## Hybrid Set

SET is a card game that involves finding lines in four-dimensional space. The Official Website has a pretty good introduction to the game. It is notable for being particularly easy to hybridize with other games. Some popular SET hybrids include Munchkin SET, Dominion SET, and, of course, SET SET.

Here we sketch a proof that SET is hybridizable with all games, including itself. This proof is probably easily generalizable to prove that for all games, A and B, there exists a game that is a hybrid of them. It is worth noting that the resulting hybrid game is not necessarily fun or easy to play: A game of Scrabble Fluxx SET SET SET may be possible in theory, but playing it in practice may not really work all that well.

Definitions: A game, G, is a hybrid of games A and B iff components of both games are integral to the outcome of G and the gameplay of games A and B are interlaced. A game, A, is hybridizable with another game, B, iff there exists a game, G, such that G is a hybrid of A and B.

Simply playing a game of A and then a game of B and declaring the winner to be the person with the highest overall point total is insufficient to make it a hybrid game, even though the results of both games are integral to G. Similarly, merely simultaneously playing games A and B, but having the winner be dependent only on the results of B would not qualify as a hybrid game. Playing a game using pieces from another game only counts as hybridizing the two games if using those pieces significantly affects the overall gameplay.

Theorem: The card game SET is hybridizable with all games, including itself.

Proof of Theorem: Let B be the game that we are attempting to hybridize SET with. Let G be the game resulting from combining SET and B. We must fulfill two criteria for G to be a hybrid game. Firstly, the gameplay of B and SET must be interlaced, and secondly the winner of the hybrid game must not be entirely independent of the winner of either of the two games independently.

Proof of Interlaceability: We can interlace the two games by simply playing them simultaneously, each according to its regular rules.1

When playing two games simultaneously, we also have to consider the relative lengths of the games.

There are three cases here:

1. SET and B take approximately the same amount of time to play.
2. A game of SET is generally shorter than a game of B
3. A game of SET is generally longer than a game of B

In the first case, simmply play both games simultaneously to completion. They should end at about the same time.

In the second case, simply log the score at the end of each SET game, shuffle the deck, and start a new SET game until the game of B completes.

In the final case, as long as a reasonable number of SETs are findable by the end of game B, simply end the SET game early and follow the same scoring rules as in the previous two cases. If B is so fast as to make finding a reasonable number of SETs untenable, play multiple games of B until the SET game is completed. At the end of each game of B, the player who has accumalated the most SETs since that game of B and the winner of game B both score a point, the winner is the player with the most points at the end of the SET game.

Proof of outcome dependent on both games: If B is a game where there is some relevant point-based scoring system (or levels, or etc.), give an extra point per SET (or some number of SETs) found by each player. Otherwise, B must at least have some method for players to win or lose. Play SET and B simultaneously several times, and give a point at the end of each game of B to both the player(s) that has found the most SETs and the player(s) that won game B. The winner is first player to reach some arbitrarily decided on number of points (ie. 10).

1 In practice, there are two possible cases, and performance is slightly better on average if you do the following optimization when playing the games simultaneously. Either B is a turn-based game or B is a game in which any player may play at any point.

• If B is turn-based, then B may be hybridized with SET by having players play SET (you may only call SET when it is not your turn) when it is not their turn, with whoever took the last SET dealing the cards.
• If B is not a turn-based, then B may be hybridized simply by playing SET simultaneously with it.