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Solution
Let a, b, c be the count of each species in no particular order. Three operations exist:
1. a++; b--; c--
2. a--; b++; c--
3. a--; b--; c++
Note that a' = (b+c) either decreases by two or is unchanged. This implies two properties:
a' cannot change from odd to even.
a' cannot increase.
The surviving species at the end (a' = 0) has to be γ, since the others have odd initial a'. And no more than 29 can be of this type, limited by the initial a' of the α species.
