Thursday, October 11, 2007

Skeptic's Circle #71: Medium

Following are the medium-difficulty problems for this Skeptic's Circle. Some math skills may be useful here.

Poor Poisoned Pinheads

You just knew one of these days the Moon Hoaxers would push Buzz Aldrin too far, and they finally have. This one started with a post at Space Files that showed a clip of Buzz explaining that the "UFOs" seen during the Apollo 11 flight had actually been identified. Well, a group of five hoaxers really didn't like this, and so they tried to take Buzz out of the picture.

They prepared a special poison which would cause retrograde amnesia in the person to consume it. It's extremely potent, and you'd have to dilute it to homeopathic portions to make it safe, but it takes a day to kick in. They sneaked into Buzz's wine cellar (Didn't know he had one? Probably because I'm just making it up), and went about spiking his bottles. However, they'd only managed to spike a single bottle before Buzz found them. In the ensuing chaos, Buzz managed to catch them all, tie them up, and get them to spill the beans on their plan, but he lost track of which bottle they'd spiked.

At this point, Buzz is so pissed off that he cares more about figuring out which bottle was spiked than whether he might inflict amnesia on his new prisoners. His cellar has a total of 874 wine bottles in it he has to check. With his five prisoners and one day for the poison to take effect, what's the shortest time it might take him to figure out which is the poisoned bottle, and how can he do it in this time?

What Wifi Woo?

Sandy Shwarc reports that there's been some scare over the ill effects of all the elctromagnetic radiation going through the air, but I'm not buying it. Personally, I think this is all just an excuse to avoid having to work. Confused? Let me explain.

Take the new Ultra-Mega-Awesome Wifi tower built the other day. It had 1000 power cords going from the bottom to the top, and not one of them was initially hooked up to anything. Worse, they're all tangled in the middle so it's impossible to figure out which bottoms of wires correspond to which tops.

Now, some poor shmuck has to go and sort them all out, and the only tools he's given are a battery and a lightbulb. They somehow expect him to to hook up the battery at the bottom to a couple of wires, then go to the top and see which wires he can connect the lightbulb to to sort them out. Maybe he could pull a few tricks like tying some wires together at the bottom or top to make long wires, but it's still going to take him quite some time.

With that job ahead of him, you can see why he'd want to believe it shouldn't be done. Maybe if we could help him out and figure out the most efficient way to solve this problem, he'll be a bit more likely to accept Wifi. The tower's pretty tall and the only way up is by stairs, so he'd probably appreciate most if we could help minimize the number of trips he has to take, regardless of how much work he has to do at the top or bottom. How can we do this, and what is the minimum number of ascents and descents required?

Manic Motor Mythbusting

Paddy K recently demolished some myths about the efficacy of so-called "green" cars, and now that he's done with that, he wants to actually demolish the cars. To do that, he's sent them to - who else? - the Mythbusters team. Now, they can't simply destroy the cars, they have to do it in an interesting way. Here's what they've set up:

Two cars are set up with a robot controller each, and this robot will have some programmed instructions. The two cars will be airlifted and dropped at a random point on a very long line marked out in the desert. Both cars will be initially facing north along the line, but we don't know which one will be in front of the other. Once they hit the ground, their parachutes will detach and remain where they landed. After this, the robots will take control of the cars and start driving according to their programming.

Now, there are a lot of fancy things that could be done in programming them, but Adam has decided to give Jamie a challenge. The first restriction is that both robots must use the same instructions. The second restriction is the the instructions are limited to the following commands:

Move 100m forward
Move 100m backward
Skip next command unless a parachute is nearby
Go to [label] (any line may be labeled for this purpose, and as many labels as necessary may be used. This line means the robot's "mind" will go to the instruction at this label and start working forward from there.)

How can the robots be programmed to guarantee a collision? Try to use as few lines of code as possible.

Paddling Pooch Problem

Well, Bronze Dog has done it again. He went and pointed out that made a completely unfounded accusation that the way many fundies practice is akin to the devil worship atheists and D&D players have been accused of. Now, they're accusing him of being a devil (as a talking dog made out of bronze, he's definitely not normal, but I don't see many mentions of devils like this in the Bible).

