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August 15th, 2005
01:44 am
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Riddles: More river crossings
Alright, I admit it: The last riddle was less of a brainteaser than it was a chance to snark.

I spent most of Friday juggling no less than three separate cars -- and two other people's errands -- in order to find a way up to my former house to pick up a clothes washer-dryer and haul it home. I was supposed to get a refrigerator, too, but simply ran out of time, and I'll have to deal with that this upcoming week (hopefully via U-Haul or a quick direct sale rather than by solving riddles just to find a vehicle capable of hauling it).

Anyway. Both of my previous riddles have been completely original, and so now I'm feeling pangs of guilt for posting such a time-worn one. It was wearing at me while I walked to work the next day, so I took some time to think up a new twist on the river-crossing theme:
#1. Alex, Ben, Charlie, Dan, and Eric are in a horror-movie plot out in the remote wilderness, being stalked by a serial killer that always murders his victim the moment that the two of them are together and nobody else is close enough to come to the victim's aid. The five survivors (the other 21 have been murdered) have figured out that the serial killer is secretly one of them, but they don't know who, so they've agreed to go back to civilization to contact police -- and all stick together so the killer won't strike again on the way.

Unfortunately, they come across a wide river filled with alligators. There is a single three-man boat on the bank. It would be easy enough to take two trips and ferry everyone across -- but if the serial killer is left alone on either bank with a victim, or is in the boat alone with a victim, they can murder one final person and escape.

"We'll just have to take our chances," Charlie says. "Without knowing who the killer is, all we can do is cross the river as efficiently as possible and hope nobody gets slaughtered. I call dibs on rowing the boat back from the first group to the second -- that way I'm never alone with the killer."

"Are you crazy?" Ben asks. "You can't call dibs on that. You're the one who suggested the trip to Dead Camper Lake in the first place. I call dibs because I suggested we go back to civilization together."

"Guys," Dave Dan interjects. "We know neither Alex or Ben are the killer, because back near the beginning of the trip we sent them off together to scout that deserted cabin. Let's leave the two of them here and the three of us take the boat across and go home."

"No," Eric said. "The deal was we all go back. I'd feel safer in a group of five anyway. And if I stick with Alex or Ben while you guys cross and someone comes back for us, I know I won't get murdered."

Alex thought for a moment. "Hang on. If you guys stop your squabbling, we can all get across without putting anyone at risk," he said.

How?


Also, as it turns out, amthrax proposed a different river-crossing tale with a similar solution in comments to the previous post ... so I'm going to present it here. Feel free to answer either or both.
#2. There are four people on one side of a river who need to cross the bridge to the other side. Of the four people, one of them is a runner who can cross the bridge in one minute, another is a brisk walker who can cross the bridge in two minutes, the third is a normal walker who can cross the bridge in five minutes, and the fourth person takes ten minutes to limp across. It's night-time and they have only one lantern.

The bridge is rickety and small, so only two people can cross at a time, and one of them must be holding the lamp so they can both see. When a pair of people is crossing, the faster person cannot leave the slower person behind in the dark, so the faster person must walk at the slower person's pace. How can you get all four people across in as little time as possible?


Hint: (The best solution + 4) is not a prime number.

As usual, comments will be screened for a few days to allow everyone to figure these out on their own. And as a reminder, if you never saw the solution to the dartboard problem, answers are visible now.

--

Edit, 8-17-05: Comments now unscreened.

Current Mood: productiveproductive
Current Music: Thanks To Gravity, "Catching Stars on Paper Plates"
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From:taral
Date:August 15th, 2005 04:31 pm (UTC)
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#1. By breadth-first search. Solution:
aabbb<
  bbb>aa
 abbb<a
    a>abbb
   aa<bbb
     >aabbb
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From:taral
Date:August 15th, 2005 05:05 pm (UTC)
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#2. 17 minutes, 5 crossings:
125A<
5A>12
15A<2
1>25A
12<5A
>125A


This one done by computer instead of by hand.
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From:baxil
Date:August 16th, 2005 03:56 am (UTC)
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taral, correct! And quick, to boot. Maybe I'll have to throw an additional problem at you to keep you busy, seeing as how you whipped up a computer program for #2 so quickly:

Given a setup similar to problem #2 with n people, l lamps, and a bridge that can hold p people at a time, what is the generalized formula for finding the lower bound for how many trips t across the river must be made?

(For n=4, l=1, p=2 -- the scenario in problem #2 -- then t=5.)
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From:necama
Date:August 15th, 2005 05:08 pm (UTC)
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We need to leave all three suspects together all the time. So, Alex and Ben cross the river, and one of them gets off on the other side, the other returns with the boat. For simplicity, I'll say Ben gets off.

Then, the three suspects all get in the boat, and cross the river. Ben then takes the boat back across the river and picks up Alex. Thus, everybody is on the other side, and the serial killer can't escape.

It takes 5 crossings.
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From:elynne
Date:August 15th, 2005 05:10 pm (UTC)
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Serial killer riddle - wow, that's an interesting twist on the "duck, fox, grain" one. I have no idea what the answer is, but I'll be waiting curiously to find out.

