Sombrero vs. Volleyball
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The two most popular statgain familiars can be pigeonholed into two categories: volleyballs and sombreros. For an analysis of the best familiar equipment for stat gains while using such familiars, see the BloodFaced Volleyball and Hovering Sombrero articles. The present page is concerned with comparing volleyballs against sombreros.
Sombrero vs. Volleyball Formula
In the formulas below, VW is volleyball weight, SW is sombrero weight, and ML is monster level.
The formula for extra stats given by the Hovering Sombrero of weight SW is:
And for the Bloodfaced Volleyball:
To find the point where the sombrero matches the volleyball, we must solve for monster level in terms of the two weights. Thus, we start with:
However, given that we have obtained both (or neither) familiar equipment, as well as assuming that any equipment/buffs we want can be switched between them (e.g., wax lips, Leash of Linguini), we can say that:
Setting that value equal to W, we substitute:
divide through by (0.1 + 0.005*W) to get:
Rearranging and multiplying through by 4, we get:
This cannot be simplified any more until either a weight (W) or Monster Level (ML) are provided. The below chart showing breakeven points plots this formula graphically.
It should be noted that with equally weighted familiars, a Sombrero will always be better than a Volleyball at 160 or more Monster Level. This is because the limit as W approaches infinity of the ML formula above is 160.
 According to Wolfram Alpha,
limit (4 * (2 + W/5) / (0.1 + 0.005*W)) as W>infinity
= 160 Monster Level.
Solving the formula for 1 pound familiars, it can also be noted that with equally weighted familiars, a Volleyball is always better than a Sombrero at 83 or less Monster Level.
Charts
 Comparing different weight Sombreros and Volleyballs is much more useful than comparing equal weight ones, but figuring out where to stop using one familiar and start using another can be tricky. The following graph attempts to make that process easier by giving a visualization of the familiars at various weights and Monster Levels, showing how many stats they give.
 The Sombrero will always be able to give more stats (up to its stat cap) as compared to an equivalent weight volleyball (or baby bugged bugbear wearing a balaclava). The following graph shows those breakeven Monster Levels, and how many stats you'd get from the Sombrero (or Volleyball/Bugbear) at the breakeven point of ML for each weight.
Examples
Example: Picking a weight such as 20 pounds gives: ML = 4*(2+20/5) / (0.1 + 0.005*20) = 4*(6) / (0.2) = 120. So when both the Volleyball and Sombrero are 20 pounds, if you are fighting monsters more than 120 ML, use the sombrero, and less than 120 ML, use the volleyball.
Why does a Sombrero always win at some monster level?
Because the Sombrero's stat bonuses grow with monster level, and the volleyball's does not.
Computer scientists define the growth of a function using Big O notation, which describes the behavior of a function as its argument gets arbitrarily large. In the case of the stat gains for volleyballs and sombreros, volleyball stat gains are constant at a given volleyball weight. They are O(v) = O(C) = O(1). However, the sombrero formula contains a nonconstant factor in its numerator, the monster level, so its stat gains are O(m), where m is the monster level. Since sombrero stat gains always grow as m increases and volleyball stat gains are independent of m, with a large enough m, the sombrero will always outperform the volleyball. This ignores the fact that the sombrero has a stat cap, because in order to reach the current stat cap of 230, a volleyball would need to weigh 230 = 2 + w/5 => w = 228*5 = 1140 pounds, which is unrealistic.
Under extreme conditions, the Wild Hare's ability to provide extra rollover adventures will provide more (net) substats per adventure than either the volleyball or the sombrero because the benefit from the Wild Hare is O(m). Since this requires an infinite supply of level 9200+ monsters, it's really only feasible for highstat characters fighting against scaling monsters such as the Sloppy Seconds Diner (cap at 11,111 ml), Fernswarthy's Basement and Crimbo monsters.