Here’s one other proof, this time by induction:

Induction Hypothesis: (Not essentially true but) Among any card set of N playing cards, each transfer sequence terminates after a finite variety of strikes.

Using the Induction Hypothesis, show the Induction Hypothesis for N+1:

Induction Step: The leftmost card of any set can’t be flipped again, as soon as it has been flipped face-up, so it may be flipped at most as soon as. Therefore, the maximal sequence in a set of N+1 playing cards can’t be longer than “maximal sequence among the many N rightmost playing cards + flip the leftmost card + one other maximal sequence among the many N rightmost playing cards”. Particularly, utilizing the Induction Hypothesis, each transfer sequence amongst a set of N playing cards terminates, and subsequently each transfer sequence amongst a set of N+1 playing cards additionally terminates.

Base case: Any set consisting of only one card permits for less than terminating sequences. Proof: the doable units are “face up” and “face down”, which terminate after 0 and 1 strikes, respectively.

Now, the Base Case proves that the Induction Hypothesis is true for N=1, and the Induction Step proves that if the Induction Hypothesis is true for some N, additionally it is true for N+1.

Conclusion: Therefore, by induction, for all integers N >= 1, the Induction Hypothesis is true.