Instructions
Capture an Ice-type Pokemon like Snorunt or Swinub. Train this Pokemon against Water-type Pokemon until it is Level 25. Teach the Pokemon four Ice-type moves like Ice Fang, Icy Wind, Ice Shard or Ice Punch.
Use "Surf" to navigate Route 223. Flee from every Tentacruel or Pelipper you encounter. Once you encounter a Mantyke, lower its hitpoints with Ice-type moves. Toss a Great Ball or Ultra Ball to catch the Mantyke.
Speak to the Fishing Guru in Hearthrome City to get the Good Rod. While you're surfing on Route 223, use the Good Rod. Eventually, you will encounter a Remoraid. Decrease the Remoraid's hitpoints with Ice-type moves. Fling Great Balls or Ultra Balls until you capture the Water-type Pokemon.
Retrieve the Remoraid and the Mantyke from your PC at a Pokemon Center. You need the Remoraid in your party to evolve the Mantyke into a Mantine.
Pick up all Item Balls on Route 223. In one of the Item Balls, you will discover a Rare Candy. Go to the "Items" section of your "Bag" and use the Rare Candy on your Mantyke. If the Remoraid is in your party, the Mantyke will evolve into a Mantine.