# So Many Possibilities!

Discrete Mathematics Level 3

Let $$S_{1}=[1,2,3]$$. For any $$S_{n}=[a,b,c]$$, define $$S_{n+1}=[b+c,b,c]$$, that is, exactly one element from $$S_{n}$$ is replaced with the sum of the other two elements. For how many values of $$n$$ does $$S_{n}=[1234,2345,3456]$$?

For example, if $$S_{133}=[1234,2345,3456]$$ and $$S_{583}=[1234,2345,3456]$$, there would be two values of $$n$$ that would contain the set $$[1234,2345,3456]$$, and your answer would be 2.

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