Combinatorics · State transitions · Invariants · Counting steps
Two operations \(F\) and \(G\) turn an ordered triple of real numbers into another triple by the following rules:
\(F\) sends \((a, b, c)\) to \((a+1,\, b+c,\, c+1)\), and \(G\) sends \((a, b, c)\) to \((a,\, b-1,\, c+1)\).
Is it possible, applying \(F\) and \(G\) finitely many times, to turn the triple \((3, 4, 1)\) into the triple \((6, 5, 8)\)?
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Serbian Municipal Competition 2000, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source