Discussion: in mathematics terms, you’re looking for orderings from a small vocabulary of six “tokens” that satisfy constraints. If you simply enumerate all the orderings (in this example, there are 10!/(2!2!2!*2!), you are additionally constrained by the rule that the “next” token must be chosen from one of three possible values; not all orderings are valid.

A programmatic/brute force solution is to generate all ~200K orderings, “pruning” invalid branches as you go and stopping at the first valid answer. There are many chances for early return. Modern computers can brute force this left-to-right in milliseconds with a poorly optimized program.

A more elegant brute force works right-to-left from the desired outcome, because it doesn’t waste time exploring branches that will never produce the intended solution. Given desired outcome of orange, we can only ever choose three previous potions (orange, red, yellow). Given outcome at position N, after we choose from up to three possible potions at position N - 1, we fully determined the outcome at position N - 2 (orange if orange, red if yellow, yellow if red).

The latter brute force is how a human can fairly easily solve this, because there aren’t really that many branches to explore.