Summon

Summons are among the most well known and powerful abilities in the Golden Sun series. They consist of the user calling upon a powerful spirit to assist the user's party. The use of any summon requires that the party have the necessary number of Djinn of the proper element on standby at the time the character attempts the summon. These Djinn are subsequently placed into recovery. In general, the power of the summon increases with the number of Djinn required to use it. A few enemies have attacks that are treated as summons.

Unlike Psynergy, summons always hit all enemies (with the exception of Coatlicue, which heals the party rather than attacking the enemy). They also temporarily enhance the summoner's elemental power in the elements corresponding to the Djinn necessary for the summon. Summons deal more damage against enemies with more HP, making them very effective against bosses.

In Golden Sun
There are a total of sixteen summons in the first Golden Sun. All of them use one to four Djinn of a single element and are available as soon as the party has enough Djinn to use them. These summons have no secondary effects.

In Golden Sun: The Lost Age
Golden Sun: The Lost Age has all the summons from Golden Sun plus thirteen new ones. The new summons all use Djinn of two different elements and are acquired by picking up special stone tablets. Each of these summons has secondary effects.

In Golden Sun: Dark Dawn
In Golden Sun: Dark Dawn, all 29 summon spirits from the first two games return, in addition to one new summon spirit, tentatively known as the Crystal Dragon. While all of the old summons were remade into 3D, some spirits, such as Moloch remain relatively unchanged, while others, like Ulysses, are radically different to the versions seen in the first two games. When Dark Dawn was first announced, trailers showing some of the summon spirits' animation sequences were released to the public. Many of these clips featured animation sequences that were later altered when the game was released. For instance, the Neptune summon was initially shown to be a whale almost identical in appearance to the whale seen in the first two games, but was later altered to have a darker color and sport horns and markings that were not present on the original version.

For a video exhibition of all 30 summon sequences in the game in sequential order, visit this link.

Damage
The damage from a summon is calculated as follows:

$$\mathrm{damage = \left \lfloor \left( \frac{\left \lfloor 256 \cdot dimRatio \right \rfloor}{256}\right) \left(base + \frac{\left \lfloor 256 \cdot multiplier \cdot hp \right \rfloor}{256}\right)\left(1 + \frac{\left \lfloor 256 \cdot \left( \frac{power - resistance}{200} \right)\right \rfloor}{256}\right) \right \rfloor + randDamage}$$


 * damage: final damage.
 * $$\lfloor x \rfloor$$ is the greatest integer less than or equal to $$x$$.
 * dimRatio: This is always 1, 0.7, 0.4, 0.3, 0.2, or 0.1 for summons. Its value depends on the location of the enemy of interest relative to the main target. For example, the main target will take 100% of the damage, the enemies next to him will take 70%, and so on.
 * base: base damage of the attack. The value depends on the summon; see the list below.
 * hp: HP of the target.
 * multiplier: the HP multiplier of the summon. This factor is what makes summons so effective against bosses; the more HP the target has, the more damage a summon will deal. For the party's summons, the HP multiplier is usually equal to $$(3 \cdot \text{number of Djinn used})/100$$. The only exceptions are the multipliers for Iris and Daedalus.
 * power: the user's elemental power in the relevant element.
 * resistance: the target's resistance to the relevant element.
 * randDamage: a randomly generated integer between 0 and 3, inclusive.