1516: Win by Induction

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Win by Induction
This would be bad enough, but every 30th or 40th pokéball has TWO of them inside.
Title text: This would be bad enough, but every 30th or 40th pokéball has TWO of them inside.


In the Pokémon franchise, human characters called Trainers capture fantastical creatures from the wild, the titular Pokémon (a shortened form of "Pocket Monsters"), and train them to battle one another. Pokémon are captured and stored in devices called Poké Balls, which shrink the creatures down to pocket size (hence "Pocket Monsters"). The anime's English dub has enshrined the phrase "<Pokémon's name>, I choose you!" into popular culture memory. When Trainers do battle, they often shout this phrase while throwing the ball to the ground, releasing the Pokémon at full size.

In this comic, a Pokémon chosen at some point was a Pikachu (the "poster child" for Pokémon, and the most publicly-known type), which does not intend to engage in the battle himself. Instead, the Pikachu chooses another Pikachu to fight for him. This process then repeats itself. Behind the Pikachu with the Pokéball is a long line of other Pikachu, suggesting that this process has been going on for a while.

Nearby stands Cueball, holding a closed Pokéball, and Megan, looking at her watch. This suggests that Cueball intends to have his own Pokémon fight the Pikachu, but is waiting to see which enemy his Pokémon must face before the battle can actually begin (waiting in vain, if the above described process repeats indefinitely), while Megan is growing impatient with the delay. Given that Cueball is holding a closed Pokéball he has not deployed yet, Megan cannot herself be his Pokémon. She could be his opponent, or a spectator.

The joke in this comic comes from analogy with the mathematical proof by induction, which is a proof about a base case, followed by a never ending sequence of steps, each step leading to the next. Induction proves an assertion is true for one case, and then infers that it must also be true for all related cases. The title suggests that the process of Pikachu choosing Pikachu will never end, effectively postponing the battle indefinitely. But the title is win by induction, by which Randall implies that we have been given enough information to reason logically whether Megan or Cueball will win. We have here turned mathematical induction on its head: part of the humor in the comic is that the logic of induction doesn't work in reverse. We cannot reason about an initial case by inferring something from a related case whose proof is dependent on knowledge about the initial case. Or perhaps the "win" referred to is precisely that the battle is indefinitely postponed.

The name "induction" comes from logic and discrete mathematics, and is thus unrelated to the physical phenomena of electromagnetic induction; but the fact that Pikachu is an "Electric-type" Pokémon could be word play connecting the two ideas.

If there were always only a single Pikachu in each Pokéball, this would spawn an unlimited number of Pikachu growing at a constant rate. Since, as the title text notes, there are occasionally two of them in a Pokéball, this would lead to exponential growth assuming each of the spawned Pikachu in this case is bearing a Pokéball! This may be a reference to the rate of twins, which is approximately 1/30 in humans.

Pikachu was used in one of the storylines of 1350: Lorenz. See all the attack moves it made here.


[There is a long queue of Pikachu extending out of the frame to the left. They are all just out from their ball, at least the last eight Pikachu's open balls lie in two parts on the ground at their feet. They are standing in front of Megan and Cueball. Cueball is holding a closed pokéball while Megan checks the time on her watch. The frontmost Pikachu, holding a closed pokéball, speaks.]
Pikachu at the front: Pikachu, I choose you!


  • For some reason Pikachu is drawn without its lightning shaped tail. Pikachu is also drawn rather stocky; this was common in early Pokémon media, but in recent times Pikachu's official design is much slimmer.
  • In Pokémon canon, Pokémon are only allowed to hold on to an empty Pokéball when stored in a Pokéball.
  • In Pokémon canon, only one Pokémon can exist in a Pokéball, contrary to the title text.
  • Randall has drawn the Pokéballs with the button that opens them in the middle of the red half. Whereas in actual (modern) Pokéballs the button is located where the two different halves meet. However, in the Pokémon canon, earlier Pokéballs are shown with buttons or timers on top, though it is doubtful this is the intention, unless the infinite line of Pikachus has been continuing for over thirty years.
  • The open Pokéballs are shown broken in two. While early Pokémon games occasionally stated Pokéballs could break, they are now always shown to be connected by a hinge. It is an uncommon visualization that the two halves are fully separate.

