Theorycrafting xkcd's "win by induction"

Quite a while ago, the webcomic xkcd posted a gag involving a pokemon battle with an infinite number of pikachus. (here) The title is an obvious reference to the mathematical method of induction, wherein if something works at a low mumber, and it is shown that something working at a given number allows it to work at the next number, it works everywhere.

But that doesn't apply here. One pikachu can be beaten with the right set, so it is possible that an infinite number can also be beaten with something specifically designed to do so. Finding this set is what this post concerns itself with.

The source material is vague as to what the moveset being repeated is, so I'm going to assume it's the same as Red's pikachu in B2W2: volt tackle/iron tail/fake out/brick break, holding a light ball. I chose this because it's the most threatening of Red's sets for our purposes: it contains a priority move, a strong coverage move, and is a full physical set with Iron Tail (which has no immunities and applies a defense drop). I assume Adamant, 252 atk/speed, and a 0IV in HP, for similar reasons. Since RNG will do its absolute worst due to the massive number of turns involved, I've split the battle into two phases for modeling. In the first phase, I assume the RNG is pretty standard, with anything that has a 60% chance or greater happening. During the second phase, random events are whatever is least favourable to me. Crits and added effects will occur every turn they are possible, and I will be completely unable to move from paralysis unless (for PP reasons) I want to be stunlocked. This makes Iron Tail especially threatening, since it's one more thing to handle to avoid taking constant -6 crits.

The defensive solution
I don't think it's possible to use Leppa Berry to keep PP up while still staying alive. Harvest would take up our valuable Ability slot and require a lot of managing due to needing sun, and the only Recycle user that can deal with Iron Tail's defense drop (Klinklang, due to clear body) has no answer to paralysis. But, since the opponent has no access to hazards or hazard removal, a pair of sufficiently sturdy Regenerator mons could simply switch back and forth clearing stat drops while toxic spikes defeats the unending tide of yellow mice. Tangrowth needs to hold leftovers and have a full physically defensive spread to fully heal off a crit iron tail, but can manage it. You do need hazards if you follow the comic's alt text (every 30th or 40th one has two, allowing for switches), but otherwise you could theoretically wait for pikachu to struggle.

Team 1: hazard setter(s)/Tangrowth/Tangrowth

The offensive solution
Instead of avoiding struggle, we could also attempt to out-heal it's recoil and KO pikachu that way. We will need to be immune to both fake out's flinches and static's paralysis to do this effectively. Struggle does 1/4 of our max HP. Baton passing both Ingrain and Aqua Ring gets us halfway there. Just using leftovers still leaves us short. There are two solutions I found for this.

Firstly is using Shell Bell. Since it recovers 1/8 of the damage dealt and we need to heal 1/8 of our own hp, we need something with equal or less hp than pikachu. The easiest way to do this would be by underlevelling a rotom (naturally immune to both paralysis and Fake Out) to lower it's HP, I found 82 is the lowest against a level 100 pikachu to always KO with +6 struggle, which heals a full 0.7% above the threshold we need. If you need to have the same level as your opponent, you would most likely want to use Gastly, with a base 30 hp to pikachu's 35. this starts off as leaving you open to paralyisis, so before baton passing ingrain and aqua ring, make sure gastly has an odd number of Skill Swap PP and the opponent has Limber. The first part may require debuffing the opponent to survive the incoming attacks, the latter bit is most easily accomplished with a Lopunny that knows Entrainment.

Team 2: BP user 1/smeargle/rotom

Team 3: debuffer/lopunny/BP user 1/smeargle/gastly

Finally, there's Poison Heal. This means we can have whatever stats we want and don't need to worry about paralysis, but it's more complicated to get the ability to where we want it (a ghost type). Nothing has both Poison Heal and skill swap/entrainment, so we first need to give pikachu Wandering Spirit so that it will take poison heal at a later point. It's probably easier to precisely tune the HP values to avoid losing HP to rounding than it is to add the extra steps of trading magic bounce and applying toxic spikes to yourself in order to also hold leftovers.

