Five Sages

Here is a new puzzle by Nikolai Chernyatiev. Puzzle. Five sages, who all know one another, are blindfolded, seated in a row in a dimly lit hall, and then have their blindfolds removed. Each sage can see both of their immediate neighbors, but no farther; the sages at the ends know that they are at the ends. After that, each sage writes down one of the numbers 1, 2, or 3. The complete information — who wrote which number, in seating order — is then announced to everyone.Before being seated, the sages may agree on a rule for choosing 1, 2, or 3 based on what they see. After the five numbers are announced, each sage must reconstruct the full left-to-right order of all five sages. I love puzzles related to information theory, and this is a lovely example. Let’s do a quick sanity check. There are 5! = 120 possible orders of the sages. The announced numbers form a ternary string of length 5, giving 35 = 243 possible announcements. That is more than enough in principle; so far, so good. AI…