Question:

<div class="solution"><p><strong>Key Idea:</strong> Shift by 1/3 and use {x+1}={x}.</p><p><strong>Step 1:</strong> <span class="math-block">$$f\left(x+\tfrac{1}{3}\right) = \left\{x+\tfrac{1}{3}\right\}+\left\{x+\tfrac{2}{3}\right\}+\{x+1\} = \{x+\tfrac{1}{3}\}+\{x+\tfrac{2}{3}\}+\{x\} = f(x)$$</span></p><p><strong>Answer: Period = 1/3</strong></p><div class="trap-box"><strong>Trap:</strong> Do not over-expand into cases. The cyclic shift proves the period in one step.</div><div class="key-concept"><strong>Key Concept:</strong> Equally-shifted fractional parts — period = common step size</div></div>