Team 4: Runerigus/Poison Heal/BP user 1/smeargle/ghost type


Let me know what other options and difficulties I may have missed.
 
Shedinja @ Life Orb
Level: 20
Ability: Wonder Guard
Lax Nature
- Final Gambit

Shedinja @ Life Orb
Level: 20
Ability: Wonder Guard
Gentle Nature
- Final Gambit

Toxapex @ Black Sludge
Level: 75
Ability: Regenerator
EVs: 252 HP / 252 Def
Relaxed Nature
- Toxic Spikes

Tangela @ Eviolite
Level: 70
Ability: Regenerator
EVs: 252 HP / 252 Def
Bold Nature
- Endeavor

Shedinja is in fact immune to that Pikachu's entire moveset, so you can switch back and forth between two of them until it runs out of PP, then go to the two Regenerators to absorb a few Struggles. Oddly, both Tangela with Eviolite and Tangrowth with Leftovers have a minimum level of 70 to ensure that all the damage from critical Struggle is wiped out by Regenerator recovery. Toxapex misses by 1 HP for the ability to drop itself to level 74 and still manage the same, lest it runs into an awful string of luck where Pikachu gets 139 consecutive max roll criticals in a row on its turns in.

------

Meanwhile, for a team with the idea of withstanding Struggle recoil:

Cottonee @ Poke Doll
Level: 1
Ability: Prankster
- Memento

Durant @ Focus Sash
Level: 1
Ability: Truant
- Entrainment

Vaporeon @ Pasta
Level: 1
Ability: Water Absorb
- Protect
- Aqua Ring
- Baton Pass

Bellossom @ Bread
Level: 14
Ability: Chlorophyll
- Protect
- Ingrain
- Swords Dance
- Baton Pass

Gliscor @ Toxic Orb
Ability: Poison Heal
EVs: 52 HP / 154 Atk / 88 Def / 216 Spe
Jolly Nature
- Substitute
- Protect

Cottonee either takes itself off the field or gets outsped and KO'd by Fake Out; in either case the Fake Out turn is spoiled and Pikachu can never hit with it again (necessary due to the unfortunate double-cost of flinching something with Truant). Then Durant survives anything with Sash and uses Entrainment to create the ultimate in passivity, switching out on what will be Pikachu's first Truant turn. The next two members alternate between Protect and their various setup moves, handing Ingrain, Aqua Ring, and +6 attack (but you really only need +4) to Gliscor who gets its Toxic Orb and heals 25% per turn. Alternate between Substitute and Protect until both moves run out of PP, and you're set. Struggle easily KOs Pikachu, and Gliscor is faster. If Pikachu decides to use its Fake Out turn, then at best Fake Out can only deal as much damage as Gliscor normally deals to itself in recoil, and the flinch prevents it from dealing any additional recoil to itself that turn, making it a no-op on the end-of-turn healing.
 
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Shedinja is in fact immune to that Pikachu's entire moveset, so you can switch back and forth between two of them until it runs out of PP, then go to the two Regenerators to absorb a few Struggles. Oddly, both Tangela with Eviolite and Tangrowth with Leftovers have a minimum level of 70 to ensure that all the damage from critical Struggle is wiped out by Regenerator recovery. Toxapex misses by 1 HP for the ability to drop itself to level 74 and still manage the same, lest it runs into an awful string of luck where Pikachu gets 139 consecutive max roll criticals in a row on its turns in.
I'm torn between, "Let's do Level 74! Who cares about impossible odds?" and, "Infinite Pikachu means infinite chances for the imperfect storm."

