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Rare Monsters are among the easiest monsters to both find and tame, as you will find them in Stage 1 and 2 monsters, as well as Normal and Hard difficulties.

All Rare Monsters can be captured with the Rare Gem at a 100% rate.

All Stage 1 Monsters that are also Rare can be captured at a maximum 40% rate with a Taming Gem

You can find a list of all Rare Monsters on their category page here

HP

All Rare monsters share the same HP pool

Level Base HP
1 130
2 151
3 173
4 194
5 216
6 237
7 259
8 280
9 302
10 324
11 346
12 367
13 389
14 410
15 432
16 475
17 518
18 562

This increase pattern (plus 21 hp, then plus 22 hp) continues on from level 1 to level 15. Level 16+ has a different pattern still awaiting further exploration.

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