# Digimon 2 – dark spires and transitivity

Digimon 2 is an adventure anime in which children (called Digi_chosen) travel to a digital world (Digi_world) accompanied by monsters from this world called Digimon, they help them overcome challenges and defeat the forces of evil. These combats take place in different ways, sometimes requiring help from several Digimon, sacrifices or even if the Digimon evolves (change temporarily or permanently to a stronger form).

In this context, we can measure the power of the Digimon by peers from comparisons of their combats, defining the winning side as the strongest among them. However, the result of the victory due to the various factors involved in the combat does not guarantee the strongest transitive property. For example, take Digimon X, Y, Z:

*Digimon X beat Digimon Y;*

*Digimon Y beat Digimon Z.*

But due to the various factors involved, it is not possible to say with certainty that Digimon X is in fact stronger than Digimon Z, and this, consequently, makes it difficult to measure the power of the Digimon, even knowing the outcome of their battles. Thus, only if Digimon X defeated both Y and Z, we could say that X is stronger than Y and Z.

In a broader picture, it would be impossible to say that any Digimon is stronger than another if both did not fight. We can however assume that if X beats Y, Y beats Z, and the result of X and Z is unknown, then X tends to beat Z, although we don’t really know if it does. That is, the property of transitivity until it fails, although it is not deterministic, reinforces some hypotheses of ordering forces between Digimon.

To facilitate the notation, we will denote X >> Y, to indicate that X won Y, and X> Z to indicate that, if the transitive property was not invalidated, then X tends to win Z.

In this case, the transitive property can be extended to varying degrees. For example, take X, Y, Z, W, J, K, P, Q. If X >> Y, Y >> Z, Z >> W, W >> J, J >> K, K >> P , P >> Q. So we can say that X> {Y, Z, W, J, K, P, Q}. But if for example, Q >> K, then the transitivity from K to Q fails. In this way, we can only say that X> {Y, Z, W, J}.

However, in Digimon 2 between episodes 24 and 32, there is an additional element that makes it possible to measure Digimon, the Black Spires. They were created by Emperor Digimon and aim to destroy the balance in the Digi_world, thus affecting, among other things, the evolution of the Digimon. Right at the beginning of this anime, Digi_mundo was full of these Black Spires and from episode 24, Digimon Arukenimon (who wants to destroy the balance in Digi_mundo) started to convert the Black Spires into Digimon of darkness. Thus, given that all the Black Spires are similar in power throughout the Digi_world, this allows us to measure a unit of power for the Digimon created from the number of Black Spires used in their formation. To facilitate the notation, we will call S the Dark Spire unit.

With this, I made a survey between these episodes, to identify which Digimon the Arukenimon created with the Black Spires, and how many were necessary to create them.

*Thundermon – Ep. 24 – 1S*

*Snimon – Ep. 25 – ?S*

*Golemon – Ep. 25 – 1S*

*Minotarumon – Ep. 26 – 1S*

*Okuwamon – Ep. – 26 – 10S*

*BlackWarGreymon – Ep. 30 – 100S*

*Blossomon – Ep. 31 – 10S*

*20 Mammothmon – Ep. 32 – 200S*

*Knightmon – Ep. 32 – 10S*

Although it is not said at all times how many Black Spires were used to make each Digimon, this can be verified by counting the number of Black Spires that start to merge or the number of hair strands that Arukenimon hurls to make this transformation (each wire is launched into a Dark Spire).

A case however remains open. The Snimon that appears in episode 25. At the time the origin of those dark Digimon had not yet been revealed in the anime, their connection with the Black Spires was secretly contributing to some conflicts between the characters. Thus, in episode 25 it shows the Digi_chooted Ken with his Digimon Stingmon facing a Snimon, and with some effort his Digimon destroys him. We know with this that it was a Dark Digimon, therefore it is formed by at least one Dark Spire.

On two other occasions Stingmon (Ken’s Digimon) faces Okuwamon (10S) and then Blossomon (10S), both times Stingmon was unable to win. Thus, we have to:

*Blossomon (10S) >> Stingmon*

*Okuwamon (10S) >> Stingmon*

*Stingmon >> Snimon (?S)*

*Stingmon >> Thundermon (1S)*

*So, given that there are no counterexamples, we can say that: {Blossomon, Okuwamon}> {Snimon, Thundermon}*

*Based on Arukenimon’s dialogue before creating BlackWarGreymon (which required 100 Black Spires), she analyzes only the power of a Black Spire and the power of 10 Black Spires. This is an indication that she has no information regarding other quantities of Black Spires, leaving the Snimon with a measurable amount of 1**S**or 10**S**. Thus, it is likely that Snimon was created with 1 Black Spire.*

*Another value analysis we can do is:*

*BlackWarGreymon(100S) >> 20.Mammothmon(10S).*

From this we arrive at the following table:

1 S | 10 S | 20.(10 S) | 100S |

Thundermon | Okuwamon | 20 Mammothmon | BlackWarGreymon |

Sorting the table from left to right by the strength of the Digimon, we realize that the strength function of a Digimon is not the arithmetic sum of the number of Black Spires used to create them. For if it were, 20 Mammothmon would correspond to 200S while BlackWarGreymon would have only 100S. But in battle, the 20 Mammothmon fought alongside this Digimon. Thus, in order to think of a comparison of power between Digimon and the number of Black Spires used, we need to suppose some function that explains the conversion of Black Spires into Digimon strength. Taking the exponential function with an exponent that satisfies our situation, that is:

*P(1)=1 and P(100) = 20.P(10)*

We have the exponent 1,30103. Thus, we can say that for a number of Black Spires N, the strength function of the created Digimon is given by:

P(N) = N^{1,30103}

Applying this function to P(10) we have that P(10) = 20.P(1). In general, we can define the following graph that represents the strength of the Digimon from the number of Black Spires used to create them.

From this graph or the expression that generates it, we get several interesting information about the power of the Digimon:

*P(6) > 10.P(1);*

*P(35) > 100.P(1);*

*P(203) > 1000.P(1).*