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LF: Damn, I hate it when puzzles turn out to be "IMPOSSIBLE"; what a waste of time!

Oh, it's not so bad. I think one should always be aware of the possibility that a puzzle has no solution. It's just a part of life, and part of mathematics too.

I could not help thinking, "What if the grid were infinite?" Presumably then there would be a solution, with aleph-nought moves.

It reminds me strongly of the old chestnut involving a convention of mathematicians and only one hotel. The hotel has a countably infinite number of rooms, but they are all full. The last mathematician arrives, and is told the bad news (I've been there -- it's a sickening feeling).

Fortunately, the last mathematician has a plan. "Move every existing guest from Room #N to Room #N+1," he says, "and then put me in Room #1."

The clerk is dubious, but to his surprise everyone is eventually accommodated.

Then in walks a physicist...

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When Life Gives You Lemons
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