# Themed Challenge 2 (Manga/Anime): The Exhaustive Battle

**Discrete Mathematics**Level 4

Pikachu and a wild Rattata have been battling for a long time. They have been fighting for so long that their accuracy is haywire, and their health is one on-target hit away from being a K.O. Any attack now, however powerful or weak, if on target, could defeat the opponent.

At this stage of battle, Rattata has \(2\) attacks. He can use Tackle, which has a \(\frac{1}{3}\) chance of being on target. He can also use Hyper Fang, but only after \(2\) moves have occurred (i.e. Rattata moves, then Pikachu moves, then Rattata can use it or save for a later move). This attack has a \(\frac{2}{3}\) accuracy rate.

Pikachu, on the other hand has \(3\) attacks. One is Quick Attack, which has a \(\frac{1}{3}\) probability of being on target. Another attack is ThunderShock, which has a \(\frac{1}{2}\) probability of being on target, but can only be used after \(2\) moves. Finally, Pikachu can use ShockWave, a move with Infinite Accuracy, which, mathematically speaking has an accuracy rate of \(1\). However, it can only be used after \(4\) moves.

In this battle, the highest probability of Rattata winning and defeating Pikachu is given by \(\frac{M}{N}\) , where \(M,N\) are co-prime integers. Find \(M+N\).