Given the sum of squares formula, the number of days to spend x candies is roughly the cube root of 3x. So I back off a few days from that estimate to be conservative and then iterate through until I find the right value.

def solve(candies):
    day = math.floor(pow(3*candies,1/3))-3
    while day*(day+1)*(2*day+1) <= 6*candies:
        day += 1
    return day-1