Location: Seafoam Islands

Method:
The stunning Ice/Flying Pokemon and first of the legendary bird trio, Articuno appears at the bottom of Seafoam Islands after a rigorous rock-moving puzzle involving Strength and Surf. Like Snorlax and the other legendaries the ball will sometimes miss if it is of inferior quality so always use Ultra Balls if you can help it. Both its Defence and Special are high so feel free to use stronger moves like Rock Slide or Flamethrower to weaken it if your Pokemon is under level 50.

Articuno Level 50

OT: Yours ID: Yours

Peck Ice Beam

Location: Power Plant

Method:
The shockingly powerful Electric/Flying legendary bird Zapdos appears at the end of the Power Plant in the room with the exit. Repels are recommended for the trek to Zapdos since the plant is quite a maze. No HMs other than Surf to reach the plant are needed. Moves like Ice Beam work well against it and Pokemon like Golem or Rhydon can hit it without being hurt by its Flying and Electric moveset.

Zapdos Level 50

OT: Yours ID: Yours

Thundershock Drill Peck

Location: Victory Road

Method:
Appearing in the middle of the Victory Road, the third and final legendary bird Moltres requires no more than Strength to access it. Like Zapdos, it is recommended you use Pokemon like Rhydon to avoid taking too much damage from its Fire and Flying movepool.

Moltres Level 50

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Peck Fire Spin

Location: Cerulean Cave

Method:
The master of the Unknown Dungeon at Cerulean City, Mewtwo requires both Surf and Strength to get to. Like the other overworld Pokemon, balls may miss when trying to catch this Pokemon. Get its HP down carefully and avoid burning, confusing, or poisoning it so you don't knock it out accidentally. Sleep and paralysis work well. As Psychic was a broken type in the first generation, no moves will be super-effective against it with the exception of Pin Missle and Twineedle (both found on Beedrill, a weak Pokemon, and Jolteon, who has terrible Attack). If you've saved your Master Ball from Saffron City, now is the best time to use it. There is no better Pokemon to catch in the game than Mewtwo.
Pokemon Mansion alludes to a few facts about Mewtwo; these include its origin, how powerful it is, and its connection to the super rare Mew, who can only be found via Nintendo event or glitch.

Mewtwo Level 70

OT: Yours ID: Yours

Swift Psychic Barrier Recover