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Is the alt text a reference to double-yolkers (eggs with two yolks)? They're only about 1 in every 1000 but it seems like an obvious reference. --Fenn (talk) 08:32, 24 April 2015 (UTC)

Makes sense to me. I didn't even think of double yolks until you mentioned it here. 09:04, 24 April 2015 (UTC)BK201
Seconded. -- 14:34, 24 April 2015 (UTC)
I'd think it's a reference to the rate of twins, which is currently almost exactly 1/30 (and on the rise) [1] 17:45, 24 April 2015 (UTC)Merkky173.245.56.186 17:45, 24 April 2015 (UTC)

The explanation currently says that doubling makes it uncountably infinite. I'm pretty sure that doubling at each step (or every few steps) is still a countable infinite set. Proof here: http://practicaltypography.com/the-infinite-pixel-screen.html (see section "The internet demands a recount", because the first attempt is wrong). We can also prove it using the same argument as when proving that N x N is countable infinite (making zig-zag), but in this case making a breadth-first search of the tree of Pikachus: map 1 to the first Pikachu, map 2 and 3 to the two Pikachus at the second level, map 4, 5, 6, 7 to the four Pikachus at the third level, map (2^(n-1))…((2^n) - 1) to the 2^(n-1) Pikachus at level n. (talk) (please sign your comments with ~~~~)

Saw this too late. Yes, I agree, and I have fixed it accordingly. --Stephan Schulz (talk) 09:28, 24 April 2015 (UTC)
The problem being that we don't have an exact number for how many steps include double Pikachus. Granted, this is just a problem of practice, not theory. 12:37, 24 April 2015 (UTC)

"infinite, but countable" {Cough.} Someone doesn't understand infinity. Perhaps they meant "enumerable". 09:29, 24 April 2015 (UTC)ū

Someone doesn't understand countability. 09:46, 24 April 2015 (UTC)
enumeration is counting, in the simplest sense. "To name one by one; specify, as if in a list". That said, the whole of infinite whole numbers CAN be counted, just not by a human and not within a reasonable amount of time. -- 14:34, 24 April 2015 (UTC)

"The front most Pikachu speaks." Hey, look, it has those little lines to show it's speaking, not the blank white space behind it. Duh. 09:32, 24 April 2015 (UTC)

Looks like Megan is looking at her watch as well. Mention in transcript/explanation? Fenn (talk) 09:34, 24 April 2015 (UTC)

Are Megan and Cueball supposed to fight each other? It seems like Cueball still has his closed Pokéball in his hands. Is it then Megan's Pokéball that has evolved into all these Pikachu? And is it because she waits for her Pokémon to be ready to fight Cueball, that she checks her watch? I do not know anything about the Pokémon game/world. But it seems to me that some part of this setup is unexplained by the above... --Kynde (talk) 11:23, 24 April 2015 (UTC)
I've not seen many anime episodes, but in the games' battles the trainers always face each other from opposing sides. Plus, there are classes of trainers that are two people, so it could be that Megan is simply with Cueball. 18:58, 18 December 2015 (UTC)

Friendly reminder: Grammatically speaking, Pokémon are like sheep or deer. Singular and plural are both written the same. One Pikachu, many Pikachu, all the Pikachu. You'd be surprised at how much rage forgetting this causes in certain corners of the Internet. (talk) (please sign your comments with ~~~~)

What doesn't make sense to me is how this could continue indefinitely – after all, each of those Pikachu must have caught its own Pikachu beforehand. I don't see any infinite loop here, just a bunch of Pikachu that already had one another caught itselves. 10:13, 24 April 2015 (UTC)

Ah, the immortal quip from Jerry Bona: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's Lemma?" Aube (talk) 05:29, 25 April 2015 (UTC)

