Tips for using potions
So now let's bring everything together. There are two main factors: move TU and entrance speed. Move TU will reduce by 5% on every 3rd bonus, while entrance speed reduces by lots in the first +5 (13.6%) then only a tiny bit from +5 to +9 (13.6% to 15.8%). Also, in this section I will focus almost solely on legendaries. For epics you want to get all the important ones to +9 first, for super epics you should get all the important ones to +6 and then with excess potions take the best to +9 and for mythics there's no decision to be made because they automatically get +3 for each shard. However, there may be some specific super epics you want +9 for combos or because you use them everywhere so you may choose to boost these up first rather than get other good ones to +6.
We know from the analysis above that +5 is a great point to reach on monsters, but since move TU only reduces every “+3” then +6 is a more sensible stopping point. +9 is the ultimate goal for any of your monsters.
General rule:
If you’re looking for some simple, general rules to follow then here it is. Get limited monsters you like to +3 and when you have potions to spare make the best +6. If you get duplicates of a monster you use so it becomes +2, +5 or +8 then use a potion to get that third bonus for the move TU reduction. You will get monsters from the normal egg pool a lot more so I’d avoid potting them at all if you can, better to save your potions for the limited monsters. However, there are many ways you can do this so make up your own mind. Some normal egg pool monsters you will love or use a lot so having them +3 will be useful or there might be a particular monster you desperately want +9.
More strategic thinking:
If you’d rather feel like you’re being more clever with your use of potions or otherwise your potions are very limited and you want to know where to stick them then here are some things to think about:
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The first +3 gives you the biggest boost. There’s very little reason to stop before +3 so a monster you want to pot should be taken to +3 at least.
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Monsters with slow speed get a bigger boost on their entrance speed so those slow monsters you like to use which have a huge impact as soon as they get a turn may deserve the potions more.
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Monsters which get their power from their passive skills that affect the battlefield (rather than their active skills) typically gain less benefit from potions because these passives receive no boost.
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Monsters which rely on using moves frequently to perform well generally benefit from potions more, e.g. Polaboss heals with every crescendo strike and it charges with number of uses so attacking more frequently makes it both a far better damage dealer and harder to kill. Also this extends more generically to any monster with low TU moves and no immunity to stun because these monsters are typically built to have a lot of impact as long as they can keep attacking.
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Monsters that gain combos or improved movesets because of reduced TU / entrance may need a specific amount of boost to unlock that, e.g. Chronotitan can be sped up so it can use bomb curse just before using OoO, which is a very effective way to get the bomb to go off.
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Monsters that care most about getting their first turn quickly to make their impact but don’t care about move TU are good targets for +6 and not bothering to ever go higher (e.g. a number of cloning monsters like Doomengine or Sanctallion)
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(More relevant to PvP) Monsters which are the ones you use in your front line for gaining control of the battle are good targets for making +6 or even +9 to take control as early as possible, e.g. repulse, OoO or cannibalise monsters.
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Specific customisation (see below)