Solving The "Impossible" Escape Riddle

Thanks to Rahul for the suggestion! A hospital has 16 rooms in a 4x4 grid. The rooms are numbered row by row and rooms and connected vertically and horizontally. Each room has exactly one patient. The patient in room 1 wants to meet all patients and then exit from room 16, but cannot meet another patient twice. How can it be done?

(Repost due to small mistake in original proof - the nxn grid needs to be even n)
Original    • Solving The "Impossible" Escape Riddle  

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