Sequence Lock: compare the questions before choosing a guess

Make a small candidate notebook

A Sequence Lock guess is both an attempt to finish and a question about the hidden code. Those purposes sometimes point to different choices. Instead of asking which sequence feels most promising, ask which possible feedback results would separate the secrets you have not yet ruled out. A short candidate notebook makes that question concrete. The game keeps the feedback history, but it does not maintain this notebook for you.

Use a separate Standard-sized teaching example: three positions, symbols 1 through 4, with repeated symbols allowed. Suppose the first guess is 112 and its feedback is zero exact matches and two value-only matches. We are taking that feedback as the example’s starting information; it is not a statement about the random secret in your open game. There are 4 × 4 × 4 = 64 sequences before this observation.

Explain why just five candidates survive

Zero exact means the first position is not 1, the second is not 1, and the third is not 2. To obtain two value matches against 112, a compatible secret must contain both a 1 and a 2. It cannot obtain two matches solely from two copies of 1: the first two positions forbid 1, leaving only the third position for that symbol. Therefore the last position is 1, and at least one of the first two positions is 2.

The possibilities are 221, 231, 241, 321 and 421. Each produces precisely the stated feedback. This explanation also shows why 222 and 111 are excluded, even though each repeats a symbol from the original guess. Matching is limited by the copies available in both sequences, and a repeated digit does not override the exact-position exclusions. Keep all five candidates until another observation separates them.

Predict every answer to the same next question

Consider submitting 123 next. Score that proposed guess against each surviving candidate, without pretending that any candidate is already known to be the secret. The table gives the predicted feedback. Exact and value-only are separate counts; their sum is not an indication of which particular guessed positions are correct.

Read the groups without overstating their power

The question creates four feedback groups. A response of zero exact and three value-only leaves only 231. A response of zero exact and two value-only leaves only 241. One exact and two value-only identifies 321. However, one exact and one value-only still permits both 221 and 421. If that is the observed result, choosing between those two requires another observation; neither has become correct merely because it appears first in the notebook.

Notice that 123 is not itself among the five possible secrets. It cannot finish this teaching puzzle immediately, but it can still provide useful information. That does not make noncandidate guesses universally better. A surviving candidate can sometimes both win and divide the alternatives well. We have evaluated this one question, not proved that it minimizes the expected number of guesses over every possible strategy.

Carry the method into a larger code

For an actual puzzle, retain a proposed secret only when it reproduces every earlier feedback row. Then compare a few prospective questions by their resulting groups. A question that puts every candidate into one group teaches nothing new; a question with smaller groups usually gives a clearer next decision. Which group actually occurs is still outside your control.

Hard uses four positions and six symbols, so a complete handwritten list can become unwieldy. Work with justified restrictions on positions and symbol counts instead of copying a long list carelessly. Restart clears the displayed guesses while retaining the same secret; New puzzle and a difficulty change create different puzzles. Do not carry a notebook across that boundary. The useful habit is making each question answer a stated uncertainty, rather than merely trying a visually different sequence.