Robert wrote:

> I'm still not sure I know what John meant, I'll have to try again later.

In Puzzle 68 I was asking this. Suppose there's just one chemical reaction you can do:



When can you turn

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

into

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

using this reaction? Here \\(a,b,c\\) and \\(a',b',c'\\) are natural numbers.

For example, you _can_ turn

\[ 4 \text{H} + 2 \text{O} + \text{H}_2\text{O} \]

into

\[ 3 \text{H}_2\text{O} \]

using this reaction. (Of course you have to use it twice, but that's okay.) You _cannot_ turn

\[ 3 \text{H} + 2 \text{O} + \text{H}_2\text{O} \]

into

\[ \text{O} + 2 \text{H}_2\text{O} \]

using this reaction. So, what's the general rule? If you want, you can abbreviate

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

as a list of 3 natural numbers \\( (a,b,c) \\), and write

\[ (a',b',c') \le (a,b,c) \]

to mean "you 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} \\)."

Michael Hong tried answering this question in Comment 4:

> **Puzzle 68.** Define \\((a, b, c) \leq (a', b', c')\\) if and only if \\(a \gt a', b \gt b' \, and \, c \lt c'\\).

This is the right _kind_ of answer, but it's not the right answer. If this were true, we would have \\( (1,1,0) \le (0,0,1) \\), meaning we could turn \\(\text{H} + \text{O}\\) into \\(\text{H}_2\text{O}\\). That would be cool, but actually we can't do that using this reaction:



Marius gives an interesting alternative answer in Comment 8. Do you think this answer is correct?