3282: Trick Play
| 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[edit]
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This is one of 46 incomplete explanations: This page was created by the math department which is salty after the last game. Don't remove this notice too soon. If you can fix this issue, edit the page! |
The Banach-Tarski paradox is a theorem in set-theoretic geometry stating that a solid three-dimensional object 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 three-or-more-dimensional mathematical object, but it is impossible to replicate in real life because it requires dividing the objects into 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 an association 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. This allows two players on the team to simultaneously attack the opponent's goal, scoring with one of the balls. 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, which is necessary to prove the Banach-Tarski paradox. Set theorists can choose to work either with or without the axiom of choice, leading to different theorems being provable, but the axiom's impact on association football is a bit less clear. FIFA is the top-level institution governing association football, although defining the Laws of the Game, the rules laying out how the game is played, is actually done by a separate institution, the International Football Association Board (IFAB). It's implied that Randall's phone number has been blocked due to him calling FIFA too many times with math questions, in a similar way to how other xkcd characters have been banned from various other institutions. It is notable that Randall called FIFA, when he should have called IFAB instead.
Strictly speaking, the answer to Randall's problem may already be covered under existing rules. According to IFAB's Laws of the Game, Law 5.3, under the section Outside Interference, if there is an extra ball on the field then the action is to "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". Both balls created clearly interfere with play, since the ball is the centre of play, and the ball wasn't going towards the goal when it was split. Even if the ball had been going towards the goal, it is clearly interference by the attacking team, so the goal wouldn't count. As such, the referees should stop play and restart with a dropped ball. All of this does depend, however, on whether one or both resulting balls would be considered 'extra' balls, which is an interesting philosophical debate in its own right.[citation needed] A ball that has split into two new balls might also be ruled defective under Law 2.2, in which case the defective ball should be replaced and the game restarted with a dropped ball.
Other offenses could also come into play. If they need to handle the ball in order to split it, they would commit a handball offense. When a ball gets split in two it could also be considered to be an object being thrown at a ball or making contact with a ball with a held object, as you cannot really split a ball in two without those two balls coming into contact. Either of these should result in a free kick to the opposing team under Law 12.1.
Transcript[edit]
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This is one of 32 incomplete transcripts: This transcript was created by William Shanks, who continues to calculate PI years later from beyond the grave. Don't remove this notice too soon. If you can fix this issue, edit the 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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Ive been here before the comments but never before even a explanation, or even a transcript! Ill do a good deed and add the transcript rq 163.120.65.183 17:34, 7 August 2026 (UTC)
I'm sorry Randall but it's IFAB (the International Football Association Board) who manages the Laws of the Game, not FIFA. The FIFA officials right now wouldn't answer anyways, they're much too busy being corrupt :P Midnight vortigaunt (talk) 18:50, 7 August 2026 (UTC)
- True it's IFAB who come up with the rules (namely LotG), but FIFA is responsible for requiring IFAB's rules to be used. Also FIFA has half of the votes on IFAB anyway, so Randall's effectively correct. DKMell (talk) 21:46, 7 August 2026 (UTC)
- Technically FIFA is a member of IFAB, so there are a lot of people who are officials in both IFAB and FIFA at the same time.--Trimutius (talk) 14:08, 8 August 2026 (UTC)
Wasn't something like this pulled off in one of the books in Robert Asprin's MythAdventures series, to score on both teams in a three-way game? RegularSizedGuy (talk) 20:12, 7 August 2026 (UTC)
- In Myth Directions, the ball got divided in normal way (not using any paradox) and they scored with both halves to different goals resulting in final score 1.5:1.5:1. --Hkmaly (talk) 02:31, 8 August 2026 (UTC)
If this play results in a goal would it be scored as 1/2 goal if one goes in and the other is blocked? 15.248.0.79 21:13, 7 August 2026 (UTC)
- But if the new balls have the same volume and mass as the original ball? Or does the origin count? The mathematicians should play against physicists with conservation of mass. Sebastian --88.217.185.170 07:35, 10 August 2026 (UTC)
I'd like to mention this famous video on the Banach-Tarski paradox: https://www.youtube.com/watch?v=uFvokQUHh08 from 2011 even if it is a little off-topic. 84.154.68.192 10:16, 8 August 2026 (UTC)
In regards to an extra ball on the field, which one is the extra ball? As I understand each ball would be constructed from and/or still be a part of the original ball. Since the procedure involves cutting the ball into infinitely many pieces I suspect that would break a rule and supersede the presence of multiple balls on the filed/pitch. 69.204.108.174 15:59, 8 August 2026 (UTC)
- Please allow me to correct you on the point “infinitely many pieces”: the wonder and beauty of the Banach–Tarski paradox lies not only in the fact that you create volume out of thin air, but that the original object is divided into only finitely many pieces (five in the classical case of a ball). – Michael L. J. 134.60.67.135 06:35, 10 August 2026 (UTC)
- Creating volume out of thin air is something that commonly occurs when inflating a football. 2A0A:EF40:F68:E001:289C:24CD:D20E:4530 09:46, 10 August 2026 (UTC)
- Indeed, the fact that a sphere could be cut into *infinitely* many pieces and reassembled into two balls isn't at all surprising to a mathematician, its just the claim that they contain the same amount of points. A line segment, a square, a disk, a ball, a cube, 10000 balls, all of 3-D space, etc all have the same amount of points, the same as the amount of real numbers there are ("the cardinality of the continuum" if you want to sound fancy). Doing it with finitely many rigid (but infinitely intricate) pieces is the fun thing. Terdragontra (talk) 13:16, 12 August 2026 (UTC)
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