The word "induction" could also be intended to have a double meaning, referring also to electromagnetic induction. Pikachu is, after all, and electric pokémon. (talk) (please sign your comments with ~~~~)

Yes, I think this is right. Something about Maxwell's equations and induction. (talk) (please sign your comments with ~~~~)
From an engineering standpoint, in my opinion, Pikachu act more like biological capacitors (stored electric charge at potentially high voltage able to deliver large discharge currents) than inductors ("storing" magnetic energy via constant current, able to deliver high voltage when interrupted, like the ignition coil for an older automotive engine). I'm not too familiar with the Pokémon in-game/in-show universe, but I would imagine the Nurse Jenny corps could use electric Pokémon such as Pikachu (or Raichu) like defibrillators for cardiac events! --BigMal // 11:42, 24 April 2015 (UTC)
There are certain moves, including some that Pikachu can learn, that appear to be based on induction (Thunder Wave and Shock Wave). Besides, they build up charge in their bodies from somewhere; I'd suspect induction from the surrounding environment is what charges them up. -- 14:34, 24 April 2015 (UTC)

There's a point floating about how infinity doesn't imply completion. For instance, the number of all even integers is infinite, yet any given integer "only has a 50% chance of being even", so the series is quite obviously incomplete. This article seems to tend towards the idea (in diction) that an infinite number of pikachu would result in a win based on a 'logical' premise, without referring specificially to the terms of it's assumption. Xerxesbeat (talk) 11:38, 24 April 2015 (UTC)

The observation proceeds from the fact that the cardinality of all even integers is the same as the cardinality of all natural numbers (and the cardinality of all rational numbers). You can say that there are as many even integers are there are integers, conterintuitive as that seems. This, however, has nothing to do with the reasoning behind induction. Suppose that there is a finite number that doesn't correspond with a Pikachu, we can pick the least number for which this is the case (just check all the lower numbers until we find the least non-pikachu number N). But there is a pikachu corresponding to N-1, and it is holding a pokeball with a pikachu. So the pikachu in the pokeball of pikachu N-1 is pikachu N, and we have a contradiction to our supposition. Therefore there is no finite number that doesn't correspond with a Pikachu, QED.Aube (talk) 05:29, 25 April 2015 (UTC)

What happens if the Pikachu in the ball is recursing - picking himself? That doesn't fit the 30-40 double yolk thing, but would explain an infinite series. Food for thought. Megan is bored, waiting for the fight to start. I thought the game was supposed to begin when the players choose, though, so I don't understand why the wait is happening at all. (talk) (please sign your comments with ~~~~)

I doubt this is an intentional part of the joke, but the strongest Ground-type moves (Earthquake, Precipice Blades, etc.) are multi-target, hitting all foes in a 1v5 situation such as Horde Battles. In theory, a strong enough super effective move from Cueball's lead would still end the battle in one turn. 12:04, 24 April 2015 (UTC)

Not Land's Wrath, Dig, or Earth Power, which are strong ground-type moves. 13:05, 24 April 2015 (UTC)
Actually, Land's Wrath is multi-target. (The ones you named are also weaker than Earthquake and Precipice Blades, so the original comment stands regardless. Although a lucky Magnitude is more powerful than any of those.) --

I normally get a hearty chuckle out of Randall's graphical musings, but this one had me scratching my head. Fortunately, ExplainXKCD always comes to the rescue! After reading this page, my first thought was: Pokéception! 13:17, 24 April 2015 (UTC)

This sentence is nonsensical: When Trainers do battle, the anime's dub has immersed the phrase "<Pokémon's name>, I choose you!" into popular culture memory, which is accompanied by throwing the ball containing the selected Pokémon to the ground, which releases the Pokémon at full size. 17:51, 24 April 2015 (UTC)

Should it be noted that the Pikachu is drawn without its tail? It would normally a have lightning bolt shaped tail that appears to the side or from behind its head. (Trivia or other note?) Azule (talk) 15:22, 24 April 2015 (UTC)

