Why Dry Bones has more powerful die than Luigi
Subtitle
Introduction: Meeting the characters
Here are the two characters that will be featured in our case study: Dry Bones (top) and Luigi (bottom). To the right, one can see each character's respective "character die".
Initially, it would seem that these dice are equal, on average, but we'll explore why that's false.
You might say: "Trent you're a moron. On average, these dice will be exactly the same." and while you would be correct, there are more factors at play here than just averages.
Let's look at the facts:
The five, present in only Luigi's block, is the true killer in our comparison of these two characters. Attached is a description of the number in case you need a refresher.
Five is the third prime number. Because it can be written as 221 + 1, five is classified as a Fermat prime; therefore a regular polygon with 5 sides (a regular pentagon) is constructible with compass and unmarked straightedge. 5 is the third Sophie Germain prime, the first safe prime, the third Catalan number, and the third Mersenne prime exponent. Five is the first Wilson prime and the third factorial prime, also an alternating factorial. Five is the first good prime.[1] It is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. It is also the only number that is part of more than one pair of twin primes. Five is a congruent number.[2]
In more relevant terms, a five alone alters the course of the character die significantly, in ways that we will explore below.
Mapping the difference
Whomp's Dominos Ruins, the first map in the game, will help us explain our point.
Players spawn in the bottom right of the map, where the wooden bridge lies, and quickly get two options on where to move towards. Assuming that our player moves first (or rather doesn't get interfered with by any other players), we can run a few simulation turns to see how our player will perform.
Simulation: Turns one and two
When rolling his character die on turn one, Luigi will have the option of 1, 1, 1, 5, 6, or 7.
In a best case scenario, we want our character to hit two important spaces in the early game, the ally space and item space.
Rolling a 6 will allow Luigi to pick up an item, enabling him to increase his board tempo and star collection, and will set him up for a turn 2 ally. Rolling an additional 6 after that will land him straight into the ally space, which will add a permanent modifier to his rolls (allowing him to quickly outpace his foes).
Yet in our best case scenario, Luigi needs to have one of these roll combinations:
6 -> 6
5 -> 7
7 -> 5
The odds are 1/6 * 1/6 for any combination, equaling a 1/36 chance of getting a perfect turn 2.
Now we take a look at Dry Bones. Again, to get optimal success on this map, Dry Bones will need to get the item space on turn one, and the ally space on turn two.
In a calculation of his best case scenario, let's see how he will roll.
Dry Bones must achieve a combo of either:
6 -> 6
6 -> 6
6 -> 6
The odds of which are 1/2 * 1/2, totaling to a
1/4th chance of getting a perfect turn two.