# Proofs: "That's some catch, that Catch-22"

Friday Proof Seminars by Yongquan Zhang. Here are the times. Join one of them for 30 minutes at least:
• 10:30 proof office hours,
• 12 pm in SC 221 (note 12 rather than 12:30)
• and 3 pm in SC 221.

## Week 7:

• Q: There is the golden, the silver and the bronze mean. What is the name of the next one?
A. No idea. We did not find any reference. Copper? It is the number 2+sqrt(5). Anyway, you can get the metallic means as follows
```Metallic[n_]:= x /. Last[Solve[ x^2-n x - 1==0,x]];
Metallic
```

## Week 6:

• Q: Where can we find more pictures of Cladny figures.
A: Here are some videos:

## Week 5:

• Q: in the 4 queen problem. Are there 4 queens on a 8x8 board?
Answer. No (this is now clarified). We look in Problem D for the solution of the 4 queen problem on a 4x4 chessboard. By the way, chess on a 4x4 board is also called Silverman Chess. It is a completely solvable game See this twitter entry.

## Week 4:

• Q: How to write down the proof of the determinant property for the partitioned case.
Answer. This is a bit difficult to write down. One thing which helps is to write down the case of a 4 x 4 matrix P partitioned into 2 x 2 matrices A,B,C,0
```   | a b e f |
P= | c d g h |   | A  C |
| 0 0 u v | = | 0  B |
| 0 0 w q |
```
Seeing that (ad-bc) ( uq-vw) = aduq - advw - bcuq + bc vw triggers to write down (sum of pattern values of A) (sum of pattern values of B) = sum of pattern values of of P. There is an other possibility: row reduce and keep track of the factors and swaps.

## Week 3:

• Q: Why is the norm of a quaternion a+ib+jc+kd equal to a^2+b^2+c^2+d^2.
Answer: This is a bit strange. Especially in number theory, one looks at the norm N(z) which is the sum of the squares and not the square root of it. One calls it the Arithmetic Norm.

## Week 2:

• Q: What is maximum of the matrix entries which a n x n matrix can have, which has entries 0-1
Answer: We don't know. We will explore this statistically in the statistics part and explore this therefore a bit experimentally. Even the most basic questions about random matrices can be impossibly hard. There is now a little exhibit with code to experiment.
• Special cases:
Something about the question about the matrix entries of a projection or reflection at a plane a x + b y + c z = 0: many of the proofs of the class that this leads to rational matrices assumed that a is not zero. But the result is true in general also if a is non-zero. As long as not all a,b,c are zero, one can find an integer basis and so an integer matrix S which produces the coordinate change.

## Week 1:

• Q: For problems 1 and 2 of Unit 3, the set of all sets is not a set (Russell's paradox). Does this mean we have no monoid?
Answer We assume to work with a set theory which is tamed. The Zermelo-Frenkel axiom system provides a shelter from the Russell paradox. We don't worry about this there.
• Q: For problem 3 in Unit 3, I got a hexagonal shape. Can this be true?
Answer: Yes, you should get a hexagonal shape divided into triangles. You actually draw here the first Barycentric refinement of the triangle. You can iterate the refinement: look here the graph. Looks a bit like code used in Skyfall 2012, The "rubic cube fighting back".
• Q: Is the space X={0} only contains the zero vector a linear space?
A: Yes, it is a bit of a singular space as it has only one element, the zero element. But all the properties of a linear space are satisfied. This space is the zero-dimensional vector space. The addition also satisfies the axiom system of a monoid. The only thing we can compute in this space however is 0+0=0. Fortunately, the Euclidean space we live in is not zero dimensional. There is no interesting life in it. The zero dimensional space is the favorate space of a Nihlilist.