One option to consider is Psych Up, removing your own stat debuffs. Steelix cannot learn Recycle but if it could a set of Earthquake/Psych Up/Recycle/Rest could stall indefinitely
Recycle Leppa Berry is a good idea. After checking what Electric or Ground types learn it, the best is Palossand. It's immune to everything but Iron Tail and has something better that resetting its stats, Iron Defense. Your defense will still be raised after healing with Shore Up, even if Iron Tail lowers defense every time. I'd pick Sandstorm as the last move just so Shore Up heals 2/3 instead of 1/2. Edit: You'll only be able to use Iron Defense instead of Shore Up to heal every 10 turns without lowered defense and if Iron Tail lowers defense every time, game over. Using Protect every other turn lets you block 8 of Iron Tail's 15 PP, so you only have to take 7. After that, get back to +6 defense and wait for Pikachu to use Struggle. It's an infinite battle, so I'm sure you have infinite time to wait for it.

Even with the worst luck, Palossand wins.


252+ Atk Light Ball Pikachu Iron Tail vs. 252 HP / 252+ Def Palossand on a critical hit: 141-166 (37.7 - 44.3%) -- guaranteed 3HKO

Possible damage amounts: (141, 142, 144, 146, 147, 149, 151, 152, 154, 156, 157, 159, 161, 162, 164, 166)


One Palossand with Recycle/Iron Defense/Shore Up/Protect is enough to endure infinity.
 
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Shoutouts to SadisticMystic and CTNC for meeting CTNC's status moves only spirit.

Just imagining a Pokemon sitting there tanking infinite Pikachu attacks without attacking (and just eating berries) is hilarious, so my vote goes to CTNC. Great thread idea Ironmage !

Sorry if using @ is generally frowned upon.

Question: I know Harvest was already sorta addressed, but, if we grant the assumption of infinite sunlight (lol), is it possible to OHKO forever with one of the Harvest mons? Trevenant can set Ingrain turn 1, run leftovers, avoid Fake Out, but can it ever be faster?
 
Using Protect every other turn lets you block 8 of Iron Tail's 15 PP, so you only have to take 7. After that, get back to +6 defense and wait for Pikachu to use Struggle. It's an infinite battle, so I'm sure you have infinite time to wait for it.
Do we get to assume that the standard trainer AI routines are in use, where Pikachu will always run out the entirety of its Iron Tail supply in a row against Palossand before doing anything else? It would never catch on and shunt worthless moves onto turns it expects you to Protect, or adopt the wild Pokemon behavior of picking moves randomly? Random-pick would be especially bad for lining up the turns where you have to Recycle the Leppa Berry back with the natural flow of Pikachu's attack cycle, let alone for those instances where you have to Recycle twice in a row after Leppa Berry refuels the PP on Recycle.

Question: I know Harvest was already sorta addressed, but, if we grant the assumption of infinite sunlight (lol), is it possible to OHKO forever with one of the Harvest mons? Trevenant can set Ingrain turn 1, run leftovers, avoid Fake Out, but can it ever be faster?
Max speed Pikachu will outrun Trevenant, but if you set up Sticky Web, you only have to handle the first one and then Trevenant can outrun the rest, KOing them with non-contact Earthquake even at +0.
 
Do we get to assume that the standard trainer AI routines are in use, where Pikachu will always run out the entirety of its Iron Tail supply in a row against Palossand before doing anything else? It would never catch on and shunt worthless moves onto turns it expects you to Protect, or adopt the wild Pokemon behavior of picking moves randomly? Random-pick would be especially bad for lining up the turns where you have to Recycle the Leppa Berry back with the natural flow of Pikachu's attack cycle, let alone for those instances where you have to Recycle twice in a row after Leppa Berry refuels the PP on Recycle.
The Pikachu being Red's Pikachu made me think AI Trainer, but in the comic each Pikachu is sent out by another Pikachu, so it might make more sense if it was random... I'm overanalyzing this... If it's random, Palossand loses eventually because one Pikachu will hit a Critical Hit Iron Tail every turn you're not using Protect and lower Defense every time. The AI is needed for a true infinite battle.

