Archimedes

Hello everyone!

Mathematics is often presented as if it's a body of knowledge which people have known forever.  But that's not the case; modern math has come about through the work of millions of clever people over thousands of years.  I don't have time to talk about everyone, of course, but I thought it would be interesting to at least discuss a few of the great mathematicians.  There aren't many mathematicians greater than Archimedes, so he seems a natural first choice.

(Portrait by Domenico Fetti (1620).  Image taken from Wikipedia.)

Archimedes of Syracuse (c. 287 - 212 BC) is perhaps most famous for a case of public nudity.  As the story goes, a king suspected his crown was not pure gold, but had instead been made with a mixture of gold and silver.  He asked Archimedes to find a way to check without damaging the crown.  When Archimedes stepped into the bath that evening, he noticed that the water level rose in response.  He realized that he could submerge the crown in water and use the change in water level to find its volume, and thus its density, which would tell him what the crown was made of.  Not bothering to get dressed, he ran through the streets shouting, "Eureka!" ("I have found it").
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Permutations

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This week, I'd like to talk about permutations.  A permutation of a list is another list with the same elements in some order.  For instance, imagine a deck of playing cards.  Any way you shuffle the cards, you get the same cards, but you might get a different order.  So the resulting ordering is a permutation of the original ordering.

It turns out that for permutations, the specific items in our list don't matter all that much.  We can rename the items in the list however we like.  For instance, we could take our deck of cards and give each a different color, or we could number them .  Then we can think of shuffling the cards as rearranging colors or permuting the numbers through .

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Towers of Hanoi (Part 2)

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Last week, we talked about the Towers of Hanoi.  Today, I'd like to look at the same problem from a totally different direction.  As before, we begin with three pegs and a bunch of disks.  This time, though, we're not so concerned with stacking all the disks on one peg.  Instead, let's look at all the possible arrangements of disks.

[IMAGE HERE SOON]

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Towers of Hanoi

Hello everyone!

I'm back, and I've got lots of exciting math to share, so let's get to it!  This week, I'd like to talk about a classic puzzle called the Towers of Hanoi.  To start, we have a board with three pegs, as shown below, one of which has a stack of disks on it.  We want to move all the disks from the starting peg to either of the other pegs.hanoi_basicHowever, we have two constraints: we can only move one disk at a time, from one peg to another, and we can never place a disk on top of a smaller disk.  So in the example below, we can move the middle disk onto the larger disk on the left, but not the smaller disk on the right.hanoi_move Continue reading Towers of Hanoi

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