31 votes
Accepted

Join six cities with roads

msh210's user avatar
  • 13.1k
21 votes

Join six cities with roads

This was not as easy as it first appeared. Here is a solution with a bit of geometric flair:
Daniel Mathias's user avatar
19 votes

Covering a 15x15 grid with rectangles

Here is a proof of the lower bound of 13: Why it works (and how I found it):
mathlander's user avatar
16 votes

Join six cities with roads

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Pumpkin_Star's user avatar
14 votes
Accepted

Walking in a random direction

Start at the origin $(0,0)$. If the two directions are $t$ and $u$, the ending location is $\pi(\cos t+\cos u, \sin t+\sin u)$ and the distance from the origin is $$\pi\sqrt{(\cos t+\cos u)^2+(\sin t+...
RobPratt's user avatar
  • 13.1k
11 votes

Covering a 15x15 grid with rectangles

Another proof for the lower bound:
loopy walt's user avatar
  • 20.5k
8 votes
Accepted

The shady puzzle that will keep you in the dark

Something like this (6 walls)? I assume that the polygon is not allowed to self-intersect, as that would not be regarded as “a room”. Otherwise a solution of 5 exists.
Kris Van Bael's user avatar
7 votes

Join six cities with roads

Not as pretty as @Daniel Mathias's but perhaps a bit more obvious:
loopy walt's user avatar
  • 20.5k
5 votes

Walking in a random direction

If we consider the angle between the two walks to be $2\alpha$, in which $0\le\alpha<\pi/2$ (since the case for $\pi/2\le\alpha<\pi$ is equivalent), then the distance between the two points is $...
Sny Smartie's user avatar
  • 2,262
4 votes

The shady puzzle that will keep you in the dark

Here is the solution that I initially came up with:
Will Octagon Gibson's user avatar
4 votes

Join six cities with roads

This is cheating, but at no point does the question say Therefore, I propose that we can do better than the solution given to us in the question. In fact, here is a layout with ...
juicifer's user avatar
  • 10.4k
3 votes

Dissection Puzzle - The Umbrella Stand

I don't know why nobody came up with this answer.
Florian F's user avatar
  • 28.3k
2 votes

Walking in a random direction

Let the smaller angle between the two walks be $\alpha$, where $0\le\alpha\le\pi.$ Using the law of cosines, the distance from where you started is $\sqrt{\pi^2+\pi^2-2\pi^2\cos\alpha}$. Taking the ...
Ronchen Luo's user avatar
1 vote

Join six cities with roads

Don't let yourself get fooled! There is only one solution. It is as folllows:
Florian F's user avatar
  • 28.3k
1 vote

Join six cities with roads

In fact, it is possible with straight lines and looks really cool: Edit: whoops, pumpkin star already got this... upvote his answer plz In fact, this is optimal, since each intersection consists of ...
Sny Smartie's user avatar
  • 2,262

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