MathDB

The numbers one through eight are written, in that order, on a chalkboard. A mysterious higher power in possession of both an eraser and a piece of chalk chooses three distinct numbers

x

y

, and

z

on the board, and does the following. First,

x

is erased and replaced with

y

, after which

y

is erased and replaced with

z

, and finally

z

is erased and replaced with

x

. The higher power repeats this process some finite number of times. For example, if

(x,y,z)=(2,4,5)

is chosen, followed by

(x,y,z)=(1,4,3)

, the board would change in the following manner:

12345678 \rightarrow 14352678 \rightarrow 43152678

Compute the number of possible final orderings of the eight numbers.

Problem Statement