I like Marius Furter's answers to Puzzles 68 and 70 in [comment #13](https://forum.azimuthproject.org/discussion/comment/18009/#Comment_18009). Here's how I'd do these puzzles.

In Puzzle 68 we're asking when we can turn

\[ a \text{H} + b \text{O} + c \text{H}_2\text{O} \]

into

\[ a' \text{H} + b' \text{O} + c' \text{H}_2\text{O} \]

by repeatedly performing the reaction \\( 2\text{H} + \text{O} \to \text{H}_2\text{O}\\).

Each time we do this reaction we decrease \\(a\\) by 2, decrease \\(b\\) by 1 and increase \\(c\\) by 1. So, we can turn

\[ a \text{H} + b \text{O} + c \text{H}_2\text{O} \]

into

\[ a' \text{H} + b' \text{O} + c' \text{H}_2\text{O} \]

precisely when

\[ (a',b',c') = (a - 2n, b - n, c + n) \]

for some \\(n \in \mathbb{N}\\).

If we also allow the reverse reaction, as in Puzzle 70, we can do it whenever

\[ (a',b',c') = (a - 2n, b - n, c + n) \]

for some \\(n \in \mathbb{Z}\\).

These answers can be converted into answers of the sort Marius gave.

I forget if someone answered my other question: which of these two relations define partial orders, as opposed to mere preorders?