(I created this variant of a classic puzzle a long time ago, before the movie HPCOS was released.)
Enough is enough. You've decided it's time to take a stand. Every year since that boy started attending Hogwarts, Dumbledore has given the House cup to Gryffindor by a "last minute awarding" of large numbers of points to him for ever more dubious reasons. The Gryffindors have even stopped trying, knowing that they'll get the cup no matter what happens. This year, Dumbledore has just announced an award of 400 points to the boy "for parting his hair in a most interesting way", taking the cup away from your own house of Hufflepuff.
So you've confronted Dumbledore, and not only your own house, but also all the Ravenclaws and Slytherins have joined you. Even the Gryffindors look embarassed. "So, young Hufflepuff. I've been partial, have I? Well, then, let's put this nonsense to rest. We'll have a contest, between you and our fine Gryffindor champion. Mr. Finch! Bring the Urns of Peano from my office!" Finch comes back carrying 3 large urns, and Dumbledore calls the two of you forward. You see that two of the urns are empty while the third is full of balls with numbers printed in a strange magical script. Indeed, you pull one out of the top and try to read the number, but it keeps going on around the ball repeatedly, yet somehow without ever overlapping itself. Finally you give up and put it back. Dumbledore explains "In this urn is a ball for every natural number. Yes, infinitely many - a wonderful magic. As you can see, each of the urns has a dispenser track on the bottom where the lowest ten values in the urn are displayed. Go ahead, take the first ball." You pick it up and find the number 0. When you drop it back in the top of the urn, suddenly the balls on the track back up a space, while the 0 ball rolls through them to its original position.
"The contest", Dumbledore explains, "is this: Each of your houses are assigned one of the empty urns. The two of you will act in turns. In each round, you, young Hufflepuff," (you suspect he doesn't even know your name) "will take two balls from the track and place them in the Hufflepuff urn. Then our fine Gryffindor here will take a single ball from the Hufflepuff urn, and move it to the Gryffindor urn. This will continue until one of you can no longer take your turn." "But, won't that take forever?" asks your opponent. "Indeed, it would", Dumbledore replies, "But I shall cast 'Zeno's Accelerator' upon you. Fix a strategy in your mind, and when I cast the spell, your hands will follow it automatically. Each round will take but half the time of the round before, until an infinite number of rounds have been completed. In this way, the entirety of the urn may be emptied in only twice as much time as the first round takes." Your opponent's egghead friend can no longer contain herself. "That's ridiculous! The energy it would take is infinite! The heat produced would kill us all!". "Honestly, young lady,", Dumbledore replies, "you should have learned by now that muggle physics takes a back seat to magic!" The Gryffindor champion still appears confused. "But, sir, if in every round, two balls are added to Hufflepuff, and I take one of them to put in Gryffindor, how will either house pull ahead? We'll just be tied in the end!"
"Oh, no, Gryffindor. You misapprehend! If even one ball remains in the Hufflepuff urn when the contest ends, Hufflepuff will be declared the winner. Let them accuse me of favoritism now!" Your opponent looks very nervous, but suddenly appears to listen to something and calm down. A glance shows his egghead friend whispering something, apparently using a teleloquela charm. Obviously the fix is in. But fortunately, it appears that you've paid more attention to your arithmancy classes than Dumbledore did, for you spot a flaw in his plan. You smile as you fix your strategy in mind.
Why does Dumbledore expect Gryffindor to win? And what strategy can you use to defeat him?