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Can You Solve the Virus Riddle?

As another long weekend approaches, it's once again time for you to save humanity by solving a Ted-Ed riddle. This time, an earthquake has created a risk of a deadly virus escaping a lab and destroying the world. Naturally, it's up to you to stop the virus from getting out, but securing the lab requires you to solve a complex riddle.

Hint: You may need to refresh (start?) your knowledge on Hamiltonian Paths. Good luck!