Task 1. Is it possible to cut $\frac{1}{2}$ meter from a wire that is $\frac{2}{3}$ meters long without using a meter or any other tool for measuring length? Explain your answer!
Solution. If we fold the string exactly in half, the segments will be $\frac{1}{3}$ meters long, and if we repeat the process once more, we will get segments that are $\frac{1}{6}$ meters long. If we cut off one of the end segments, we will get $\frac{2}{3}-\frac{1}{6}=\frac{4-1}{6}=\frac{3}{6}=\frac{1}{2}$ meters. Therefore, from a string of length $\frac{2}{3} \mathrm{~m}$, without using a meter, a piece of length $\frac{1}{2} \mathrm{~m}$ can be cut.