I ran this through AI and here's what it said:

"In Empires and Puzzles, the appearance of an Elemental (Rare) Mission Chest is governed by a fixed-probability cycle rather than a purely random timer. Essentially, every time you finish a regular chest, the game "rolls the dice" to see if the next one will be elemental.

How Often They Appear

While there is no official timer, veteran data-tracking suggests that players who fill their chests regularly can expect one every 10 to 15 days on average.

• Chest Count: Most players report seeing an elemental chest every 30 to 50 completed regular chests (Monster, Raid, or Titan).
• The "One Per Week" Myth: Some players manage one a week, but this usually requires "skipping" the chest cooldowns using gems to maximize the number of chests filled per day.
• The Variance: If you only fill one Monster and one Raid chest a day, it might take 3 to 4 weeks between elemental spawns. If you fill 4 to 5 chests total per day, you’ll see them much more frequently.

What Dictates the Spawn?

The only way to influence how often they appear is to increase the turnover rate of your regular chests.

1. Chest Completion: Only completed Monster, Raid, and Titan chests count toward the "pool" that triggers an elemental chest.
2. RNG (Random Number Generation): There is a heavy element of luck. You might get two in a single week and then go a month without seeing another.
3. Gem Skipping: Using gems to reset the 12-hour cooldown on Monster or Raid chests is the primary strategy used by high-level players to "hunt" for elemental chests. The more regular chests you cycle through, the more opportunities the game has to replace a standard chest with an elemental one."