Sequencer

Sequencer is a numeric set collection game. Plotinus won the 2019 Mashtermind award for best General Mish Mash game for designing & hosting this game.

Summary

Players get a hand of cards with numbers on them. On your turn you can either steal-and-continue somebody else's sequence or play a new sequence of your own. A sequence is a set of numbers that all have something in common with each other, like all being even or all being tetrahedral numbers. It is okay if you can't write a closed form for your sequence as long as you can tell us what the rule is. When a sequence has 7 or more cards in it, it is converted into points and can't be stolen anymore.

Game Flow

• You start with 7 cards
• On your turn, play some cards in only one of these ways:
  • Start one new sequence with at least 3 cards: "I'm playing 3, 9, 18 for divisible by 3." You are welcome to include a closed form if you can but you must translate this into English for the rest of the players.
  • Continue one sequence you previously started: "I'm adding a 3 to my primes"
  • Steal one sequence from somebody else: "I'm adding 4, and 16 to Plotinus' powers of 2"
• When the sequence has at least 7 cards in it, it is finished. You get n points, for finishing it, where n is the number of cards in the sequence, and the sequence is removed from the game.
• After your turn I replenish your hand up to 7 cards so you can be thinking about your next move while the other players go.
• If you cannot think of anything at all to do on your turn, and it's your very first turn of your very first game, then you may post your hand for others to help you. After your first turn is over, you may not reveal your hand.
• If all players pass consecutively the game is over and whoever has the most points wins.
• If the deck runs out of cards and the players run out of cards then the game is over.

Invalid Sequences & Exceptions

• Most sequences are valid if you can tell us what the rule is. The rule does not have to be number theoretic; cosmetic rules are fine. They can be as complicated or as simple as you like, though for complicated ones it is worth checking the deck to make sure there are at least 7 such numbers. You don't need to collect the numbers of the sequence in order. There are just a few exceptions:
• A sequence is a set of numbers that all have something in common with each other. Numbers may not be excluded from sequences they would naturally belong to, for example "primes except 7".
• The rules for a sequence must apply to at least 10 unique cards in the deck.
• Variants of "all integers", "numbers less than 1001", "random numbers" -- these are all equivalent to "player 1 starts the game with 7 points for no reason"
• Repeats: if someone has already put 15 in the divisible by 5's sequence, you can't put another instance of 15 in it. This goes even for sequences that naturally contain repeats. fibonacci only gets one 1, or a naturally repeating sequence like [1, 2, 1, 2, 1, 2, 1, 2] would be disallowed.
• The variables used in closed forms must represent integers (ℤ). For example you may not say that 5 is a member of the sequence n2 because the square root of 5 is not an integer.
• Sequences that are the same or equivalent to a sequence that we already have, whether it is in active play or in the Finished category. For example if we already have the "divisible by two" sequence, we cannot also have the "even numbers" sequence. If we have "divisible by 3" then we cannot also have the "n = 0 mod 3" sequence. Once a sequence has been used it cannot be reused until we start a new game.

Bingos (putting down 7 cards at once)

• Bingos are worth 8 points.
• Bingos must be neither too common nor overengineered. An attempt to clarify what this means follows:
  • Your sequence should match less than or equal to half of the unique cards (55.5) in either the current deck or in the range [1,100], which may be less tedious to verify.
  • Your sequence shouldn't have more than 2 working parts, for example: "the number is prime mod 7" has two parts and is allowed. "The digit sum of x2 is less than 5 and x3 ends in a prime number has four parts and is not allowed."
  • If you are a following a meta rule to generate rules for sequences, then the meta rule shouldn't be able to generate sequences that are bingos for more than 1/64th of hands.
    • An example meta rule that would generate a bingo all of the time is roots of the 7 degree polynomial (x - my first card)(x - my second card)(...)(x - my seventh card) = 0.
    • Another meta rule for generating sequences "look up my hand in a large database of sequences" is also not allowed for bingos, so you should be able to show enough of your work that we can see you worked forwards not backwards.