I agree it looks weird, but can it be written off as it's being obscured by itself? (talk) (please sign your comments with ~~~~)
I would say not. Look how the left arms are all a bit obscured by the body. This indicates that the Pikachu are turned slightly toward a side view. That would mean the back end would more visible, including the tail. Azule (talk) 09:34, 27 April 2015 (UTC)

In Pokemon games from Gold and up, pokemon are able to hold items, including pokeballs. While in the game, once a pokeball is filled it is no longer available to select as an item, this comic would seem to imply the possible 'inception' scenario of having a pokemon hold an active pokeball (as the games have already shown that a pokeball can go into a pokeball). -- 14:13, 24 April 2015 (UTC)

ahem... "pokeception" short for "pocket inception" - I can't be the first one to coin this (?) - Brettpeirce (talk) 16:33, 24 April 2015 (UTC)

With Megan looking at her watch and Cueball holding the ball, I think we're meant to understand that Megan IS the Pokémon Cueball intends to use against Pikachu. 19:12, 24 April 2015 (UTC)

Since Cueball has a closed ball in hand he has yet to choose a Pokemon. Tjus Megan cannot be his. She must have thrown the first Pikachu ball. Should be changed in explanation.Kynde (talk) 20:31, 24 April 2015 (UTC)

It is possible the "win by induction" is from the Pikachu's opponent inferring the series in infinite, and conceding. 19:56, 24 April 2015 (UTC)

Am I missing something or does Randall not quite understand how Pokemon works? (Or is intentionally misrepresenting it for the sake of the joke) Pokemon don't come out with their own pokeball with them-- the pokemon aren't magically created. In theory, if someone were to give a pokemon its own pokemon, a chain could occur, but it would be limited to the number of pokemon previously caught. The pokemon are born in the wild and are captured inside pokeballs-- not created from them. (talk) (please sign your comments with ~~~~)

If a Pikachu can catch another Pikachu in a Pokéball, then there is no reason why the Pikachu it just caught, did not think about this before, and that it had done the same. So when it was caught and put into the Pokéball, it already had a Pokéball with another Pikachu. Of this has occurred enough times you get the result of this comic. No one said this would go on forever, that is something we have interpreted from the comic. It does not come directly from Randall! --Kynde (talk) 05:36, 25 April 2015 (UTC)

Bother this. I send out Quagsire. Use Earthquake. Please do not wait.Greyson (talk) 05:18, 25 April 2015 (UTC)

No mention of the exponential growth? If every 40th pikachu releases 2 and each of those also release their own pikachu then there is an average growth rate of the pikachu able to release another pikachu of 41/40 = 1.025. 19:48, 25 April 2015 (UTC)


Two other possibilities: one, in a bit of googling, it would appear that there is a type of Pokémon evolution called induced evolution, which involves stones of some kind? Alternately, we can use the term induction in the sense of soneone being inducted into a group. In this case, Megan has trained her Pikachu to be a Pokémaster. (Perhaps by arranging for it to be inducted into a rarified "gym"? I confess, I know nothing about the show.) 13:11, 24 April 2015 (UTC)

Considering Team Rocket's Meowth, which could talk and act just like a human, I think it would be possible for a Pokémon to become a trainer, maybe even fight without a trainer, knowing the intricacies of type effectiveness and what not. Also, the infinite Pikachu could have been made at a Day Care, but that would take an infinite time, and therefore can't be what happened in this comic. 18:58, 18 December 2015 (UTC)

I'm surprised no one mentioned that Pokémon is a game a long time before becoming a show. Although it was because of the animated series that Pikachu became "special" among the hundreds of other cute critters.

Also, no mention to the russian matryoshka dolls? Come on... Closest other xkcd I recall is https://xkcd.com/878/ (talk) (please sign your comments with ~~~~)

Axiom of choice

Could this be to do with the axiom of choice from set theory? From my understanding, it's a fundamental axiom of set theory that says 'given a set of sets, it's possible to choose one element from each of those sets'. "Choosing" is in this case a specific operation that can be performed on an element.

