Many partnerships can achieve the same goal
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Example showing different partitions for moving 'off' and 'on'.
Initially discs 1 and 2 are grouped, moving the top four discs off the base.
Then, as the top four discs move onto the base, discs 2 and 3 are grouped.
Disc 1 moves 4 times in the off phase and 2 times in the on phase. In the terminology of the proof, disc 1 is a height 3 disc on a base of disc 2 (moving off); it
is a height 2 disc on a base of disc 4 (moving on).
The total number of moves is conserved, off and on, since disc 2 moves 2 times, as a height 2 disc (moving off) and 4 times as a height 3 disc (moving on).
For completeness, disc 3 is a height 2 disc and moves 2 times, off and on. However it has 1 free peg moving off, and 2 moving on (but uses 1).