Tuesday 30 August class

Don't forget to read the previous  post if you haven't already!

What we did today:

• We discussed the student solutions to the "silly stories" #1 That's a Meanie Genie, #3 The Fountain of Knowledge, and #10 Dot of Fortune.
It turned out that The Fountain of Knowledge had two different ways it could be solved. There might be another... maybe.

• The Pigeonhole Principle:
If you have more pigeons than you have pigeonholes, and you put all the pigeons into pigeonholes, then at least one pigeonhole will have more than one pigeon in it.
That's all there is to it. Kind of obvious, but very useful. (Good examples of the power of the Pigeonhole Principle at that link!)

From the reading from Mind your decisions, fun applications of the Pigeonhole Principle, we discussed up to #8.

The Pigeonhole Principle is a theorem about the Natural Numbers. The Natural Numbers will be the subject of our investigations for the next few topics. Make sure you know what they are, because later on we will be talking about other sets of numbers!


Homework:

• Journal assignment: write up the solution or partial solution that your group arrived at for each of the problems we worked on in class, explaining your reasoning, and then describe the correct solution for that problem, explaining its reasoning and whatever you learned from that process. As an example of describing a reasoning process, I've created a sample journal entry (admittedly rather detaled!) to show the kind of thing I mean.
If you want to discuss the problems or any aspect of them while you are working on this, please feel free to post a question to the Piazza discussion board!
• Homework problems:
  Make sure that you record your reasoning (including anything that didn't work out!) and not just your answer, and explain how you know your answer works. In each case, we are asking for the smallest number necessary to make sure you have what you want. [Explanation of that last part in case it's not clear.]
Suppose you own 15 black socks and 10 blue socks. The socks are all mixed together in your sock drawer, not paired. In the middle of the night with the light burnt out, you want to take enough socks so that you know for sure you have a pair of matching socks.
1) How many socks must you take so that you know that you have a matching pair?
2) How many socks must you take so that you know that you have a pair of black socks? This is not the same as the first question. Think it through. If you get stuck, here is a Nudge.
3) How many socks must you take so that you know that you have a pair of blue socks?
 

• Reading for tomorrow: We will continue with the Pigeonhole Principle a bit (to see its proof), and then begin with another topic from number theory:
Fibonacci numbers: from Math is Fun. This is where you will maybe need to compute square roots. You may want to look through the questions they provide at the end of that webpage: pay special attention to #10.

Next up will be prime numbers and prime factorization of the natural numbers.