gavinksong

This problem is extremely hard. One idea for finding a solution via computer...

I looked through the first few replies, and I now realize that this problem is even weirder than I initially thought. But here's another thought: plasmid's method of assuming that the first two points are the closest/farthest doesn't work on a finite plane, so maybe it's wrong in the infinite case. But at the same time, maybe not, because stating that the first two points are the closest/farthest clearly warps the joint probability distribution of their positions on a finite plane (because obviously they are more likely to be closer together if they are the closest vertices of the triangle), but not on an infinite plane. Probability and infinity are pretty fragile things it seems.

Ohhhhhh, this one is weird. I didn't read the thread for this one, so forgive me if I am redundant, but here are my thoughts...

This problem is pretty similar to a host of puzzles that are variants of a classic puzzle known as "The Hardest Logic Puzzle Ever". I think one presented on BrainDen's official site. What Koren is doing is employing a device called a counterfactual, which is the general solution to the aforementioned puzzle.

I think this is pretty simple. Isn't Koren correct?

Mind = Blown

I don't have access to a computer, and I don't know how to use the spoiler mask on a phone. Hopefully, this post doesn't really warrant one. Could someone point out how to mask things using only text? Anyways, Karthick, I think you should be careful with your assumptions. I'm not saying they are wrong, but your approach seems to assume that the optimal placement of circles will be the one that wastes the least space. Note that the optimal placement is actually the one where the maximum distance of any circle from the "center" is reduced, which isn't necessarily the one where the circles take up the least amount of total space. PS. BMAD - You're really creative; you know that? I admire your prolificity in puzzle making.

Woah.

Phil actually has a point. It is hard to actually draw any solid conclusions without making the unrealistic assumption that all of the explorers are intelligent and think at the same speed. A more complete version of the same puzzle is actually posted on the official site, http://brainden.com/logic-puzzles.htm, under "Masters of Logic Puzzles (Dots)" - although the story is more like "Head Bands", another similar puzzle also listed on the same page.

How does this work exactly? And how do you tell the nines from the sixes?

This is correct. One down, two to go. These should be pretty easy after looking at Anza's solution.

Well, the pizza is always the same size, regardless of how many pieces you cut it into.

I had random endpoints in mind, but I suppose either of the other methods might make interesting problems too. Disclaimer: since this is an original problem, I am not quite sure of my own solution. I just tried solving the problem with the "random midpoint" and "random radius" methods. I think they may be easier (if I'm not wrong). I encourage all of you to try solving the problem using all three methods outlined in Anza Power's link. Whoever finds all three solutions wins a free compliment.

I had random endpoints in mind, but I suppose either of the other methods might make interesting problems too.

Nice try, harey! But you should not think it is so simple! The cuts are not necessarily through the center.

Hey, kids. Here's a pizza. I'm going to make N random straight cuts. Enjoy! How many pieces do you expect?

That is exactly right. You didn't need make the assumption that it was a square or use coordinates, but using those formalities is probably good practice... Good job!