In the hot pursuit, eventually Bronze Dog found himself paddling in the middle of a circular lake, with a fundie waiting on shore for him. This fundie can run about four times as fast as Bronze Dog can swim, though fortunately he never learned how to swim and won't enter the water. If Bronze Dog can make it to land, and the fundie isn't waiting right at the shor for him, he should have no problem outrunning him.

The fundie isn't going to listen to anything Bronze Dog might say, so tricking him is out of the question. Bronze Dog is pretty fit and can paddle for quite a while, but he'll still likely run out of energy before the fundie gives up, so he's going to have to do something. Assuming the fundie makes his best effort to catch Bronze Dog, what can Bronze Dog do to guarantee he'll be able to get to shore safely?

Action/Adventure Akusai

Akusai recently "psychically" predicted a phone-call from his long estranged cousin. After hearing that his father had gotten this call, he decided to go out and surprise this cousin with a visit. Well, things didn't go quite as planned on his journey, and, long story short, he found himself at the mercy of a deranged hermit.

This hermit offered to play Akusai in a game, with his freedom at stake. The game is played on a circular table, and the players take turns laying circular disks down at any location on the table. If a player can't lay down a disk anywhere without overlapping a disk that has already been placed, they lose. The hermit gives Akusai the choice of playing first or second. Which should he choose, and what strategy should he use in the game?

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John M said...

Manic Motor Mythbusting is quite a clever puzzle, I've not seen one like that before. I can do it in 8 lines.

Infophile said...

Yup, I believe eight lines is as good as it gets for that one. However, if we add a slight modification and say that each line of code takes the same amount of time to execute, there's a way to optimize further.

exarch said...

I don't seem to be able to get the Poor Poisoned Pinheads to figure it out in less than 4 days, and I get the feeling it should be possible to do it faster than that ...

For What Wifi Woo?, can the little lightbulb be seen from the bottom of the tower if it's lit? Because that saves at least half the trips going back up to do more puzzling/labeling.

The Manic Motor Mythbusting is interesting. Yeah, I think eight lines is the minimum.

Paddling Pooch Problem seems like a case of continually swimming towards the exact oposite shore of the lake from the fundie. This will likely result in a circular swimming path starting in the center, with him essentially racing around 3/4 of the lake.

Action/Adventure Akusai sounds like the pizza puzzle, except you're not allowed to overlap at all. And the guy is going to try to counter you. Come to think of it, it's nothing like the pizza puzzle. I think much depends on the size of the disks in relation to the table.

Infophile said...

Poor Poisoned Pinheads - You're right, there's a better way.

What Wifi Woo? - It can, but it's possible to do it in the same number of trips either way. Also, I should have noted that it's possible to touch the battery directly to the light and use them as one unit if you wish.

Action/Adventure Akusai - Nope, there's a solution that works regardless of the sizes of the disks.

John M said...

Action/Adventure Akusai: you need to go first, play dead center, and thereafter play symmetrically.

exarch said...

Concerning the poisoned wine-bottle, I reduced the number of bottles for the second tasting to 33 or less, but that means there's still a 15% chance that a 3rd tasting session is needed. Unless I missed a combination ...

exarch said...

Nope, got it. 2 tasting sessions is enough. I *did* miss a few combinations. If my calculations are correct, you could even test as many as 992 bottles in two days.

Infophile said...

Yup, 2 sessions is the best possible, although my calculations will take you up to a possible 1024 bottles in that time.

exarch said...

Hmm, I was thinking that for the second tasting-session, you needed to end up with 31 bottles instead of 32. But now I can't remember what the reason for that was.

Anonymous said...

What Wifi Woo: WHat is the objective of the puzzle? I mean, is the schmuck going to have to label the wires (1,2,3...) downstairs and upstairs to identify which one is which? To an outside observer, how would the identification have been accomplished?

Infophile said...

Pretty much. He has to label each wire at the top and bottom to match them up, so for any given wire at the top and bottom, it's possible to figure out which top or bottom it connects to.

miller said...

Ok, haven't figured out the transparencies, but the paddling pooch, I can figure.

So the key is to always stay on the opposite side of the lake's center from the fundie. Bronze dog's velocity has a radial component and a tangential component. He moves outward in a spiral such that his motion around in a circle matches the fundie's, and such that the radial component is maximized. This spiral path may only reach as far as R/4 from the center, but as soon as Bronze Dog is closer than piR/4 to the edge, he may swim directly to shore, and the fundie will not be able to reach him.