Solution to Clueless

Starting off

As hinted by the mention of "clues" and the fireworks-related words, as well as the choice of colors used, this is a puzzle about Hanabi. The encoding of the cards by letters is sequential, as shown in the table below. (This can be confirmed by checking that, say, U = red 5 appears only once in any given scenario, whereas one scenario has A = red 1 three times.)

This leaves a set of logic puzzles to be solved (with the first scenario being an easy one for the solver to learn the ropes).

Endgame rules

Solvers will need to be familiar with the rules of Hanabi, especially the conditions for the end of the game (each player once after the last card is drawn), and the fact that 5's restore a clue.

This puzzle is specifically about Hanabi endgame situations in which the players have painted themselves into a bit of a corner. The rules for each of the given scenarios are:

• The players know what cards they are holding (hinted by the flavor text).
• The players start with 0 clues (hinted by the title and flavor text).
• The first player starts (written explicitly).
• The number of cards in the deck is n, where the denominator of the "chances" number is n! (n factorial).

These conditions are enough to give a unique path to a perfect score of 25.

The numerator of the chances is not needed in order to solve the puzzle, but serves as a check-sum; it is the number of deck orderings for which the players do win (assuming that unneeded cards in the deck are trash). For example, in the first situation (where the deck has two cards), the players win only if red 4 is the first card in the deck, while the second card in the deck can be any trash card.

Solutions to the logic puzzles

For solving the puzzles, the following two heuristics are quite helpful.

• The pace of a game-state is defined by current score + cards in deck + number of plays - max score. It is independent of the number of clues, and decreases by 1 for every discard before the last round. It is impossible to win if the pace is negative. Moreover, at pace 0, every player must play in the last round.

• Playing a 5 restores a clue, and it is an important way to avoid discarding (since players with no usable cards can spend clues to stall).

• We assume players never misplay because it is strictly better to discard a card than to misplay it.

Most of the scenarios start at a very low pace. For example, the scenario labeled FIVES WIN THIS starts at pace 0, meaning that no player may discard for the entire scenario.

We now present solutions to the logic puzzles. They are shown in video format on YouTube, also embedded here. To make the names easier to refer to, we call the first, second, third, fourth players as Alice, Bobby, Carol, David respectively, following a cryptography convention.

We also provide the solutions in various text-based formats below: in the HTML itself, as a JSON file for use on hanab.live, and as a plain text file with a complete proof that the solution path is actually unique. (In the text files below ".." means to spend a clue while "kt" is an abbreviation for "known trash".)

Extraction

By now, the solvers have likely noticed that the number of turns exactly matches the number of letters in the "plan", which is the last unused piece of information. Solvers can take the cards played on each turn (treating clue and discard turns as empty slots) and overlap them with the plan for each scenario. This yields the following letters.

This spells the message FINAL PROPOSAL. The answer is nine-letters, and the usual nine-letter answer for this clue is ULTIMATUM.