Fort!

[sniffnoy]

Oh, I made some mistakes in the "week 1" entry - Aaron vs. Sara was not for whatever position I said it was, it was for some other position. Anyway...

So I walk into Kate's room and I see the ? boxes. They had put them up during O-Week to go with the video game theme, but they had since come down. And she was saying about how she thought I would like them, and I should hang one in my room. So I did - it hung low to actually walk beneath, but oh well. I get Kate to come see it - she notes all the boxes in my room and says I should do something with them, like build a tunnel. Of course, the boxes are not very big, and just a straight tunnel isn't very fun. Build a fort? The room's not very good for doing that...

And so Kate goes down to the lounge and takes the pillows off the couches from the upper lounge and pushes two couches and a chair together and makes a pillow fort. Girl Alex sees this and wants another - they make one with two more couches and one of Girl Alex's sheets. Then she suggests making a tunnel between them - Kate suggests instead moving the original fort to connect with the new one, which we do. But it's not secure! I push into position another chair and a bookcase so it's only got one entrance. Now it is complete! I put in some empty soda bottles for weapons, and later Chris adds the tongs from the fireplace.

I stayed inside to keep the fort secure for a while, but it got really hot in there and I left - so since then I've just been making sure someone is guarding it, from inside or out. I also added a sign on the door saying "Tuftsians only!"

Also Ian got a video camera and interviewed me about it and its defenses. Kate really built it but I've largely been responsible for sealing up cracks and such.

Yay, fort.

Also yay for a simple combinatorial proof of Cauchy's Theorem, but boo for stupidly burning my thumbs.

Actually, I suppose that last part requires some explanation, but I don't feel like writing it right now.

OK, here's my list...

Things to write about:
Thumb-burning
New RHs and what's going on with House Associates
Also add the big "YAAAAA!" to the Vassar entry

OK, enough of that...

-Harry
(Also I still owe Nick $15 and I have yet to resolve my debt to Chandi)

Comments

Re: Cauchy?

[sniffnoy.livejournal.com]

No, the group theory one. I'd only before seen the inductive proof in Hirstein. There are really too many Cauchy's Theorems. I suppose the question is, why are they all known simply by that name and not more specific ones? I don't know how a "simple combinatorial proof" of that Cauchy's Theorem would be possible.

Re: Cauchy?

[sniffnoy.livejournal.com]

The book didn't write it in terms of group action, though you certainly could - I don't think that it would really make anything clearer, though. Here, I may as well just repeat it.

Take G a finite group, p|o(G). Let S={(x_1,x_2,...,x_p)|x_i∈G, x₁x₂...x_p=1)}. So |S|=o(G)^(p-1), p divides |S|. Note that as inverses commute, S is closed under cyclic permutation. For a,b∈S, let a~b if a is a cyclic permutation of b and divide S up into equivalence classes. Note that the number of elements in an equivalence class is just the period of one of its elements (if we repeat it over and over), and so must divide p, i.e. it's either 1 (for something of the form {(x,x,...,x)}) or p. So if we write |S|=a+pb, where a is the number of 1-element equivalence classes and b is the number of p-element equivalence classes, we see p|a. Since (e,e,...,e) is in an equivalence class by itself, and p does not divide 1, there must be some other (x,x,...,x)∈S, i.e. some nonidentity element with x^p=e.