One specific detail about the axiom is that all sets under consideration must be nonempty; that is, they must contain at least one element. So I think this is analogous to the situation of a Pokemon trainer owning multiple (full) Pokeballs: his Pokeballs are a collection of non-empty sets from which he is now trying to choose a single element ("Pikachu, I choose you!").

Under normal circumstances, he can do this without invoking the axiom of choice because he knows the names of all his Pokemon and so can select one from each set. In this case, he could prove his ability to make the choice simply by releasing all of his Pokemon from their balls one at a time. (The Pokemon's name is actually irrelevant, because simply releasing the Pokemon counts as a choice).

However, the situation becomes more complex if it turns out that his Pokemon also possess Pokeballs, because now his ability to make the choice is uncertain. In this situation, there could be infinitely many Pikachus, and so he can't definitely select a Pikachu from all the Pokeballs under his control. In a situation like this, a mathematician would invoke the axiom of choice.

However, it seems that Cueball is actually having a go at it using an inductive method of choice: first by choosing a Pikachu, then having each Pikachu choose a Pikachu. If the number of Pikachus carrying Pokeballs is finite, then eventually, this will demonstrate that the choice can be made and so the axiom of choice is unnecessary. However, if it's infinite, then this will generate a neverending stream of Pikachus. In the latter case, the game never begins, because you can't begin a Pokemon battle until all participants have chosen Pokemon. Most likely, the other players would simply abandon the game, which Cueball could claim as a victory. Hawthorn (talk) 13:52, 24 April 2015 (UTC)

I think you are confused about the AoC. AoC states that given any collection of elements, you can choose an element from EACH set. If you are choosing a pokemon from a collection of pokeballs, it's equivalent to choosing one full pokeball from the collection and you are picking an element from a single set, which doesn't involve the AoC (this is something you can always do as long as the set is non-empty). In the example in the comic, AoC is not needed because there is already a natural ordering (ignoring the alt-text, which would make the set a partial ordering), so it's trivial to construct a choice function for any subset (choose the "least" pikachu in the sequence). On the other hand, if we have infinite pikachus running wild, we would need the Axiom of Choice (preferably its equivalent, the Well-Ordering Theorem) to assert that they can be ordered so that all of them except one is captured in a pokeball held by another pikachu.Aube (talk) 05:10, 25 April 2015 (UTC)
I was hoping a real mathematician would get involved. ^^ Do you think that this mathematical definition of 'choice' is the one being referred to in the comic, though? Hawthorn (talk) 13:47, 25 April 2015 (UTC)

Why don't those Pikachu have tails? Have they been sliced off? Is this some kind of mutation?-🐼🐯😺🐱 -- FlyingPiggy (talk) (please sign your comments with ~~~~)

Its all moot anyway. Pokemon can't talk but to say their name. Yourlifeisalie (talk) 14:45, 28 April 2015 (UTC)

Team Rocket's Meowth... 18:58, 18 December 2015 (UTC)

The 1 in 30 or 40 could be a reference to the fact that twins account for around 1 in 30 child births in the US, following in this vein, induction could be wordplay on the act of inducing labour in pregnant animals. 21:22, 28 April 2015 (UTC)

Pikachu (the one the main character has) doesn't like living in a Pokeball. Maybe this comic explains why? 23:29, 30 April 2015 (UTC) <<Why are all the IP addresses wrong?

4th Pokemon movie (Pokémon 4Ever: Celebi - Voice of the Forest; 2001) takes place in the past (relatively to Pokemon anime canon). There is old-fashioned pokeball used by young prof. Oak and it looks similar to one in the comic. Watch that part here. 17:20, 1 May 2015 (UTC)

Maybe each Pikachu will take exponentially shorter time so the total time is finite like GOD in GEB? For example, with the initial Pikachu taking one moment to summon the next, the meta-Pikachu taking half moment, the meta-meta-Pikachu taking quarter moment. 22:28, 7 February 2022 (UTC)