Week the Third

This week's most interesting activity was probably Friday's paper-folding puzzle (though, really, I doubt that's news for anyone taking the course. Who doesn't like spending 45 mins of a 1 hour lecture in Uni folding paper? It's awesome.) And of course, because it is Uni, also a bit mind-boggling at first. The puzzle is this: take a strip of paper, and fold it evenly over from left to right, and then continue doing so until you think you've amassed enough information on the art of paper folding, have found the meaning of life, cannot possibly fold it once more even if your entire future depended on it, or simply get bored (well, the last one not so much.) The point of this puzzle is to see if you can figure out the pattern in which the paper folds (whether, when unraveled, the folds are concave or converse) given the number of times it's been folded.

For example, assuming concave = 1 and converse = 0:
1 fold gets: 1
2 folds get: 011
3 folds get: 0011011
4 folds get: 001001110011011
and so on.

It took (I'm willing to admit) quite a few minutes to make sense of these observations. When I finally did, I noticed that (a.) the midpoint created by that first fold always stays the same and (b.) if you work outwards from that center, every new fold on one side is an inverse of the same-distanced fold on the other side! (genius, I know. You can do a mental 'facepalm' if you so desire.) Still, how can you get a pattern from that? Well, after looking at it a bit more in the hopes of prodding my brain into a state of usefulness, I garnered observation (c.) working again from the center, the left side is exactly the same as the pattern you get when folding it 1 time less, or: with n number of folds, the pattern on the left is the same as that of (n-1) folds. The pattern on the right is (as by observation (b.)), mirrored. Yay problem solving! Danny was right; using Polya's approach (outline the problem, concoct a few plans on how to solve it, and then attempt) is more successful than jumping headfirst into the folding.

Useful thing to remember:
A(antecedent) implies C(consequent) is false only if A is true and B is false.

Danny's quote of the week:
In talking about the usefulness of parentheses and variables: "We want to make sure there's still variables left for our grandchildren, but if we're just going from 2 variables to 4, I don't think that makes a huge difference on the store of variables in the world, so we'll probably be ok." And it's true; if ever in need, I'm sure we can move on to the greek alphabet or, who knows, maybe even use kanji.