Hey Michael,

> This companion and conjoint business is confusing but from what I've been able to put together, its basically the two ways of creating a feasibility relation from a monotone function? It seems to be related to whether the feasibility relation is based on "at most" or "at least" or both or none? This is still very distant to me and I think I need more examples to get some intuition. Has anyone worked the answers to this puzzle?

Maybe we can find a few companions and conjoints together? I already tried to find a few.

John mentioned that one of the problems has *neither* a conjoint nor a companion, so finding that one and proving it doesn't have any is going to be hard.

> >**Puzzle 206.** Suppose you are trying to feed hungry children with the loaves of bread you bought in the previous puzzle, and you can feed at most three children with each loaf of bread. Describe this using a feasibility relation \\(\Phi : \mathbb{N} \nrightarrow \lbrace 0,1,2\rbrace \\). Here \\(\mathbb{N}\\) is the set of natural numbers \\( \lbrace 0,1,2,3,\dots \rbrace \\) with its usual ordering.
>
> \[
\Psi(x, y) =
\begin{cases}
\texttt{true} & \mbox{if } (x \leq 0 \mbox{ and } y = 0) \mbox{ or } (x \leq 3 \mbox{ and } y = 1) \mbox{ or } (x \leq 6 \mbox{ and } y = 2) \\\\
\texttt{false} & \mbox{otherwise.}
\end{cases}
\]

I can see a conjoint that gives rise to your answer.

The companion, if it exists at all, is less intuitive. If it doesn't exist do you see how we'd go about proving it doesn't?