In an ancient cipher known as the "Scytale," a strip of papyrus was wound around a cylindrical rod, turn by turn, without gaps or overlaps. Then, with the rod in a horizontal position, the text of the message was written on the papyrus line by line. After that, the strip of papyrus with the written text was sent to the recipient, who had an identical rod, allowing them to read the message. We have received a message encrypted using the "Scytale" cipher. However, its author, concerned about making the lines straight, drew horizontal lines during writing, which remained on the strip as dashes between the letters. The angle of inclination of these dashes to the edge of the strip is $\alpha$, the width of the strip is $d$, and the width of each line is $h$. Indicate how, using the available data, to read the text.

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Study the geometry of one turn of the tape on the cylinder's development.

## Solution

Consider one turn of the tape on the development of the cylinder (cut along a horizontal line).

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-04.jpg?height=495&width=309&top_left_y=547&top_left_x=883)

According to the problem, the height $CE$, dropped to the side $AD$, is equal to $d$, and $\angle CAE = 90^\circ - \alpha$. Therefore, $AC = d / \cos \alpha$. Since the height of the line is $h$, then on one turn there are $d / h \cos \alpha$ lines. Therefore, to read the text, the tape needs to be cut into segments of $d / h \cos \alpha$ lines and placed side by side.