Sombrero vs. Volleyball

The two most popular stat-gain 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 Blood-Faced Volleyball and Hovering Sombrero articles. The present page is concerned with comparing volleyballs against sombreros.

In the formulas below, VW is volleyball weight, SW is sombrero weight, and ML is monster level.

Sombrero vs. Volleyball Formula

With the rollout of NS13, the formula for extra stats given by the Hovering Sombrero of weight SW is:

stats = sqrt(SW)*[1 + sqrt(ML - 4)]/10

And for the Blood-faced Volleyball:

stats = sqrt(VW)

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:

sqrt(VW) = sqrt(SW)*[1 + sqrt(ML - 4)]/10

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:

sqrt(VW) = sqrt(SW)

Setting that value equal to W, we substitute:

W = W[1 + sqrt(ML - 4)]/10

and then, since W/W = 1:

10 = 1 + sqrt(ML - 4)

followed by:

9 = sqrt(ML - 4)

then:

81 = ML - 4

and at last:

85 = ML

So, if you are fighting monsters of 85 ML or higher, bring along your Sombrero as opposed to your equally weighted Volleyball.

Different Weights

At what weight does using a sombrero make sense at ML monster level, instead of using a volleyball of weight VW? The answer is simple:

SW = 100VW/(1 + sqrt(ML-4))²

To determine at what monster level a certain weight volleyball is equal to a certain weight sombrero you can use the following formula:

ML = (sqrt(100VW/SW)-1)² + 4

Charts

• Volleyball vs Sombrero
• Tuned Bander vs Sombrero
• 

Examples

In the examples below, q is VW/SW, and is useful in establishing at what Monster Level the Sombrero is more beneficial.

Example: When the volleyball/sombrero weights are equal, VW/SW=1, so this formula reduces to m=100-20+5=85, as in the first section.

Example: Suppose we want to compare a Llama Lama's stat-gains to the stat-gains from an equal-weight sombrero-type (say, a Baby Sandworm). The llama has the stat gain of a volleyball of half its weight, so q=1/2. This means that ML=100(1/2)-20(1/2)^1/2+5=55-20(1/2)^1/2=40.85. (Of course, this isn't the whole story, since a glimmering roc feather gives stats and agua de vida doesn't, but it means that llamas are generally preferable to sandworms at least when fighting monsters of level 40 or below.)

Example: Say we want to compare a Frumious Bandersnatch with its appropriately tuned equipment to a sombrero, and that we only care about the familiar's effect on our main stat. To make our lives easier, we'll say that the sombrero has the same weight as the bandersnatch after equipment is taken into account. A bandersnatch gives double the main stat experience as a same-weight volleyball; if you look at the volleyball-weight formula, you'll see that this means it's like a volleyball that's four times as heavy (it gives 2(VW^1/2)=(4VW)^1/2 experience at weight VW). So q=4, which means ML=100(4)-20(4)^1/2+5=365. That is, the sombrero only starts matching an equal-weight bandersnatch at monster level 365, which is unlikely to occur very often in an ascension context.

Why does a Sombrero always win at some monster level?

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^1/2) = O(C). However, the sombrero formula contains a non-constant factor in its numerator, the square root of the monster level, so its stat gains are O(m^1/2), 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.

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 4000+ monsters, it's really only feasible for high-stat characters fighting against scaling monsters such as Hamburglaris Shield Generator (cap at 5000 ml), Fernswarthy's Basement and Crimbo monsters.