Thursday, March 27, 2008

More Crimes Perpetrated by Infinity

Here's another brain teaser involving infinity...

One afternoon at 4pm, an old man was pulling a bucket of water up from his well, and was delighted to discover that ten golden coins were in the bucket, and they were strangely numbered 1 through 10. He took them inside, and stacked them lovingly in his cellar, and puzzled over their origin.

At 4:30, he returned to the well, and drew the bucket again, and shouted for joy when he saw ten more coins, these labelled 11 through 20. Unbeknownst to him, back in his cellar, a curious mouse had found his stack of coins, and stolen one from the bottom. When the mouse heard the old man returning, he fled to his hole, holding coin number 1. The old man entered the cellar and added coins 11 through 20 to his pile.

At 4:45, he could not resist going out again and checking the well once more, and sure enough, he discovered coins 21 through 30. He hurried inside with them, by which time the little mouse had managed to work coin number 2 out from under the pile. Since there were still 18 coins remaining, the old man didn't notice that any were missing as he put coins 21 to 30 on top of the pile.

The old man made further trips to the well in shorter and shorter times, after 7.5 minutes, then 3.75 minutes, etc, each time making the trip from well to cellar in half the previous time (he was a spry old man), each time adding ten new coins to the top of his ever-growing pile. Also each trip, the mouse managed to creep back out and sneak a single coin from the bottom of the ever growing hoard.

(Take a moment here to convince yourself that infinitely many trips to the well will have occurred by 5pm... and kindly suspend disbelief!)

At 5pm, the old man was very tired, having moved infinitely fast, but was also satisfied that he had gathered the infinitely many numbered coins that the well seemed determined to offer. He returned to his cellar to view his spoils, and was shocked at what he saw!

What did the old man see?

And the really crazy question: What would be different if a sparrow had been stealing coins from the top of the pile instead of a mouse from the bottom. It matters! Think of the numbers on the coins.


Daniel said...

Initial hunch is that he'd return to see that the weight of so many coins and so much patter of old man feet had completely destroyed the house down to the foundation, and promptly realized that any coins left would shortly be spent to repair said structural faults.

Drew & Erin said...

Assuming the man made infinite trips, then the mouse stole an infinite number of coins- so upon his ultimate return, the man would find all of the coins embossed with a fancy symbol for infinity. If however, the sparrow stole from the top, then the stack would still contain coins marked with integers.

Rachael said...

I like Drew & Erin's answer - it's smart and cute. Maybe it's the "correct" answer for this problem?

But why should the coins ever come to be marked with infinity symbols? Or, why - if the sparrow stole from the top - should you ever reach coins once again marked with integers when taking from an infinite pile?

In any case, he must be left with 9/10ths of his pile because only 1/10th is ever taken -- although saying so makes me feel a bit like Hofstader's Achilles, about to be spun around by Mme. La Torte.

Chris said...

The answer by Drew and Erin is pretty much what I had in mind (although I imagined that no coins were stamped with "infinity"; I'd imagined the old man's cellar was empty). The mouse is stealing coins number 1, 2, 3, 4, ... so if the old man has 9/10ths left, which numbers would those be exactly? The mouse eventually takes every number even though at every step he only takes a tenth!

On the other hand, the sparrow takes coins numbered 10, 20, 30, ... lots are left inbetween in the old man's collection.

Weird huh? Does that mean the mouse gets "more" coins? Are there "more" multiples of ten than there are integers?

PS - hahah, patter of old man feet!

Rachael said...

Maybe my answer should be instead, that the man and the mouse can never make an infinite trips to and from the gold pile in a finite amount of time. This gets rid of the paradox, and we can say something more reasonable -- that the man's pile usually contains 9/10ths of what he's gathered (there's a moment when it contains a little more before the mouse steals the coin) while the man and the mouse continue to modify it's contents infinitely.

Rachael said...

Is that cheating? ;)

Chris said...

Of course you're right, the reason logic breaks down and turns nine tenths into none is because we allow infinite coins, infinitely fast movement, etc. However, there's a more satisfying (to me) answer...

This mouse/sparrow difference corresponds to a subtle mathematical difference. Since the mouse steals from the bottom of the pile, he ignores the order in which the old man gathers the coins and sort of "does his own thing". Thus the old man gathers lim(10n) coins (limit as n goes to infinity that is), and SEPERATELY the mouse steals lim(n). These amounts are both infinite, so we end up with zero!

The sparrow on the other hand is always stealing from the man's current 'batch' of coins, so it is one limit together; lim(10n-n) instead of lim(10n) - lim(n). In this case, that's lim(9n) which is infinity, so the man still keeps infinite coins!