proportion (or percentage) of those who end up with lung cancer.
   2. When we say "given that a male is a nonsmoker", we mean that you need only look at the smokers (there are 19+1256=1275 of them). Then calculate the proportion (or percentage) of those who end up with lung cancer.
   3. The last calculation is actually the sensitivity, if we think of "smoking status" as a test for lung cancer. With a .81 sensitivity, it's actually a pretty darned good test! The specificity is the probability of getting the negative lung cancer folks right: and that's also very high (1256/1400=.90).

      So it's kind of ironic, but smoking status is a very good test for lung cancer among American men, with sensitivity .81 and specificity .90.

      Big picture: don't smoke.

5. Problem 5 was pretty easy, if you were ready:
   1. part a was just producing a picture of the model of Covid infection as a directed, labeled graph;
   2. part b was just applying the formula from page 1. The biggest problem I saw was people using "32" rather than ".32" in the formula. The numbers $r$ and $r_v$ are always between 0 and 1.

6. Problem 6: Pascal's triangle.
   1. It's easy enough to generate the first six rows of Pascal's. I was sorry more couldn't seem to do it right....
   2. This student bravely merely indicated (without words) the presence of both of these types of numbers on the figure of the triangle. I liked that! Be brave.
   3. While this student went through and drew each of the eight possible Facebooks, I really intended that one merely point to the power of two from the third line (1 3 3 1) -- eight.

      First of all, Pascal gives the total number of Facebooks. We can think of each of the three potential friendships raising their hands or not -- two choices -- as to whether they're part of the Facebook or not. There are 2x2x2=8 ways of doing that (each friendship makes their choice independently of the others).

      Then Pascal tells us exactly the makeup of the Facebooks:

      • 1 way to have no friendships
      • 3 ways to have one friendship
      • 3 ways to have two friendships (leave one out)
      • 1 way to have all the friendships.
      How many ways? That's Pascal's job, answering that question....

   4. Finally, I asked how many ways to invite three of five friends -- and, as we saw above, the answer's in Pascal's triangle: 10 (if we follow my scheme from above, we might say it's the fourth entry in the fifth row -- starting 1 5 ...).

7. It seems that folks aren't doing the readings. I think that they will help you understand the material, and do the problems. But they'll only help if you actually read them! And I will continue to test over them (e.g. problem 2 on the exam).

But our question of the day (which we didn't get to) was this:

• We did some puzzles together, testing our understanding of Euler paths and circuits as well as encountering a new concept: Euler's formula.

  You might print out or open this and give the following a try: but I did make a video discussing these problems below (linked to below).

  1. Page one and two: can you find a way to pass through each door in these houses without passing through any door twice?

  2. Euler's formula is the focus of the third and fourth pages, and it refers to planar graphs. It turns out that there's a nice relationship between the number of edges, vertices, and faces of a planar graph. (We need to define faces (or regions) -- see the definition on the page.)
     • The graphs depicted do not appear to be planar -- or are they? They need to be explicitly planar for the formula to apply, and these can be drawn without false intersections -- the definition of planar.
     • Does the tetrahedron satisfy the formula?
     • How about Euler's Konigsberg graph? It's clearly planar.

• Here's the
  • Video
  • Whiteboard work