Wednesday, September 1, 2010

100 Prisoners, 100 Boxes

Herr Professor Doktor Heinrich von EvilFiend has been removed from his post teaching group theory at Heidelberg and placed in charge of the notorious Schloss Konundrum, an innovative POW camp for Allied Airmen.

One day, he calls in the one hundred prisoners in his charge.

"I have a challenge for you, mein Freunds.

"Zere are one hundred of you. Ze castle administration has assigned you each an identity card with your name on it.

"Zere are, in ze post room, one hundred locked postboxen for ze receipt of Red Cross parcels and letters from home containing files. Again, zey have your names on them.

"Ze guards have placed your identity cards randomly in the postboxen, one card in every box."

He sniggers.

"In accordance wiz der pigeonhole principle. Ha ha ha.

"You will proceed to the post room. One by one, you will enter, and each open fifty boxen, looking for your own card.

"If you find your card, you vill be released into the exercise yard. Ze guards will close all ze boxes, and the next prisoner vill be admitted.

"If you do not find your card, ze game is over for ze day.

"Ze next day, ve play again, but ze random arrangement of the cards will have changed.

"If one day, every one of you without exception manages to find your own card, you will have von ze game. And I vill release from my personal supply a supplementary ration of tea to ze whole camp.

There are brief gasps at the generosity of the offer.

"Ve vill play zis game every day throughout ze month of August.

"If, by ze end of ze month, you have not von ze tea,

He pauses dramatically, and continues:

"You Will All Be Shot."

Wing Commander Francis Moustache-Tips is the first to enter the post room.

Alone, he opens the first fifty boxes. His card is in the thirtieth box he opens, and he is released into the yard. The guards close all the boxes.

Prisoner number two, Air Sergeant Cockney Caricature, opens boxes at random. He finds his card in the twentieth box he opens, and is released into the yard.

Together, they wait for prisoner number 3, Special Operative Angus McCryptanalyst.

But he, and the other prisoners, emerge crestfallen.

"I opened my fifty boxes, Wing Commander, but none of them had my card in it.

The game is over for the day.

That evening, our heroes are sitting round their illegal wireless listening for news from home, drinking hot water with a spot of milk.

Moustache-Tips confesses his despair:

"The way I see it, opening fifty boxes out of one hundred to see if one of them contains your card isn't too different from tossing a coin.

"What are the odds that we can toss a coin and get heads one hundred times in a row, and never get tails?

"I wouldn't like to put money on it, even if I did get thirty-one goes."

They try the experiment, and the results are terrible. In thirty-one tries they never get more than six heads before the coin comes up tails, let alone one hundred.

"We're doomed, aren't we, Wing Commander?", says old Jock.

"I've been thinking.", says McCryptanalyst. "and I think we may have a chance."

A week later, they are all sitting around the wireless with a fine supply of German tea.

Even Herr Professor Doktor von EvilFiend is in a noticeably sunny mood, having finally proved the practical utility of his subject in the so-called real world.

What was McCryptanalyst's idea, and how much of a chance did the gallant airmen have?

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