1. **Problem statement:** We need to find the side length of the smallest square plate on which a 22-cm chopstick can fit along the diagonal without any overhang. 2. **Formula used:** The diagonal $d$ of a square with side length $s$ is given by the Pythagorean theorem: $$d = s\sqrt{2}$$ 3. **Given:** The chopstick length is 22 cm, which must fit exactly along the diagonal, so: $$d = 22$$ 4. **Find:** The side length $s$ of the square. 5. **Calculation:** Using the formula, $$22 = s\sqrt{2}$$ 6. **Isolate $s$:** $$s = \frac{22}{\sqrt{2}}$$ 7. **Rationalize the denominator:** $$s = \frac{22}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{22\sqrt{2}}{2}$$ 8. **Simplify:** $$s = 11\sqrt{2}$$ 9. **Approximate:** Since $\sqrt{2} \approx 1.414$, $$s \approx 11 \times 1.414 = 15.554$$ 10. **Round to nearest tenth:** $$s \approx 15.6$$ cm **Final answer:** The side length of the smallest square plate is approximately **15.6 cm**.