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What Is Influence?
Whereas Power is a single value, Influence is determined on a per-faction basis. You can see your current influence levels on the bottom left of the game screen; your current influence for each faction is shown beneath its icon.
In order to be able to lay a card you must have both enough raw Power and enough Influence with the relevant Faction(s). For instance Thunderstrike Dragon has a Power cost of 6 plus an influence requirement of 3 Primal. So if you have, say, six Power but your influence is 2 Primal and 4 Fire then you wont be able to play it. Sapphire Dragon also has a Power cost of 6 but its influence requirement is only 2 Primal, so in this example you would be able to play that card.
Some cards have multiple influence requirements. For example Arcanum Monitor requires that you have both 2 Fire and 2 Time Influence. I'm not aware of any cards currently that require more than two types of influence.
How To Get Influence
The main way to gain Influence is via playing Power cards. Whenever you play a Sigil you gain one Power and also one influence with the relevant Faction.
As well as Sigils there are other types of Power cards: Monuments, Seats and Seals. These each increase your maximum Power by one as usual but give you an extra Influence point in each of two factions. These cards have preconditions for their ability: for example you must have a unit on the battlefield or must have a Sigil in your hand.
There are other cards that increase your influence when played, notably some of the Strangers.
So it's possible for your total Influence to be greater than your maximum Power. I'm not aware of any way for Power to be greater than total Influence.
Influence is of vital importance when building a deck. At one extreme you could make up a Monochrome deck from just a single Faction. That would mean you never have to worry about Influence but would significantly restrict your choice of cards and strategy.
At the other extreme you could have cards from all the Factions in one deck. That would give you maximum flexibility but you run the risk of not being able to play cards because you haven't yet drawn the right Influence.
Most decks (with the notable exception of Strangers) take a middle way with two Factions usually being the best option for balancing flexibility with the need for Influence.
This page written for Eternal version 1.19, Card Set 1 Last updated: 3-04-2017