Fastest way to get both kings to the other side of the board? Say that both players collaborate to get White's king to the eighth rank and Black's king to the first. What's the shortest sequence of moves that achieves this?
It can be proven that 10 moves, or 20 plies, is the least possible.
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1. f3 f6 2. Kf2 Kf7 3. Kg3 Ke6 4. h3 h6 5. Kh4 Kf5 6. Kh5 Kf4 7. Kg6 Kg3 8. Rh2  Rh7 9. Kxh7 Kxh2 10. Kh8 Kh1

Here is the proof, listed in a logical order.
Observations Of Proof

Both kings must make at least seven moves to get to the other rank, so that's 14 plies off the bat.
Each side must move a pawn move to release a king, which costs 2 plies in further total.
Each side must move a piece to unguard a square for the other king, also costing 2 plies.
Finally, at least one more pawn move by both sides is required in order to let each king bypass the pawn structure, and that amounts to 2 plies.

In conclusion, 14+2+2+2=20 plies, so it is indeed the minimum possible.