Under Falling Skies

You must place one die in each column; only one die may be placed on an unexcavated space ahead of the excavator and that die’s value must be at least the number of spaces ahead. In the Rooms Phase you may pay 1 energy to move the excavator to that die’s space, and the excavator may only move up to the die’s value (illegal placement otherwise). (pp. 4, 7)

Ships ignore spaces they pass through; after you move all ships in the column simultaneously, resolve effects on their resulting (landing) spaces starting from the top. Also, placing a die in an AA gun column reduces ship descent by 1, and if the die itself is the AA icon the ships do not descend or trigger effects. (p. 5)

A jet fighter destroys all ships on explosion spaces with a number ≤ the fighter’s value; purple ships destroyed this way return to the mothership to respawn, while white ships are removed from the board. (pp. 6–7)

You cannot add values of separate rooms — each room resolves separately (unless they are spaces of the same multi-space room, in which case you sum the dice). To advance several research spaces you need one room with sufficient value (or a multi-space research room). (pp. 6–7)

White ships spawn after purple ships; they return to the mothership if they hit your city and are removed when shot down. If all white ships are already on the board, you ignore any effect that tries to add more. (p. 9)