Thursday, October 11, 2007

Skeptic's Circle #71: Easy

Following are the easy problems for this Skeptic's Circle.

Personalized Perfume Peril

Bad has been captured by perfume manufacturers he recently angered. They're utterly convinced that their personalized perfume works, so they've arranged an elaborate puzzle to force Bad to admit this as well. They've thrown him into a dark room, in which 100 disks of perfume samples are placed. Each disk has a different scent on each side, taken from "completely different" people. The sides are color-coded red and blue for easy counting later, but it's too dim in the room for Bad's eyes to be able to make out the colors. Bad's task is to sort the disks into two piles, each with the same number of red sides facing up. If he succeeds, he'll be set free, but if he fails, he'll be shot. He'll also be shot if he tries anything "clever" such as balancing disks on their edges or throwing them out the window.

Bad knows that the task is hopeless, so he pleads with his captors to give him a hint, just one little hint. Eventually, one of them takes pity on him and gives him the following hint: In the initial set-up, 48 of the disks are placed with the red side facing up. With this information, is it possible for Bad to come up with a plan to separate the disks into two piles with the same number of red sides facing up? If so, how?

Creative Cake Capers

Christian of Med Journal Watch has been attempting to combat rumors that overweight women should lose weight when they get pregnant. In his quest to inform women that this isn't always going to be the case, he's stumbled upon one particularly tricky customer.

This particular woman has quite a sweet tooth, and figures that since she should expect to lose some weight, she can afford to binge a little. She's got a nice big rectangular cake which she plans to eat, but Christian is able to convince her down to only consuming half of it. However, when they get out the cake they find that her husband has already cut out a rectangular slice from it. The woman wants half of the full cake, but she settles on half of the remainder.

The woman knows that each slice takes away a small amount of the cake and so she won't let him use more than one. She also won't settle for anything less than half the cake, but Christian wants to make sure she gets no more than half. So, the problem is, how can Christian cut the cake perfectly in half with a single cut?

And no, making a big horizontal slice through the center of the cake isn't an option, as the cake has icing on the top and thus isn't symmetric in that direction.

Popping Placebo Pills

After a recent article by Mark Hoofnagle on the diagnosis of Chronic Lyme Disease, an enterprising researcher decided to conduct a placebo-controlled study to see if there was any benefit to the use of the antibiotics typically prescribed for treatment of it. During the conduction of this experiment, the grad student assigned to sort out the bottles of pills (some placebos, some real) ran across a problem when the record sheet was smudged and she couldn't identify whether five of the bottles had the placebos or real pills. Counting up the identified bottles, she figures out that one of the bottles should have real pills, while the rest should be placebos.

She does some quick research and figures out that the easiest way to tell apart the pills is by weight. The real pills weigh 1 gram more than the placebos, so a few comparative weighings should be able to sort them out. However, when she gets to the lab to weigh them, she finds that a lab class is going on and there's a huge line-up to use the scale. The pills need to be in before the class is over, so she'll have to put up with it. Asking the instructor for permission to step in, he relents, but allows her only the time to perform one weighing. Is it possible for her to sort out which of the bottles contains the real pills with only one weighing? If so, how?

Perilous Peace Problems

The Factician recently dismantled some biased thinking which led to the bizarre conclusion that peacetime is more dangerous than war, and for his trouble, he received a mysterious package in the mail one day. Opening it, he finds a corked wine bottle with a pill inside of it and a note. The note reads:

So, you think you're a smart guy pointing out accidental deaths, huh? Well, here's the situation: When you opened this box, a specially-prepared poison was released into the air. The pill in the bottle is the antidote for it, but I've got a little challenge for such a smart guy. I want to see if you can get the pill out of the bottle without removing the cork or breaking the bottle. If you do either of those things, I can't guarantee you won't have an "accidental" death of your own.

The Factician suspects they're just bluffing about the whole thing, but he decides to go ahead with it anyways, as he's already come up with the solution. What does he do?

Crazed Canting Christians

Romeo Vitelli tells us a story of some strange convulsing women, which is apparently a miracle. Personally, I'd chalk up curing something like this to be more miraculous, but I guess that just goes to show I don't have faith.

Anyways, it seems that a group of 20 of these women decided that it if their strange behavior led to their death, they'd go straight to heaven. So, they set up a weird ritual suicide type of thing, where the 20 of them get out on a 100-meter long raft in the middle of the ocean, each randomly selecting a direction to face and a starting point from marks laid out every meter (the first a meter from one end, up to one a meter from the other end).

