Joachim's current form is dictated by his Joachythms, which changes after fighting a number of battles.
They are represented by two coloured bands; Golden for his Golden Bat form, and Rainbow for his Invisible form.
Whichever band is currently in the centre of the graph represents his current form. When both bands converge in the centre, that is when he is able to assume his Grand Papillon (or Great Question form, if he has acquired the Mask of Question).
They follow a calculable pattern, but can be influenced by any of Joachim's unique equppiables; Gold Band, Prism Band, and the Electro Band, as well as his unique consumables; Bat's Gold, Strongoids and Getitup! pills - which reset the pattern.
For the purposes of this calculation, imagine that at 0, Joachim is in his standard form.
- Golden Bat will occur after multiples of 12 (i.e, after 12, 24, 36, and 48 battles).
- Invisible will occur after multples of 20 (i.e, after 20, and 40 battles).
- Grand Papillon/Great Question will occur after the lowest common multiple of 12 and 20; 60 (i.e, only appearing after 60 battles).
Using that same calculation, but with Gold Band equipped, Golden Bat will occur after multiples of 6 (i.e, after 6, 12, 18, 24, etc, battles)
Again, but with Prism Band, Invisible will occur after multiples of 10 (i.e after, 10, 20, 30, 40, etc, battles).
Finally, when wearing the Electro Band, Grand Papillon/Great Question will occur after the lowest common multiple of 6 and 10; 30 (i.e after 30 battles).