1. Exercise 1.4.2.

2. True or False? Every integer $n>2$ occurs in some Pythagorean Triple. (Justify your answer.)

3. Explain why there are only finitely many distinct Pythagorean Triples $(a,b,c)$ with $a=100$.

Solutions.tex.
Solutions.pdf.

1. Give a geometric proof that $\sqrt{3}$ is irrational. (Hint: It might be easier to show that $1+\sqrt{3}$ is irrational, then deduce that $\sqrt{3}$ is also irrational.)

2. Use the Euclidean algorithm to find an integral solution to $270x+168y = 6$.

3. What is the height of a regular tetrahedron of side length 1?

1. Exercise 2.2.2.

2. Exercise 2.2.3.

3. Let $P$ be a polyhedron. Suppose $F_1, \ldots, F_k$ are the faces of $P$ that meet at vertex $V$, and that $A_1, \ldots, A_k$ are the angles of these faces at $V$. Define the defect at vertex $V$ to be ($360$-(sum of the angles $A_i$)). (For example, in a cube there are three squares meeting at any vertex, so the defect at any vertex is ($360-$($90+90+90$)) = $90$ degrees.) The total defect of $P$ is the sum of the defects at all of the vertices of $P$. Exercise: find the total defect of each of the Platonic solids.