If it's another player in charge of the Pikachu army, predicting Iron Tail is like Rock Paper Scissors, except for one thing. To quote Bart Simpson, "Good old rock nothing beats that!" You're throwing rock, and nothing beats it 99.999999999...% of the time, but either Pikachu would win eventually, one of the players would leave out of boredom long before the point there's a Googleplex'th of a percent that it should of happened by then, or the universe would end. I want to be happy with knowing that, but it is guarantee to end eventually.


Edit: I'm not doing the math to see how much better it works, but against random moves or other Players, I think Palossand may be better off using Sandstorm instead of Protect to make Shore Up heal 2/3 instead of 1/2. Also, even though we're assuming the worst luck possible will happen eventually because infinite time means infinite possibilities, I still think Sand Veil increasing evasion is worth mentioning because it makes getting KOed less likely and therefore makes the attempt last longer.


Question: I know Harvest was already sorta addressed, but, if we grant the assumption of infinite sunlight (lol), is it possible to OHKO forever with one of the Harvest mons? Trevenant can set Ingrain turn 1, run leftovers, avoid Fake Out, but can it ever be faster?
No one said the infinite Pikachu army was in the current games. (If anything, it was probably Gen 6 due to the comic coming out in 2015. I'm really overanalyzing this...) If it happened in a Gen 5 game, then Drought would cause infinite sunlight. To make things even better, Ninetales can learn Hypnosis to make the start easier. Trevenant is a Gen 6 Pokemon, but Tropius also has Harvest and can boost its speed without help from Baton Pass with Dragon Dance. I'd say that sounds quicker, but either way is almost no progress for defeating the infinite.



Edit: I'd make another post for this idea, but I'm not sure if doulbe posting would be allowed for a new idea a day later. (I'm guessing it would be too soon.) In Gen 5, you could also use Sand Stream for infinite Sandstorm and win without using any moves against Trainer AI by switching between Hippowdon to absorb Volt Tackle, a Regenerator Pokemon weak to Electric like Slowbro to take the other moves, and another Regenerator Pokemon for switching to after Volt Tackle runs out of PP.
 
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Arguably, the best gen 5 Harvest user for this could be Exeggutor, who could indirectly skill swap it to something else (it's technically transfer-only, but can be bred from an imported mon). Infinite weather also potentially allows for dry skin replacing poison heal (it's much easier to set up, since it can be skill swapped off of jynx)
 

cityscapes

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i'm pretty sure something like this can work
:xy/mew:
BIG MAN ON CAMPUS (Mew) @ Leppa Berry
Ability: Synchronize
EVs: 252 HP / 252 Def / 4 SpD
Bold Nature
IVs: 0 Atk
- Roost
- Iron Defense
- Recycle
- Reflect Type

reflect type is cool here cause it gives you both para immunity and resistance to all of pikachu's threatening moves. it's not 100% reliable cause you lose if you get haxed before reflect type works (and you might need a guy in the front to absorb fake out), but once you get the setup it does work infinitely.

252+ Atk Light Ball Pikachu Iron Tail vs. 252 HP / 252+ Def Mew on a critical hit: 75-89 (18.5 - 22%) -- possible 5HKO
252+ Atk Light Ball Pikachu Volt Tackle vs. 252 HP / 252+ Def Mew on a critical hit: 135-159 (33.4 - 39.3%) -- guaranteed 3HKO
 
If it's another player in charge of the Pikachu army, predicting Iron Tail is like Rock Paper Scissors, except for one thing. To quote Bart Simpson, "Good old rock nothing beats that!" You're throwing rock, and nothing beats it 99.999999999...% of the time, but either Pikachu would win eventually, one of the players would leave out of boredom long before the point there's a Googleplex'th of a percent that it should of happened by then, or the universe would end. I want to be happy with knowing that, but it is guarantee to end eventually.
"Should of"?

"Should have".
 

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