Mewtwo, The Genetic Pokémon. It was created in a lab by a scientist after years of gene splicing and DNA engineering. It's DNA is very close to Mew's. However it's size and stature is different.. Because it's Battling Abilities are at the Ultimate Level...it only thinks about defeating it's foes. It tends to remain motionless to conserve energy so it can unleash it's full power in battle. It's said to rest quietly in an undiscovered cave, it was created solely for battles. Although the Scientists managed to create Mewtwo, they couldn't give it a compassionate heart. It was turned vicious due to the experiments and gene splicing it was subjected to.
A Good Moveset for Mewtwo would have to be this:
Strategy Using Mewtwo
Bit of an obvious Mewtwo set but effective nonetheless. Bolt-Beam can hit a lot of Ubers for super-effective damage and hit's a large majority of Pokémon for neutral damage. With Calm Mind, Recover and it's naturally good defences Mewtwo can last a long time and with it's amazing offensive stats it can go for a sweep at anytime.
The set has it's flaws however, the two main problems are when faced against a sponge with Toxic (such as LG Deoxys and Blissey), Hazers and opponents with Psych Up or Calm Mind as well as the ability to match Mewtwo's stat boosts. To overcome this weakness some people replace Thunder/bolt with Taunt, which can stop all three of these potential problems. Some people also use Substitute, although it can still be Hazed and matched with Calm Minds or Psych Up it can block Critical Hits and Toxic. Either way, when sacrificing Thunder/bolt to make room for Taunt or Substitute you leave Mewtwo exposed to Water types as well as other Pokémon who resist Ice Beam (such as Metagross).
EVs: 204 HP / 220 Spd / 84 SAtk
With 220 Speed EVs Mewtwo can outrun most Pokémon in the Uber Metagame, Timid/Jolly Latias/os, Lugia, Jolly Rayquaza and many other opponents. A lot of HP EVs are beneficial, as well as being able to create 101 Substitutes it generally aids when walling attacks with Recover. Since Mewtwo has Calm Mind it doesn't need that much additional Special Attack but it's a sweeper so the leftover EVs may as well be placed there, with a Modest Nature it gains a huge Special Attack boost anyway.
EVs: 204 HP / 96 Spd / 208 SAtk
With 320 Speed Mewtwo can still outrun most opponents, Jolly Rayquaza as well as Latias/os, and Lugia if they don't have a Speed boosting nature.
Other Optional Sets
Mewtwo @ Leftovers
Not many opponents can switch into this set, if you can out-predict your opponent this set can open up the game for you. Focus Punch catches Blissey, Tyranitar and Snorlax, Shadow Ball hit's the large number of Psychic types, Ice Beam catches Groudon and the Dragon types and Thunder/bolt hits Ho-oh and Kyogre. Selfdestruct can be used in place of any of these moves, using Selfdestruct you can pretty much KO any opponent who doesn't resist the move.
Mewtwo @ Choice Band / Leftovers
Choice Band Mewtwo has plenty of potential, much like the mixed sweeper above it requires a lot of prediction but pays off if you can use it effectively. Earthquake does nice damage against Tyranitars and Metagrosses, HP Rock can KO Ho-oh, Ice Beam does a lot of damage against Groudon and Selfdestruct can pretty much OHKO anything that doesn't resist it. This set is a lot more effective with Choice Band but Choice Band can be dropped to allow for more versatility.
Mewtwo @ Leftovers
Also functions like the mixed sweeper but with Substitute the need to predict isn't as necessary. Bolt-Beam can beat most opponents and Sub-Punch can beat opponents who can sponge Mewtwo's Bolt-Beam, although this set will encounter some problems with bulky Psychic types (like LG Deoxys).
Mewtwo @ Leftovers
Pretty much the set stated above except with a focus on Attack instead of Special Attack and Shadow Ball instead of Thunder/bolt. Very defensive Water types might cause some minor problems but in general this set can take out most opponents if it can get a Sub in place.
