Here is the puzzle. A tuple of five numbers $\langle a,b, c, d, Z\rangle$ is called consistent if:
- The numbers $a, b,c,d, Z$ are all between 0 and 1, inclusive.
- $a\times b = Z = c \times d$.
We can compare tuples, too: one tuple is smaller than another if each of its entries is less than or equal to the corresponding entry in the other tuple: $a_1 \leq a_2$, $b_1 \leq b_2$, etc. Naturally in that case, the other tuple is called bigger than the first.
The problem I'm trying to solve is this: Given two consistent tuples $T_1$ and $T_2$, find the smallest consistent tuple $T$ that is bigger than both $T_1$ and $T_2$, or prove that there isn't a unique one. I'm looking for a generic rule that I can apply for any pair of consistent tuples.
So far, I've noted that there's always at least one consistent tuple bigger than both $T_1$ and $T_2$, namely $\langle 1,1,1,1,1\rangle$, so the only problem is whether there is a unique smallest consistent tuple.
I also note that an assignment like $a \equiv \max(a_1,a_2)$ and $b \equiv \max(b_1, b_2)$ and $Z = \max(a_1,a_2)\times \max(b_1, b_2)$ works for $a, b, Z$ but might not be extensible to $c$ and $d$, since we have no guarantee that $\max(a_1,a_2)\times \max(b_1,b_2) = \max(c_1,c_2) \times \max(d_1,d_2)$ and in fact that will hardly ever be the case. The question is what to try next.