At a cue to start, each woman starts convulsing forward at 0.1 m/s. If she bumps into another woman, both will immediately turn around and start walking in the other direction. They'll keep walking until they inevitably all fall off one end and (hopefully) meet their end. If the woman are miraculously set up in the right configuration, what's the maximum time it might take for all of them to fall off the raft?

Hidden Handbook Hassle

After a perilous journey into the land of the woos, Skeptico managed to escape with the Woo Handbook. However, he's now on the run from woos who want it back, and he needs to pass off the book to a fellow skeptic. He's under close observation, so he won't be able to make personal contact with this other skeptic, but they've arranged a plan to get the hand-off to take place. The plan was to have them both lodge at the same hotel, and during the night, hire one of the employees there to pass it off.

But they ran into a problem with this plan, as it turns out that everyone that works at this particular hotel is a rabid kleptomaniac and would steal anything in their hands before passing it off to another guest. Each room did come equipped with a small portable safe though, and these are equipped with tracking devices to make sure no guests would run off with them. It also fortunately means that the employees wouldn't run off with them, so the trick is to transport the handbook within a safe.

Of course, there's still a catch. The safes are closed through a clasp, which a padlock can be put on to steal it shut. The padlocks can only be unlocked with keys found in the hotel rooms and safely wired down, and each key is unique to each lock. So even if the handbook were passed off in a locked safe, Skeptico's friend would have no way to unlock it. Is there any way to solve this problem without either Skeptico or his friend leaving their room and thus risking being caught by a rabid woo?

Weird Water Woo

PalMD recently made a post discussing hydrogen peroxide woo, and, true to the nature of events these two weeks, has been kidnapped by a crazed woo and forced to solve a logic puzzle if he wishes to live. He's locked in an empty room and given a glass that's around half full of water. His task is to determine precisely whether the glass is half full, less than half full, or more than half full. There are a few ways to do this, but some of them are pretty tricky and inaccurate if you don't have a very steady hand. What are some good methods?

Screwy Scarfe's Secrets

The guys at Holford Watch recently exposed Chistopher Scarfe as the fraud he is. Unfortunately, they didn't realize that Scarfe is also an insane supervillain, and they were promptly captured and imprisoned in his mountain fortress. They managed to escape from the fortress (Scarfe forgot to lock the cell door), but on the way out they came across a rickety bridge they'll need to cross.

It's night, and they only have one flashlight among them which anyone crossing the bridge will need. The bridge is unable to support more than two people at any time, so they'll have to make multiple trips to get everyone across, passing off the flashlight as necessary. The guys each incurred various injuries in the escape, so they're all able to move at different rates. One guy is pretty much uninjured and could make it across the bridge in two minute. Another of them is a marathon runner and could easily do it in a single minute. A third stepped on some caltrops on the way out, and it will take him four minutes to cross. The fourth had his leg broken in a fight with a guard dog, and it'll take him eight minutes to cross (if there aren't actually four guys behind this blog, pretend there are). Of course, if two are crossing together, they have to cross at the speed of the slower person.

Scarfe's hot on their tails, so they want to get across the bridge as quickly as possible. How can this be done, and how long will it take them?

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Lord Runolfr said...

OK, the Perilous Peace and Hidden Handbook puzzles I can solve easily. Not so much the others... yet.

Anonymous said...

For the hidden handbook, will more than one padlock fit on a safe?

Infophile said...


exarch said...

I've heard Personalized Perfume Peril before. One pile of disks will be 48 disks.

The Creative Cake Capers doesn't sound like the kind of riddle you can answer without the appropriate drawing skills / illustrations. Much depends on the section that's already been removed, and any cut you make will almost certainly use the existing gap to divide it in three.

For Popping Placebo Pills, I wonder if taking a bottle of pills away on either side of the scale counts as a second weighing? If not, I think I have the answer.

The Perilous Peace Problems, well, there's an obvious answer that just seems wrong but might not be (i.e. drilling a hole).
And what exactly qualifies as "removing the cork" (directly or indirectly)?

The Crazed Canting Christians
is just math.

Anonymous already gave the answer to the Hidden Handbook Hassle.

For Weird Water Woo: scales?

As for Screwy Scarfe's Secrets, it should take them 15 minutes to get everyone across.

Infophile said...


Creative Cake Capers - There's actually a simple method that can work regardless of the nature of the piece removed. It does require a little math trick to realize that it works, though.

Popping Placebo Pills - That would actually count as a second weighing.

Perilous Peace Problems - I'd count drilling a hole as a form of breaking the bottle. As for removing the cork, if you pull the cork out of the bottle (or something causes that to happen), it's been removed.

Weird Water Woo - No scales are present in the room.

Iggy said...
This comment has been removed by the author.
Bret said...

