Let's say I'm getting ready for a remodeling project, and I'm getting ready to order 2x lumber for framing. Specifically, I'll be ordering No. 2 or better 2x lumber. What percentage of my order should I expect to be scrap (i.e. twisted, warped, crap)?

I've dug through the junk at places like Home Depot and Lowe's, and found it difficult to find a relatively straight board. Do big box stores just keep re-stacking the picked over garbage, so the percentage of junk increases over time? Should I expect this level of waste when ordering from a lumber yard?


Question: If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.
Answer: Any multiplicative homomorphism must send $1$ to an idempotent.

The only idempotents of $\mathbb{Q}$ are $0$ and $1$ (the roots of $x^2-x=x(x-1)$), so $\varphi(1)=0$ or $\varphi(1)=1$.

If $\varphi(1)=0$, then $\varphi(a) = \varphi(1a) = \varphi(1)\varphi(a)=0$, so $\varphi(x)=0$ for all $x$.

If $\varphi(1)=1$ and the map is additive, then prove inductively that $\varphi(n)=n$ for all positive integers, hence for all integers; deduce that $\varphi(q) = q$ for all $q$.