docs: add nextGuess optimisation hints to readme

README.md

@@ -0,0 +1,50 @@
+# Spec
+## Tips to improve `nextGuess`
+A better approach would be to only make guesses that are consistent with the
+answers you have received for previous guesses. You can do this by computing
+the list of possible targets, and removing elements that are inconsistent with
+any answers you have received to previous guesses. A possible target is
+inconsistent with an answer you have received for a previous guess if the
+answer you would receive for that guess and that (possible) target is different
+from the answer you actually received for that guess.
+
+You can use your GameState type to store your previous guesses and the
+corresponding answers. Or, more efficient and just as easy, store the list of
+remaining possible targets in your GameState, and pare it down each time you
+receive feedback for a guess.
+
+The best results can be had by carefully choosing each guess so that it is most
+likely to leave a small remaining list of possible targets. You can do this by
+computing for each remaining possible target the maximum number of possible
+targets it will leave if you guess it. This you can do by computing, for each
+remaining possible target, the answer you will receive if it is the actual
+target, and then compute how many of the remaining possible targets would yield
+the same output, and take the maximum of all of these. Alternatively, you can
+take a more probabilistic approach, and compute the average number of possible
+targets that will remain after each guess, giving the expected number of
+remaining possible targets for each guess, and choose the guess with the
+smallest expected number of remaining possible targets.
+
+Unfortunately, this is much more expensive to compute, and you will need to be
+careful to make it efficient enough to use. One thing you can do to speed it up
+is to laboriously (somehow) find the best first guess and hard code that into
+your program. After the first guess, there are much fewer possible targets
+remaining, and your implementation may be fast enough then.
+
+You can also remove symmetry in the problem space. The key insight needed for
+this is that given any guess and an answer returned for it, the set of
+remaining possibilities after receiving that answer for that guess will be the
+same regardless of which target yielded that answer. This suggests collecting
+all the distinct answers for a given guess and for each answer, counting the
+number of targets that would give that answer. Since there are much fewer
+answers than possible targets, this can save significant work.
+
+For example, suppose there are ten remaining candidate targets, and one guess
+gives the answer (3,0,0), three others give (1,0,2), and the remaining six give
+the answer (2,0,1). In this case, if you make that guess, there is a 1 in 10
+chance of that being the right answer (so you are left with that as the only
+remaining candidate), 3 in 10 of being left with three candidates, and a 6 in
+10 chance of being left with six candidates. This means on average you would
+expect this answer to leave you with 1/10 * 1 + 3/10 * 3 + 6/10 * 6 = 4.6
+remaining candidates. You just need to select a guess that gives the minimum
+expected number of remaining candidates.