Thoughts on Test 2

First of all, what's your hurry to leave? You have an hour and 15 minutes: some of you who left early could have used some of that time to revise, add additional thoughts, improve the clarity of your solutions, ask me a question or two, draw a picture to illustrate an idea, etc.

I wanted you to say something about duality in problem 1.
Even mention of diagonalization would have been nice: this student went much further! Good job.
The integers and the natural numbers are the same size: we know this because we can find a one-to-one correspondence. We know that the naturals are certainly not BIGGER than the integers because the naturals are a subset of the integers.
I loved this answer: even though the author of it did not pretend to imply that such a correspondence exists, the author indicated that it is the ideal situation -- that businesses should shoot for a one-to-one correspondence. Great job!
This is a tiling that I found on the net. It shows rotational and translational symmetries, but not reflection (or flip) symmetry.
A couple of people had the idea of using randomness to assure the lack of symmetry -- great idea! Alternatively, one could have created a small pattern which shows no symmetry in the center of one block, and leave the rest of the plane white; or one could have created something like a spiral -- say, a logarithmic spiral!;)
For problem 7, I wanted to see something like the pattern indicated below.
It was the discovery of irrational numbers, rather than rational or whole, that caused such consternation.
Each bus is assigned its own prime number (no sharing -- but that's easy, since there are an infinite number of primes). Suppose we have a bus with prime p associated. Now the passenger in seat n will go to room pⁿ, and have a peaceful night's sleep.
If you never mentioned the infinite number of busses (i.e., if your "solution" didn't change if you had 20 busses or an infinite number of busses), than it was obviously wrong!
I forgot a very important word in this problem: I meant to say "regular" pentagons! That changes the problem greatly, and as several person showed, it IS possible to tile the plane in (irregular) pentagons!
Since the regular pentagon tiles the dodecahedron, you should be able to flatten out the dodecahedron to start a tiling of the plane. But when you flatten the dodecahedron, you find that a gap opens up between a three-some that no regular pentagon can fill.
Details (that I didn't expect you to know): The interior angles of the regular pentagon are 108 degrees, and if you add up three of them you get 324 degrees -- but you need 360 degrees to complete the pattern, so you're 36 degrees short: that's not enough room to wedge a regular pentagon into!

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