Mention of punched cards reminded me of some 1960s trivia.

There were two formats for binary code on punched cards for IBM 700-series machines. In "row binary" two 36-bit words filled each of the 12 rows from bottom to top. In "column binary" successive thirds of a word ran downward in each of 3 successive columns. Bob Morris wondered what is the order of the permutation performed in row-to-column conversion.

The order of the row-to-column permutation turns out to be 172. There are five cycles of that length plus an oddball  of length 4. My souvenir card shows the oddball.

If you convert a cycle from row binary to column binary, the output is the same as the input.* Each cycle is also invariant under 180-degree rotation.The 4-cycle is a simple parallelogram. One of the 5-cycles lines up in periodic diagonals. The other cycles show no easily described structure.

The images of the two described cycles should be viewed in a fixed-width font. The punches in the 4-cycle are numbered in cyclic order. 

I keep a few souvenir punched cards. One bears that 4-cycle.

* Well, almost the same. The first word on the card contains control information whose layout differs in the two formats.

Doug