# A new element

#### absoluzation

In one of Santa's research laboratories, the scientists have discovered a new chemical element and have named it Xmasium (by analogy with the famous elements Rubidium, Caesium, and Francium). Xmasium exists in three different types: There is α-Xmasium, β-Xmasium, and γ-Xmasium. If two Xmasium-atoms of different type collide, they sometimes merge into a single atom of the third type. If two Xmasium-atoms of the same type collide, they repel each other and nothing else happens.
Ruprecht puts 91 α-Xmasium-atoms, 25 β-Xmasium-atoms, and 4 γ-Xmasium-atoms into a cooking pot, covers it, and then leaves for lunch. When he returns, he notices that all Xmasium-atoms in the pot now are of the same type.
What is the largest possible number z of Xmasium-atoms in the pot at that moment?

I think the answer rests on the fact that number of α and β odd, and that all atoms are of the same type that leaves all of type γ.

Attempting to only have α would result in a pattern like
95 25 4
96 24 3
95 23 4
...
95 1 4
96 0 3
95 1 2
96 0 1
95 1 0
...
From here the only move would be to alternate β and γ, while α decreases to 0. A similar pattern results trying to end up with only β

Hence we are left with only γ left. We have a pattern like
95 25 4
70 0 29
69 1 28
68 0 29
...
0 0 29

Hence we are left with 29 γ particles left.