Matthew wrote:

> John mentioned that one of the problems has *neither* a conjoint nor a companion [....]

I think I might have been wrong about that.

My intuition for companions and conjoints wasn't so good at first: witness my surprise in [comment #9](https://forum.azimuthproject.org/discussion/comment/20337/#Comment_20337)! But I think that surprise wound up improving my intuition.

So, let me try to explain my intuition for this stuff. It's easiest with an example:

> **Puzzle 207.** Suppose you buy loaves of bread and then use them to feed hungry children. Compose the feasibility relation \$$\Psi : \{0,1,2\} \nrightarrow [0,\infty) \$$ from Puzzle 205 and the feasibility relation \$$\Phi : \mathbb{N} \nrightarrow \lbrace 0,1,2\rbrace \$$ from Puzzle 206 to get a feasibility relation \$$\Psi \Phi : \mathbb{N} \nrightarrow [0,\infty) \$$ describing how many children you can feed for a certain amount of money (given the fact that you plan to buy at most two loaves).

Suppose we want \$$\Psi\$$ to be the companion of some monotone function \$$F \colon \{0,1,2\} \nrightarrow [0,\infty) \$$.

\$$F\$$ is a function that takes a number of loaves of bread and gives an amount of money. So, the obvious first guess is a function \$$F\$$ such that

\$$x\$$ loaves of bread costs \$$F(x)\$$ dollars.

\$$\hat{F}\$$ will then take this function and 'pad it out' to give a feasibility relation. If you have _more than_ \$$F(x)\$$ dollars, it's still feasible to buy \$$x\$$ loaves of bread. And if you have \$$F(x)\$$ dollars, it's feasible to buy _less than_ \$$x\$$ loaves of bread. So the feasibility relation \$$\hat{F}\$$ is true for more pairs \$$(x,y) \$$ than those for which \$$F(x) = y\$$.

In fact, it's true exactly for pairs with \$$F(x) \le y\$$!

This isn't hard to see. The idea of a companion is that

$\hat{F}(x,y) = \mathbf{Bool}(F(x), y )$

but if you unravel all the jargon and notation, this just means

\$$\hat{F}(x,y) = \text{true} \$$ iff \$$F(x) \le y \$$

So, to find a function \$$F\$$ whose companion is \$$\Psi\$$, we need a function such that

You can buy \$$x\$$ loaves of bread for \$$y\$$ dollars iff \$$F(x) \le y\$$.

But to repeat myself: if we're seeking such a function, the obvious first guess is a function such that

\$$x\$$ loaves of bread costs \$$F(x)\$$ dollars.

For Puzzle 207, I think such a function exists.