Is it possible to represent triples of numbers from some small set $X$ uniquely using pairs of numbers from $X$? Given that I have a small range of numbers in ascending order with no duplicates, e.g.,
$$23, 24, 25, 26, 27, 28, 29, 30, 31, 32,$$
and three numbers are chosen from this range, let's say $30, 26,$ and $23,$
is it at all possible to represent the fact that we've selected $30, 26,$ and $23$, just using two numbers within the same range, in such a way that we can then reverse it to find our original three numbers?
The function will always know the numbers within the range and the order they appear in. In such a way we could even assign index values to each number for the range.
i.e.
$0 \implies 23$
$1 \implies 24$
$2 \implies 25$
etc...
 A: In general, no, but we can as long as the "small range" includes no more than $5$ numbers.
Since their quantities don't matter, there's no harm in relabeling the numbers in the set $[n] := \{1, \ldots, n\}$, where $n$ is the number of elements in the set. Now, we can reframe the question as asking for an surjective map
$$\{ \textrm{$2$-element subsets of $[n]$} \} \to \{ \textrm{$3$-element subsets of $[n]$} \} .$$ The domain has ${n \choose 2} = \frac{1}{2} n (n - 1)$ elements, whereas the codomain has ${n \choose 3} = \frac{1}{6} n (n - 1) (n - 2)$ elements, and so there is a surjective map, and hence such a "representation", iff ${n \choose 2} \leq {n \choose 3}$, and a little easy algebra shows that this is true iff $n \leq 5$. In particular, in the special case $n = 5$, we can (for example) simply assign to each $3$-element subset of $[n]$ its ($2$-element) complement.

This solution assumes, by the way, that we don't care about the ordering of the elements in the triple. If it does matter, an argument similar to the above shows that this is possible iff $n \leq 3$ (provided that we are then allowed to encode information in the order of the pair we choose), but for $n = 3$ this is nothing more than a specification of order (as we have no choice in the elements themselves) and for $n < 3$ this is vacuous.
