What if terrorists detonate a nuke in DC...

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They have one cabinent member out of town for the State of the Union speech.

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If two-thirds of the Senate is killed, can the other third still meet quorum?

Quorum is a majority of living Senators.

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About half a dozen of US Congressmen are foreign born. What if the sole survivor is Rep. Lincoln Diaz Balart, for example. He can elect himself speaker as long as he wants, he'll never be President. Of course, if Diaz Balart and Mel Martinez are the sole two survivors, they might pass a constitutional amendment to make themselves eligible.

They can pass constitiional amendments with a surviving Senator, but it still has to be ratified.

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It's obvious what would happen. Governors would appoint Senators. The Senate would choose a Pro Tem leader.  The Pro Tem leader would swear in as President.