Cardistry
You are holding a deck of 52 cards all facing the same way.
A move consist of separating the top two cards and then flipping them as a whole (not individually i.e. after flipping the top card goes in the second and the second card goes to the first place) and then cutting the deck anywhere you want. (A cut is to split the deck in two parts not necessirly in half and then switching the piles and recombining them to make a complete deck)
Now can you start with a deck with all cards facing the same way and after performing any number of moves end up with a deck with the cards facing in alternate directions.
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1 solution
Normal deck : Intial state (all cards facing the same direction)
Alternating deck : Final state (cards facing in alternate direction)
If you are able to convert a alternating deck into a normal deck then you should also be able to convert a normal deck into a alternating deck and vice versa.
This is because the forward process is identical to the backward process.
And the alternate deck actually does not change after performing a move (it stays an alternating deck). So it is impossible to perform the backward process.
And since the backward process is impossible so is the forward process.