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A basic qualm I have with most logical deduction games like this is the player elimination that occurs when a player is wrong. They see the three and are eliminated and usually have to sit out the last ten or so minutes while others continue with clues.
I experimented with some random variants. One was to have the guesser call their 3 suspected numbers out loud, and if no one corrected them by saying they owned one of the 3 cards, then that player won. Otherwise play continued. But the issue was that this typically revealed 2-3 letters to other players which is A LOT and usually meant the player(s) who heard could then declare and win the game.
My fave of what I experimented is below.
Non-player Elimination Variant:
- When a player/guesser thinks they have the 3 murder letters, they declare so and submit the three letters on a sheet face-down in the center.
- The player who's turn it is finishes his turn. The turn doesn't pass to the player he asks. Instead it goes to the left of the guesser who asks one final question. Then it goes left again until the player to the right of the submitter has asked a final question (special investigations can be used at this time).
- At any time, players can also declare they have the final 3 and submit their paper with guesses.
- When all players have taken their final turn, players should flip over the guessers' papers and see what everyone guessed. They should also flip over the 3 cards.
The player who got all 3 correct the soonest is the winner (ties broken by time), otherwise follow other scoring rules from the rulebook.
This has some small issues if someone guesses too early, but usually, I've found that this works because when people start guessing, players are close enough that they can deduce almost all of it, especially with a special extra round (where they can all use their special investigations too).
I've also found this to be fairly exciting/tense up to the reveal to see who is right/wrong, and it ends the game on a climactic note.
Anyway, hope this helps. This game is underrated and unpopular, but I wish it wasn't because its enjoyable, logical, and elegant.