Sublime Illusions

Hello 2016! As WordPress politely informed me today, it’s been 8 months since my last post! 8 months! Where has the time gone? I would say I’m again making a New Year’s resolution to blog more, but it’s no longer January…no, not even close. It’s already April!

On my end, the past 8 months was full of agonizing over grad school essays, mad rush towards writing results, mad rush to submitting applications, and, finally, a month of waiting and more waiting, which I say is probably worse than all the rushing and stress of the months prior. But it’s April! And I have a host of amazing, amazing schools to choose for grad school. But then, in typical fashion, comes a month of worrying over choices, of comparing every minute detail between the schools. But no more! I will be going to the place that will best fit my research interests, where I will hear about problems and solutions from some of the best researchers in the field. Where the feeling of excitement over solving hard challenges doesn’t fade, where people work and sleep at odd hours, and where nerdiness prevails. I hope that wherever I end up will make me feel with absolute certainty that I made the right decision by turning my back to the allure of Silicon Valley with all its sparkling glory two summers ago, exchanging that lifestyle for a different type of fun and a different type of nerdy.

But enough about me, to ring in my new New Year’s Resolution (hey, it’s snowing outside so we could believe it’s New Year’s if we try reaaaally hard), I’m going to talk about some of my favorite blogs. You might have heard of some of them:

1. Shtetl-Optimized: This is probably the blog you have heard of/read before. But there’s a reason why it’s so popular and a reason why it’s on the top of my list. Scott Aaronson’s humor, very candid discussions, eloquence, and breadth of scientific discussion on his blog is unparalleled. Here’s a snippet of a post he made in January (regarding his daughter’s third birthday): 

   Lily: Why is the world spinning?
   Me: Because of the conservation of angular momentum.
   Lily: Why is the … consibation of amomomo?
   Me: I suppose because of Noether’s Theorem, and the fact that our laws of physics are symmetric under spatial rotations.
   Lily: Why is…
   Me: That’s enough for today Lily!  Link to post

2. Gödel’s Lost Letter and P=NP: I think I’ve quoted several articles from this blog already. If there is a major scientific breakthrough in computing, then an article will appear on this blog within the next few hours and it is usually one of the most comprehensible and yet informative articles on the topic. This blog has a combination of news-like topics which discussions of esoteric problems in computing. It has a bit of everything, all discussed in an engaging and readable way. Plus lots of puzzles!
3. FiveThirtyEight: I guess some would say this is more website than blog but it has that blog feel to me. Nate Silver’s great. His analysis is great. I’ve been reading his analysis for the primaries when I should have been working on my thesis.
4. Terence Tao’s Blog: I usually don’t like very technical blogs (ironic isn’t it?) But Terence Tao explains concepts wonderfully, especially if you want a summary of a long and intricate paper (like the explanation of Zhang Yitang’s work on the Twin Prime Conjecture and related background). Beware, not for the faint of heart.
5. Theory, Evolution, and Games: I had an interest in evolutionary game theory at some point in the past. I still do, and this blog has some interesting articles.

It’s been a while since I posted a paper and puzzle since I last made a blog post on a paper and puzzle. Other things have been occupying my time, but hopefully, I’ll be going back to my usual schedule with the Paper and Puzzle a Day posts.

So, first, a paper. I’ve been looking into some computational geometry problems and one particularly interesting paper that I’ve been reading is that of what are called “hinged dissections.” The paper’s title is appropriately titled, “Hinged Dissections Exist.” A hinged dissection is one where all pieces of a dissected shape are connected with a series of “hinges.” For example, the following figure shows a hinged dissection of a triangle into a square discovered by Dudeney in 1902.

Dudeney’s hinged dissection of triangle into square.

The paper shows that any finite collection of polygons of equal area has a hinged dissection such that it can folded into any polygon that is in the collection. It’s quite an intricate paper, and I’m still reading some of the details but it’s quite interesting so far. I may post more in future Paper and Puzzle a Day posts about this paper.

Now, for the puzzle. Since we are talking about computational geometry, I will use a computational geometry puzzle for today.

A tangram consists of the following pieces that form a square.

https://www.thepcmanwebsite.com/media/tangram_puzzles/tangram_puzzles_15.shtml

Even though all of the following figures are made from the same seven pieces, why does the equivalent black and white pieces seem to differ by a piece?

Puzzle courtesy of https://www.archimedes-lab.org/workshoptangram4.html.

Now, the solution to Day 5’s puzzle:

The trick to this puzzle is not to be bogged down with probability calculations. There is a way to ensure that at least 5 prisoners are released regardless of what the distribution of hats is (in which case, the fact that the hats are distributed randomly is irrelevant). Each prisoner decides beforehand what her partner is. Then, one partner decides to say the color hat that is opposite the color of her partner’s hat and the other says the same color as that on the partner’s head. When the hats are distributed, the partners find each other and carry out the strategy. If the hats on both partners’ heads are the same color, then the partner who says the same color will answer correctly. If the hats are different colors, then the partner who says the different color answers correctly. Therefore, one partner in each pair will always answer correctly.