A. Number Sequence System.

In this system, as in the constant order system, we understand that the numbers on the leaves will be taken down starting at the ``upper'' side of one pat, passing through to the ``lower'' side of that pat, and then from ``lower'' side to ``upper'' side through the other pat. It will always be the ``lower'' side of the flexagon that will be closed together in flexing. We can describe the operations rotation (R), turning over (T), and flexing (F) as follows:

$$\begin{array}{llllllllllll} \;\;\;\;a\pm 1 & a\pm 2 \;... \; ... ...;a\pm k\pm m\pm 1 \;... \; a (\pm N) & \;(\, I\; ) \end{array}$$

$$\begin{array}{lllllllllll} \rightarrow \: a(\pm N) & a(\pm N)... ... & a\pm 2 \; ... \; a\pm k & \qquad \qquad (T) \\ \end{array}$$

$$\begin{array}{lllllllllll} \rightarrow \, a\pm k\pm m\;\;a\pm... ...a\pm 1 \;;\;a \; a-1 ... a\pm k\pm m\pm 1 \;\;(F) \end{array}$$

The letter ``I'' indicates the initial position.