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I've just read this question, which highlights how the goblet was fooled.
But why do the teachers allow it? DVK makes the analogy that it is like a program with a bug. But if a bug presents an obviously incorrect output, why not just ignore it?
I remember the headmasters and headmistresses discussing it, and concluding harry should compete... but I can't remember or fathom why they wouldn't dismiss the result (or at least Harry).
In what can only be described as the stupidest plot-twist ever, it turns out that in a competition designed to be entered by children, the Goblet is set to mete out a severe punishment to anyone who refuses to take part.
My reading was that since competing in the competition (the first task of which involves facing a giant man-eating dragon) is the lesser danger, this punishment must be either deadly or mind-destroying:
‘Finally, I wish to impress upon any of you wishing to compete that this Tournament is not to be entered into lightly. Once a champion has been selected by the Goblet of Fire, he or she is obliged to see the Tournament through to the end. The placing of your name in the Goblet constitutes a binding, magical contract. There can be no change of heart once you have become champion. Please be very sure, therefore, that you are whole-heartedly prepared to play, before you drop your name into the Goblet
It appears that the three heads lack the ability to change or cancel the results, even if the Goblet throws a wobbly:
Bagman wiped his round, boyish face with his handkerchief and looked at Mr Crouch, who was standing outside the circle of the firelight, his face half hidden in shadow. He looked slightly eerie, the half darkness making him look much older, giving him an almost skull-like appearance. When he spoke, however, it was in his usual curt voice. ‘We must follow the rules, and the rules state clearly that those people whose names come out of the Goblet of Fire are bound to compete in the Tournament.’
and
‘Empty threat, Karkaroff,’ growled a voice from near the door. ‘You can’t leave your champion now. He’s got to compete. They’ve all got to compete. Binding magical contract, like Dumbledore said. Convenient, eh?’
and
Don’t you?’ said Moody quietly. ‘It’s very simple, Karkaroff. Someone put Potter’s name in that Goblet knowing he’d have to compete if it came out.’
