An important thing happened today: you made the first class conjectures, which we will add to the list of tasks for class to do.
Ms Pace discussed the second half of Conjecture 1.1. She believes it is false in general. We had a nice discussion about her ideas for constructing a counterexample, and eventually wrote out some conjectures of our own to make clear our ideas.
Conjecture A (Pace): Let ABCD be a rhombus. Then angle BAC and BDC taken together make a right angle.
Conjecture B (Barker-Lewis): Let ABCD be a rhombus. If angle BAC is congruent to angle BDC, then ABCD is a square.
Conjecture C (class): It is possible to construct a rhombus which is not a square.
Mr Reihman then took up 1.6. He has a very promising outline, but the argument has two unsupported steps (usually called “gaps”). The first concerned the totality of the interior angles of a triangle, and the second was about an arrangement of supplementary angles making a pair of parallel lines. I hope that we will see this again on Wednesday.
Ms Lewis gave an argument for Conjecture 1.7, which was accepted by the class.
Then I rambled on a bit about the Riemann Hypothesis, conditional results, tuberculosis, Hilbert, the Clay millenium prize problems, and other things.
Oh! I would like to meet with each of you briefly in the next two weeks. See the bit of the syllabus which talks about “assessment interviews.” That should happen soon.
Have a good and productive weekend. See you Wednesday.