# 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.)

Red Yellow Green Blue White
Rank 1 A B C D E
Rank 2 F G H I J
Rank 3 K L M N O
Rank 4 P Q R S T
Rank 5 U V W X Y

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".)

1 2 PUFF-UPS      (json) (proof) P Alice discards a stale card and draws [P] red 4 U Bobby stalls by spending a clue F Carol plays [F] red 2 and draws any card F David plays [K] red 3 on their last turn U Alice plays [P] red 4 on their last turn P Bobby plays [U] red 5 on their last turn S Carol plays [Y] white 5 on their last turn FIVES WIN THIS      (json) (proof) F Alice plays [W] green 5 and draws [U] red 5 I Bobby stalls by spending a clue V Carol plays [P] red 4 and draws [T] white 4 E David plays [O] white 3 and draws [N] blue 3 S Alice plays [U] red 5 and draws [S] blue 4 W Bobby stalls by spending a clue I Carol plays [I] blue 2 and draws any card N David plays [N] blue 3 and draws any card T Alice plays [S] blue 4 on their last turn H Bobby plays [X] blue 5 on their last turn I Carol plays [T] white 4 on their last turn S David plays [Y] white 5 on their last turn A BIG FUEL VESSEL      (json) (proof) A Alice plays [A] red 1 and draws [S] blue 4 B Bobby discards a stale card and draws any card I Carol discards a stale card and draws [G] yellow 2 G David stalls by spending a clue F Alice plays [S] blue 4 and draws [Q] yellow 4 U Bobby plays [X] blue 5 and draws any card E Carol plays [G] yellow 2 and draws [F] red 2 L David plays [L] yellow 3 and draws [K] red 3 V Alice plays [Q] yellow 4 and draws [P] red 4 E Bobby plays [V] yellow 5 and draws any card S Carol plays [F] red 2 on their last turn S David plays [K] red 3 on their last turn E Alice plays [P] red 4 on their last turn L Bobby plays [U] red 5 on their last turn FIRE EPIC BLOSSOM      (json) (proof) F Alice plays [W] green 5 and draws [K] red 3 I Bobby stalls by spending a clue R Carol discards a stale card and draws [E] white 1 E David stalls by spending a clue E Alice plays [K] red 3 and draws [U] red 5 P Bobby plays [P] red 4 and draws any card I Carol plays [E] white 1 and draws [Y] white 5 C David plays [J] white 2 and draws [S] blue 4 B Alice plays [U] red 5 and draws any card L Bobby stalls by spending a clue O Carol plays [O] white 3 and draws any card S David plays [S] blue 4 on their last turn S Alice plays [X] blue 5 on their last turn O Bobby plays [T] white 4 on their last turn M Carol plays [Y] white 5 on their last turn PYROPHORIC ROOFTOPS      (json) (proof) P Alice discards a stale card and draws [P] red 4 Y Bobby plays [F] red 2 and draws [U] red 5 R Carol stalls by spending a clue O David plays [K] red 3 and draws [C] green 1 P Alice plays [P] red 4 and draws [H] green 2 H Bobby plays [U] red 5 and draws [M] green 3 O Carol stalls by spending a clue R David plays [C] green 1 and draws any card I Alice plays [H] green 2 and draws [T] white 4 C Bobby plays [M] green 3 and draws [E] white 1 R Carol plays [R] green 4 and draws [J] white 2 O David plays [W] green 5 and draws any card O Alice stalls by spending a clue F Bobby plays [E] white 1 and draws [Y] white 5 T Carol plays [J] white 2 on their last turn O David plays [O] white 3 on their last turn P Alice plays [T] white 4 on their last turn S Bobby plays [Y] white 5 on their last turn

## 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.

PUFFUPS
..FKPUY

FIVESWINTHIS
W.POU.INSXTY

ABIGFUELVESSEL
A...SXGLQVFKPU

FIREEPICBLOSSOM
W...KPEJU.OSXTY

### Matching: POS

#### Pyrophoric rooftops

PYROPHORICROOFTOPS
.F.KPU.CHMRW.EJOTY

### Matching: PRO

_ _ _ _ _ _ _ _ _

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

## Author’s Notes

After playing maybe too many hours of Hanabi during quarantine, I knew I wanted to write a Hanabi puzzle for the Mystery Hunt as well, especially after seeing a Hanabi duck konundrum in the inaugural teammate hunt.

When I played Hanabi at summer math camps as a teenager, the only conventions we would have were play left, discard right, (forward) finesses, and the "fake finesses" (a worse form of the bluff). Plays and saves weren't even clearly differentiated. Thus our win rate was very low, and it was not until I played with a more well-developed convention set that I started appreciating the complexity (and fun) of Hanabi.

So, I wanted to develop a puzzle that would showcase at least some of that complexity. Since it's so difficult to use "conventions" as a puzzle mechanic, I settled on the idea of using the notion of pace for a logic-only puzzle (without regards for efficiency, a measure of how much information is communicated by clues). The idea that the number of discards is limited, and clues can be used not just to indicate cards but also to stall for time, gave me a mechanic that I thought would be new and interesting even to people who may have casually played Hanabi before. It was a lot of fun during test-solving to watch groups slowly realize (and then prove) that no cards could be discarded in the second scenario.

For extraction, choosing the plans was a fun exercise in Nutrimatic, since if I used a "random" string, it would be too easy to figure out which letters overlap just by looking at which cards were still left to play. Various strings that appeared in one version or other, often that barely didn't work:

• `TRUST MY LOGIC`
• `A GENIUS LEVEL IQ`
• `AN EXPLOSION FINESSE`
• `CRISIS SAVING SKILLS`

I also originally considered using tap code for the encoding of the letters, but this seemed like an un-thematic complication that was quickly scrapped.

I didn't really want to use the crossword cluephrase `FINAL PROPOSAL`, but the current answer has the unfortunate property that it is almost uniquely determined by the first two letters, so this was necessary to prevent solvers from short-circuiting the puzzle too prematurely.

The title of the puzzle was suggested by Danny Bulmash.