Shrinking Squares.An empirical exploration

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Start with a sequence \(S=(a,b,c,d\) of positive integers and find the derived sequence \(S_{1}=T(S)=(|a-b|,|b-c|,|c-d|,|d-a|\).Define a sequence \(S,S_{1},S_{2}=T(S_{1}),S_{3}=T(S_{2}),...\).

Let \(S_{i}\) minimizes the sum of the four elements.What is the sum of the four elements of \(S_{i}\)?

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