MathDB

22

Part of 2020 LMT Spring

Problems(1)

Spring 2020 Team Round Problem 22

Source:

8/22/2020
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 xx, yy, and zz on the board, and does the following. First, xx is erased and replaced with yy, after which yy is erased and replaced with zz, and finally zz is erased and replaced with xx. The higher power repeats this process some finite number of times. For example, if (x,y,z)=(2,4,5)(x,y,z)=(2,4,5) is chosen, followed by (x,y,z)=(1,4,3)(x,y,z)=(1,4,3), the board would change in the following manner: 12345678→14352678→4315267812345678 \rightarrow 14352678 \rightarrow 43152678 Compute the number of possible final orderings of the eight numbers.