(Quiz week 5, sep 15th, Question 2) Consider the Yu-Gi-Oh! Trading Card Game. You have monsters that battle each other in order the reduce to 0 the opponent's life points, in short. An example of a monster card, is Subterror Guru, showed in the picture bellow. In the scope of our exercise, monsters have 5 features: attibute, level, type, attack and defense points. Also showed bellow, there is a spell card named Small World. The text of small world is "Reveal 1 monster in your hand, choose 1 monster from your Deck that has exactly 1 of the same Type, Attribute, Level, ATK or DEF, and banish the revealed monster from your hand face-down. Then add, from the Deck to your hand, 1 monster that has exactly 1 of the same Type, Attribute, Level, ATK or DEF as the monster chosen from your Deck, and banish the card chosen from the Deck face-down. You can only activate 1 "Small World" per turn."

In short, Small World card allows the player to search one monster from another. Based on the information of the 1st monster, we search a 2nd monster which shares exactly 1 of the 5 features with the 1st; then, based on the information of the 2nd monster, we search a 3rd monster which shares exaclty 1 of the 5 features with the 2nd. An example is shown in the picture bellow, where we start with the monster card D.D. Crow, which shares the level 1 with Subterror Fiendess (and no other other feature), and then, we finally search Subterror Guru which shares the attribute earth with Subterror Fiendess (and no other feature).

So, monster cards are the nodes, and the feature shared among them (any of the 5 listed above) are the links of our network. This way, we can start at a given node, and set a path to another. The table bellow contains the information of  6 monster cards:

$$\begin{array}{|c|c|c|c|c|c|} \hline \text{Monster} & \text{Attack} & \text{Defense} & \text{Level} & \text{Attribute} & \text{Type}\\ \hline Subterror\ Fiendess & 800 & 500 & 1 & Earth & Spellcaster \\ \hline Subterror\ Archer & 1600 & 1400 & 3 & Earth & Fairy \\ \hline Subterror\ Guru & 1600 & 1800 & 4 & Earth & Dragon \\ \hline Maxx\ ``C" & 500 & 200 & 2 & Earth & Insect \\ \hline D.D.\ Crow & 100 & 100 & 1 & Dark & Winged-beast \\ \hline Ash\ Blossom & 0 & 1800 & 3 & Fire & Zombie \\ \hline \end{array}$$

Based on the information of the table, we are able to collect all the shared features between the monster. Also, considering the constraint of Small World card, that a monster (node) must share one, and only one, feature with a another monster (neighboor), we've built a network as follows bellow. For instance, node A shares level 3 with a neighboor e attribute earth with others; so the monster card corresponding to node A, has attribute earth and has level 3. It can only be Subterror Archer, and so on.

Small World spell card applies the Small World Theory, where we can connect a node to another through  a small chain of acquaintances (except for the constraint of sharing only 1 feature, that is specific for the game). Considering that the Small World spell card searchs only paths with size 2, and based on the network built, judge the statements bellow:

1. It's possible to use Small World spell card to search Subterror Guru from Ash Blossom.
2. From any other monster card, it's possible to search Maxx "C" using Small World spell card.
3. The average of the maximun distance to reach any node in the whole network is 2,33.
4. Only 1/3 of the nodes are not applicable to search any other node by using Small World spell card.

The correct statements are:

1. I, II and III.
2. I, II and IV.
3. I, III and IV.
4. II, III and IV.
5. None of the above.

Original idea by: Fábio Assunção.