Creative Cake Capers
Cut the cake diagonally? Corner to corner.

Popping Placebo Pills
Label the bottles A, B, C, D and E, place 1 pill from A, 2 from B, 3 from C, 4 from D, 5 from E. Weigh the pills, and the weight should be 15w + extra, where w is the weight of the placebo pills. the extra weight should be a multiple of 1g, where 1g = A, 2g = B, 3G = C, 4G = D, 5G = e.

VWXYNot? said...

Does pushing the cork in count as removing it? Or do you have to melt the bottle / burn the cork out?

Infophile said...

Cutting the cake diagonally won't work, for instance if the lice removed is on one side of the diagonal.

For the bottle problem, pushing the cork in is in fact the classic solution. The other common solution is to somehow destroy the cork or drill a hole through it without taking it out of the bottle. Melting the bottle, however, is an interesting new one.

Francis said...

If the cake is A * B and the slice is C * D, then one half the remainder is (AB - CD)/2. A straight slice parallel to side A such that A*X equals (AB-CD)/2 means that X (the point along side B at which to cut) equals (AB-CD)/2A. That reduces to B/2 - CD/2A.

This should work but it does require precise measurement. But I didn't see any prohibition against rulers.

Infophile said...

The problem with that method is what happens with very big pieces cut out of the middle of the cake. In that case, a cut determined by that method would overlap with the part that's been removed, and then wouldn't have exactly the right value.

Mike said...

For the pill in the bottle puzzle. This is probably wrong but i think I have the simplest solution. Just pull the cork or break the bottle, then take the pill. Assuming the poison gas thing is real, then you dont really have to concern yourself with some silly challenge.

Infophile said...

Well, if you really want to get down to it, the way I set the problem, up, the most likely explanation is that there is no poison gas, and instead it's the pill that's poisonous. But that kinda ruins the logic puzzle, now doesn't it?

narc said...

For the pill in the bottle puzzle: Get a large syringe full of water. Insert the syringe through the cork and inject the water into the bottle. Wait until the water dissolves the pill, then turn the bottle upside down and use the syringe to draw out the liquid. Drink said liquid while basking in your own glory.

exarch said...

In other words, pushing the cork into the bottle, or destroying it, don't count as removing the cork.

exarch said...

Okay, concerning water woo, you'd still need aditional props, but no scales:

Mark the current water level, take something flat to cover the top of the glass, and turn upside down. Observe the new level. If it the water level doesn't reach the mark, it's half empty, if it goes beyond the mark, it's half full, if it's exactly on the mark, it's exactly half a glass.

Infophile said...

There's still a way to do it without any props.

Anonymous said...

For water woo, it involves tilting the glass right?

If the water level is exactly at half, when tilted - the surface of the water will meet the the top edge and bottom edge of the glass precisely. This of course assumes you have a uniform glass and a very steady hand =)

Infophile said...

Eh, even if you don't have a steady hand, you can just spill a little and declare "less."

Anonymous said...

Please elaborate on the Hidden Handbook Hassle solution...

Anonymous said...

Crazy Canting Christians:

Correct configuration would require 10 women standing on the left side of the raft (marks 1m, 2m,...10m), and the other 10 standing on the right side of the raft (marks 90m, 91m,...99m). Both groups of women start walking toward each other. The time it takes for all of them to get off the raft would be determined by when the two most interior women (10m and 90m) get off. Each of them will bump each other and one from their respective groups alternatively till the 9 women from each group go off the board. The math then is simple:

time = 40/0.1 + 9 (0.5 + 0.5)/0.1 + 50/0.1 = 990 secs.

Infophile said...

Right answer, but there's a much more elegant way to look at and solve the problem.

miller said...

For Crazed Canting Christians, the key observation is that if we ignore the identities of the women, then each bump is effectively the same as if the women walked right through each other. So as long as we have a woman at either 1m or 99m facing towards the center, it will take 99m / (0.1 m/s) = 990 s to end.

miller said...

For Creative Cake Capers, can we assume that the rectangle cut out always shares a corner with the original cake?

If so, the cake can be (mentally) divided into two rectangles. Find the centers of those two rectangles, and cut on the line created by them.

If not, I've no clue. Also, this may result in three pieces, but maybe that's ok?

Infophile said...

Got it for Crazed Canting Christians, and for Creative Cake Capers, you can't make that assumption, but you're getting ridiculously close to figuring it out.

miller said...

I've got it!

For Creative Cake Capers,

You cut a line that goes through the center of the original cake and through the center of the removed rectangle. No matter the angles, you will always get one half of the original cake minus one half of the removed rectangle.

Infophile said...