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## #1 2014-03-26 02:56:03

eigenguy
Member
Registered: 2014-03-18
Posts: 78

### For the honor of Hufflepuff

It has gone too far. Sure, the first year that Dumbledore had stolen the house cup from Slytherin by a last minute awarding of large numbers of points to his favorites, nobody minded except Slytherin. Who cares if a bunch of slimey gits have their faces rubbed in the mud? But every year since, it has been the same. Another awarding of huge points to Harry and his friends for evermore ludicrous reasons. The Gryffindors didn't even try anymore. They just sat back and waited to see what excuse Dumbledore would use to give the cup to them. This year, when your own house of Hufflepuff had a 400 point lead over Gryffindor, he awarded 405 points to Harry "for parting his hair in a most amusing fashion." Even the other teachers had the grace to look embarassed.

You could stand it no longer. So you stood in protest of this favoritism. Everyone except the Gryffindors have joined you. "Unjust, am I?", Dumbledore roared. "We shall see about that! Since you accuse me of dishonest judging, let us settle this with a challenge! Yes, yes. You shall serve as champion for the Hufflepuffs, and Harry shall serve as champion for the Gryffindors! Snape! Go to my office and bring the Urns of Peano here." As you wait, Dumbledore sits back with a sly smile. You know he has a trick up his sleeve. Presently, Snape returns with three large urns, each with a dispenser at the bottom and a short track. For one of the urns, the track contains balls numbered from 0 to 6. The tracks for the other urns are empty. Dumbledore explains, "These urns have a marvelous enchantment upon them. They can each contain an infinite number of these balls. Indeed, this urn already contains a ball for every natural number." Suddenly, the History Professor Binns interupted, "But 0 is not a natural number! The natural numbers are the numbers of the Greeks!". "Please! Do pull yourself out of your stuffy old books!", replied the Arithmancy Professor, Septima Vector. "0 obviously belongs in the Natural numbers! You can't start counting without it!" The argument became more raucous, until Dumbledore was forced to expel them from the hall.

"Getting back to business. The maker of these urns included 0. Now, if you will please take a ball and drop it back into the top of the urn?". You remove the ball marked 1. The higher numbered balls all roll down, allowing a ball marked 7 to imerge from the dispenser. You drop the 1 ball into the top. Ball 7 rolls back into the dispenser, and balls 2 through 6 move back up the track. Ball 1 appears, passing though the other balls as rolls back to its original spot. Dumbledore continues "Another enchantment on the urns causes the lowest numbered balls to alway be displayed. Now as you can see, the other two urns are empty. The middle urn will be assigned to Hufflepuff, while the Gryffindors will have the urn on the right. The challenge is this. You, Hufflepuff, will take two balls at a time from the full urn and place them in the Hufflepuff urn. Each time that you do so, Harry will take a single ball from the Hufflepuff urn and add it to the Gryffindor urn. You will continue until the original urn is empty." Hagrid pops up. 'Surely, ye cannat be thinkin' of 'aving 'arry 'ere spend the rest o' 'is days movin' little balls around. Even I knows that'd take forever!" "Ah, do not fear!", Dumbledore replies, "for I have an answer. I shall cast Zeno's Infinite Acceleration upon them. All our brave champions must do is to fix their plans of action in their heads as I cast the spell. The spell will cause each repetition of these actions to proceed at twice the pace of the previous one. As Professor Vector could explain if she were still here, this will cause the entire contest to finish in only a short time."

Professor McGonagall suddenly appears quite agitated. She quickly hurries to Dumbledore and whispers to him. "Hmmm, yes. I see. Still, it will be most entertaining to watch!" Looking at her face, he goes on, "No, I suppose not." Turning to the assembly, Dumbledore says "Minerva has reminded me that if we don't want our two champions to erupt into balls of fire, a further step will be needed." Before he can continue, Snape speaks up, "Headmaster, it so happens that I have two potions of Caratheodory's Entropic Isolation upon my person. If our contestants do not waste their time, they might be able to finish before the potion wears off. Possibly, anyway." "Splendid", Dumbledore replies, "well, then, if our champions will just step forward, we can be underway!". Harry looks confused. "Professor Dumbledore, if I take one ball out of every two in Hufflepuff, how would I ever win? We will always have the same number of balls!" "Ahh, Harry. I am afraid your task is harder than that! Did I forget to explain? Gryffindor must do more than just match Hufflepuff in balls. If, when we are finished, there is even a single ball left in the Hufflepuff urn, then Hufflepuff will be declared the winner! Let them accuse me of favoring Gryffindor now!"

Harry is clearly panicked, but Hermione leans over and starts whispering to him. Suddenly he relaxes and a smile spreads over his face. Clearly there is more to this contest than it seems. You wonder, how does Dumbledore expect Harry to win? And how can you defeat him?

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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## #2 2014-04-03 08:35:56

eigenguy
Member
Registered: 2014-03-18
Posts: 78

### Re: For the honor of Hufflepuff

Sigh

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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## #3 2014-04-03 09:14:17

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

### Re: For the honor of Hufflepuff

The puzzles section isn't the most popular, so some threads get unnoticed. I'll try to solve it.

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## #4 2014-04-27 05:00:04

ManiticMath
Member
Registered: 2014-04-24
Posts: 29

### Re: For the honor of Hufflepuff

So the numbers on the balls add up to the total score right?

When ever I see a brand new new math problem, I want to solve it. Then I want to burn it. Then I want to rebuild it in a new, more streamlined way.

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## #5 2014-04-27 05:02:22

ShivamS
Member
Registered: 2011-02-07
Posts: 3,648

### Re: For the honor of Hufflepuff

You probably won't get the answer to the question, since eigenguy isn't here.

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