When the moons are in certain phases, gains for a particular stat are increased by 25%. Stat gain fractions are rounded randomly. If Grimace is full, Muscle gains are increased and HP recovery from resting at your Campground is doubled. If Ronald is full, Mysticality gains are increased and MPrecovery from resting at your Campground is doubled. If both moons look exactly the same, Moxiegains are increased. See moons for exact details. The moons have a 16 day cycle, out of which there are 2 Muscle Stat Days, 2 Mysticality Stat Days, and 2 Moxie Stat Days.
The stat gain bonus does not occur when under Hardcore or Ronin restrictions, although the resting bonuses still occur.
Before this change, much strategic gameplay in the Kingdom revolved around Stat Days. This moon cycle allowed players to maximize stat gain for a particular stat through careful gameplay. If a player was attempting to maximize, say, Muscle, they could perform Muscle-gaining activities on days during which Grimace is full. These things might include: using large stat items like Gourd potions or the chest of the Bonerdagon, selecting which food or booze items they consume, or donating at The Dungeoneers' Association gates. Probably the best and most common strategy is to save up adventures and use them on a stat day, without exceeding the 200 Rollover limit.
This is a simplified version of the table at Full Moon Table.