Mewtwo @ Leftovers
Pretty much a physical variation of the Calm Mind set (with Bulk Up instead of Calm Mind obviously). Ice Beam can be used instead of Recover to take out Groudon, who can cause physical sets some problems. Much like the Calm Mind set you'll also have problems against Toxic, Psych Up and Hazers.
Other Optional Moves
Psychic, Reflect, Light Screen, Safeguard, Toxic, Thunder Wave, Flamethrower, Counter, Barrier, Amnesia, Snatch.
Mewtwo has a lot of options although most of them are better served on another Pokémon. Psychic gains STAB but most Ubers either resist it or are hit harder by Ice Beam or Thunder/bolt. Reflect, Light Screen and Safeguard present passing options as well as making Mewtwo a much tougher Pokémon but most of these moves are available and better served on other Pokémon who are designed to fill a support role. Toxic allows Mewtwo to out-stall most opponents, especially when combined with Taunt and Recover. Thunder Wave hurts the majority of opponent's sweeping abilities. Flamethrower can benefit from Groudon's Drought although a lot of Ubers resist it, it's main purpose is to KO Metagross. Counter can be a surprising move to use against physical attackers, with it's great speed stat and Recover it can regain it's HP rather quickly after being hit. Barrier and Amnesia, much like Reflect and Light Screen, makes Mewtwo much stronger defensively however critical hits are inevitable and Mewtwo is still exposed to status attacks. Snatch is a situational move that can be used to steal an opponents stat-boosts, but since Mewtwo has it's own stat-boosts there's very little point, the best case scenario with the move would involve Snatching Groudon's Sword Dance.
Strategy Against Mewtwo
Mewtwo has a ridiculously versatile move pool as well as versatile stats, opening up an amazing number of effective possibilities and leaving Mewtwo very difficult to Counter. LG Deoxys can survive most of Mewtwo's attacks barring Shadow Ball and is an effective option to scout Mewtwo's set, however if Mewtwo has Taunt it can make countering difficult. Blissey can sponge most of Mewtwo's special attacks and is immune to Shadow Ball but has difficulty surviving against physical variations of Mewtwo and can be out-stalled by a Taunt or Substitute variation of Mewtwo, it's a similar story for Regice except Regice isn't immune to Shadow Ball and can only match Mewtwo's Calm Minds with Psych Up. Snorlax, much like Blissey and Regice, can survive against most Special variations of Mewtwo and is immune to Shadow Ball but unlike Blissey it can't be out-stalled as easily. A Metagross with adequate defence can survive most of Mewtwo's attacks except for Flamethrower however Metagross's only form of self-recovery is rest so it can easily be beaten down by moves that hit it for neutral damage (Focus Punch, Shadow Ball and Thunderbolt).
Against physical Mewtwos Groudon can easily survive most of it's un-boosted attacks but has difficulty surviving against Ice Beams. Other non-Uber physical walls can also wall physical Mewtwos (Forretress and Skarmory for example) but are at risk against Mewtwos with special attacks.
Generally all Mewtwos are much easier to beat after being hit with Thunder Wave or Toxic, Tyranitar's Sandstorm also aids in beating down Mewtwo's HP. Mewtwo has good defensive stats but in comparison to many other Ubers it's defensive stats are average, it can be beaten down by strong sweeper if that sweeper isn't at risk of being KOed (Metagross can beat it down with Meteor Mash, Tyranitar can hit with Crunch, etc) but if it's given time it can easily come back with Recover. Opposing Mewtwo's using a set with Shadow Ball can KO it if it lacks a Shadow Ball to fire back with. Deoxys-FR can also KO Mewtwo with Shadow Ball and Choice Band.
A good Contest Moveset for Mewtwo would have to be this for the Beauty Contest best with Modest, Mild, Rash or Quiet Nature:
Strategy Using Mewtwo
If you do the attacks in this following order you should have very few problems; 1st - Hail, 2nd - Ice beam, 3rd - Rain dance, 4th - Water pulse, 5th - Hail
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