A square with side $1$ is cut from the paper. Construct a segment with length $1/2024$ using at most $20$ folds. No instruments are available. It is allowed only to fold the paper and to mark the common points of folding lines.
Let the square be $ABCD$ and fold diagonal $AC$ as the first move. Assume that you have a point $P$ along $AB$ such that $AP=\frac{1}{n}$ then we can construct...
Construction 1: A point $Q$ along $AB$ such that $AQ=\frac{1}{2n}$ in one fold
Simply fold $A$ to $P$ and mark the midpoint.
Construction 2: A point $R$ along $AB$ such that $AR=\frac{1}{n-1}$ in two folds
Fold at $P$ parallel to $AD$ and let it intersect $AC$ at $X$. Then fold $DX$ meet $AB$ at $R$.
Now let $\rightarrow$ denote the first construction and $\mapsto$ denote the second $$1\rightarrow\rightarrow\rightarrow\rightarrow\rightarrow\rightarrow\rightarrow \frac{1}{128}\mapsto \frac{1}{127} \rightarrow \frac{1}{254}\mapsto \frac{1}{253} \rightarrow\rightarrow\rightarrow \frac{1}{2024}$$
