4. Nuke New York, Prevent World War III - Fail-Safe
The Dilemma: You are the President of the United States, and by accident (somehow!?), a nuclear strike has leveled Moscow. World War III is on the brink as a result, but you have one lifeline - if you nuke a major US city, the war can be prevented.
The Solution: The President (Henry Fonda) goes through with the nuclear strike, reducing New York City to rubble in the film's final moments, killing millions.
How It Should Have Been Solved: It's an absurd scenario from the outset, and though the POTUS does go through with it in the end, it's really not as difficult as it sounds - the film really blows the anguish out of proportion (the cold, heartless robot says). Sure, you're going to be executing millions of innocents, but it's about utilitarianism in action; a failure to obliterate a city will result in certain nuclear obliteration of the US, whereas the willful destruction of NYC might at least pacify the Russians enough to stop a far greater loss of life. Twisted, but true.