So far, we have health and mana equations for the two phases; first you have enough health to canni-dance the manna robe every tick, then once you run out of surplus health you click every other tick as passive health regen builds up.
First phase:
h(t) = 2000 - 60t + 29t = 2000 - 31t
m(t) = 3500 - 50t + 20t = 3500 - 30t
Second phase:
h(t) = starting health + 29t - 30t = starting health - 1t ; we're just not going to worry about losing one health per tick
m(t) = mana when starting phase two - 50t + 10t = mana when starting phase two - 40t
In order to get the right starting mana value when starting phase two, we need to calculate when the first phase ends, which is when h(t) = 0:
h(t) = 0 = 2000 - 31t
t = 2000/31 = 64.5 ticks
After 64.5 ticks, mana will be:
m(t) = 3500 - 30 * 64.5 = 1565
Or using variables:
t = health_to_burn / 31
m(t) = 3500 - 30 * (health_to_burn / 31)
Where 30 is the modified mana burn rate (50 mana burned per tick, 20 mana regained from manna robe), and 31 is the modified health burn rate (60 mana burned per robe click, 29 mana regained through passive regen)
So we can rewrite the mana equation for the second phase to take into account how much mana you've spent at the end of phase one
m(t) = 3500 - 30 * (health_to_burn / 31) - (t - health_to_burn / 31) * 40
Now we can solve for the time when we hit oom:
0 = 3500 - 30 * (health_to_burn / 31) - (t - health_to_burn / 31) * 40
(t - health_to_burn / 31) * 40 = 3500 - 30 * (health_to_burn / 31)
40t - 40/31 * health_to_burn = 3500 - 30 * (health_to_burn / 31)
40t = 3500 - 30 * (health_to_burn / 31) + 40/31 * health_to_burn
40t = 3500 - 30/31 * health_to_burn + 40/31 * health_to_burn
40t = 3500 + 10/31 * health_to_burn
t = 3500/40 + 10/(31*40) * health_to_burn
t = 3500/40 + 10/(1240) * health_to_burn
t = 3500/40 + health_to_burn / 124
So under these assumptions, to add an additional tick before you run out of mana you can either add 40 mana or 124 health.
In the final section, I'll convert the raw numbers back into variables so it's easier to play around with the assumptions to see how it changes this conversion rate.
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