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| Trick Play |
 Title text: I've been trying to find out whether the Laws of the Game take the axiom of choice or not, but FIFA officials keep blocking my number. |
Explanation
The Banach-Tarski paradox is a theorem in set-theoretic geometry stating that a solid three-dimensional object — in this case, a sports ball — can be taken apart into several pieces (usually five) and that those pieces can then be reassembled into two identical copies of the original object. The paradox has been proven true for any object at the theoretical level, but it is difficult to replicate in real life because it requires the objects to be made of infinite collections of points, rather than a finite number of atoms. The paradox has been mentioned before in 804: Pumpkin Carving.
Here we see a recap of football (soccer) game between what are presumed to be two college departments. The game starts as normal with one ball, then Stefan Banach and Alfred Tarski on the mathematics department team use their namesake paradox to make it into two balls, allowing for two of the team to attack the opponent's goal with both at once, and scoring with one of them. This would be difficult to do with a real ball.[citation needed]
The title text refers to the axiom of choice, an axiom of set theory that says that given an infinite number of non-empty sets, it's possible to choose one element from each of these sets. The axiom of choice, which is not accepted in all set-theoretic work, is necessary to prove the Banach-Tarski paradox. FIFA is the top-level institution governing football, but implementing the Laws of the Game, the rules laying out how the game is played, is done by IFAB instead. It's implied that Randall's phone number has been blocked due to calling FIFA too many times with math questions, as other xkcd characters have been blocked from calling official bodies. It is also possible he was blocked only because he contacted the wrong organization, as he should have contacted the IFAB instead.
Strictly speaking, the answer to Randall's problem would already be covered under existing rules. According to IFAB Laws of the Game, Law 5.3, under section Outside Interference, it is covered what happens if there is extra ball on the field: "stop play (and restart with a dropped ball) only if it interferes with play - unless the ball is going into the goal and the interference does not prevent a defending player playing the ball, the goal is awarded if the ball enters the goal (even if contact was made with the ball) unless the interference was by the attacking team". This second ball that was created clearly interferes with the play as around the ball is where the play is, and ball wasn't going towards the goal. Even if the ball was going towards the goal, it is clearly interference by the attacking team, so goal wouldn't count. As such, referees would stop play and restart with a dropped ball. Though technically it can be a free kick under the Law 12.1 instead. Even if it is not handball and they manage to split the ball without using hands, when a ball gets split in two it can still be considered as object being thrown at a ball or making contact with a ball with a held object, as you cannot really split ball in two without those two balls coming into contact.[citation needed]
Transcript
- [A diagram of a little more than half of a football field is shown. Multiple individuals are on the field. The two teams are represented as 'O's (of ten players, presumably excluding the goalie who would be in their respective off-panel goal-area) and 'X's (all eleven active players being visible in this scope of the field of play), and there's a goal at the top of the diagram. Arrows (passes between separate players) and dotted lines (dribbles by a single player, shown by an 'O' in the each of the positions for both the start and end of this movement stage) are drawn between some of them. In the middle of the diagram, an O player simultaneously passes to two of the other players, one diagonally up to the left, the other diagonally up to the right. Each of those receiving players kicks their ball towards the goal. The X goalie is near where the left ball enters the goal area, in the correct position to receive/save it, but the right ball is consequently unopposed as it goes into the goal.]
- [Caption below the panel:]
- The math department team's opponents hate it when Banach passes to Tarski.
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