Mark Levi's solution:
Suspend the sneaker by its lace to make a pendulum. Using the stopwatch, record the number of swings it makes in one minute, and call this n1.
Double the lace to halve its length and again record the number of swings in one minute. Call this n2.
Now sqrt{2}=n2/n1
For better precision, record the number of swings over a longer period of time.
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