So you can unscreen this comment, since it doesn't have even a whiff of a spoiler or solution in it. ;)
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From:necama
Date:August 15th, 2005 05:28 pm (UTC)
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For riddle #2, the runner is going to get quite a bit of exercise! Each crossing, except the last, is of the form (slower person + faster person) followed by a return trip to the other side of somebody with the lamp. It is always faster to send the faster person back, so the round trip time is the sum of the slower person's speed and the faster person's speed. The only real parameter to vary is the faster person's speed. In order to minimize that, use the runner each time.

Thus, it will take (10 + 5 + 2 + 2*1) = 19 minutes (the runner only needs to run back twice).
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From:baxil
Date:August 16th, 2005 03:43 am (UTC)
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necama - Correct on #1! For #2, remember the hints listed in the post ... :)
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From:thrames
Date:August 15th, 2005 07:44 pm (UTC)
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#1: Everyone strips down naked one by one, wherein clothes and one's person are checked for weapons. Anyone found with weapons is deduced as the murderer and tie him up, arms behind his back in such a manner that his shoulders are nearly dislocated. If the group is feeling particularly scared, they can go so far as breaking both of his arms and hands. Hell, they might even snap his ankles, too, so he can't run.

Of course, this assumes that no one brings things resembling weapons to Dead Camper Lake unless they're a serial killer. If everyone brings knifes, hatchets, etc. for camping purposes, well, they throw them into the alligator-filled river (only when the sun is up, though) in order to prevent slaughter at the hands of a serial killer. If it's night, they should make a fire and all camp around it until morning with EVERYONE staying awake or two people going to sleep at a time until morning. When morning comes across, they tie a rope to the boat that is sufficiently long to stay attached to the boat across the full width of the river. Then, two people (not Ben or Alex) get into the boat and go across, with one of them getting off at the end and the other staying in the boat. Then, the other person gets into the boat and both go across and both hop off. The serial killer won't kill while on the boat since he cannot get away (the rope is being held by Ben and Alex so they can pull him back and beat the holy hell out of him; that, or they keep a hatchet around to go crazy on his ass once they get him back). Once the boat gets across, the two get off with the guy on the bank. Then, the boat is pulled back by Ben and Alex, who then hop in and row across, safe and alive. (okay, I admit that this is a far better solution than my previous one)

2#Runner and Brisk Walker across, Runner goes back, Runner and Normal, Runner goes back, Runner and Crip back with Crip on Runner's back (assuming that it would slow the Runner down to Normal's pace). Time to get across: 14 minutes.

If you don't like that one, well, do the same thing, except without the Runner carrying the limping guy. In which case, time is 19 minutes.
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From:baxil
Date:August 16th, 2005 10:36 am (UTC)
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I'm unscreening this one early because I got a kick out of the halting-a-horror-movie-situation advice.

#2: Alas, no carrying. And the 19-minute answer, while very common and feeling intuitively correct, isn't actually the fastest. Took me a while to figure out how to break that barrier myself. I provided a(nother) hint in the original post -- read carefully.
From:(Anonymous)
Date:August 15th, 2005 10:31 pm (UTC)
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'Tis me again; let's see if I can get the first new one right.

Andy (tastes like chicken) and Bob(-B-Que) go across first, leaving Chuck, Dash (of salt & pepper) and Eddy(ble) to watch each other like hawks. Bob goes back, and Chuck, Dash, and Eddy go forth in the boat, leaving Bob. Andy takes the boat back, collects Bob, returns, and the five set off on their merry way...

Only to be killed and eaten by Grue, Andy's ex-wife and newly-turned cannibal and her new husband, Harry('d at lunch) just outside of the nearest city. But then, this is a C-movie, where the copious amount of ketchup and Jaws-esque fear-inspiring music matter more than any sense of that pointless thing called plot.

...But that's another story. One I'm not sure my Intestines can take the telling of.

Did I get it logically right? (and hire a new screenwriter for that ending.) And how do they avoid getting eaten by the alligators?
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From:baxil
Date:August 16th, 2005 08:59 am (UTC)
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Anonymous person who may be (but is unlikely to be) the longtime contributor formally known as "Anonymous": Correct on #1! My only question is, are you implying that you believe Chuck is the serial killer, or did you mean to name him "ground" Chuck? ;-)

As for how they avoid getting eaten by the alligators ... well, it's a horror movie, they're almost all going to die one way or another. So chances are good they get attacked while in transit -- quite possibly eliminating their margin of safety and letting the killer strike again.

But, hey, it's the *plan* that's important. It establishes that Alex is being a Conscientious Horror Movie Survivor. (Which, naturally, puts him right in the crosshairs.)
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From:amthrax
Date:August 16th, 2005 04:19 am (UTC)
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Alex and Ben cross; Alex returns; Charlie, Dan, and Eric cross; Ben returns; Alex and Ben cross. Of course, in a bizarre, unexpected, and totally uncalled-for horror-movie twist, Alex and Ben were probably in on it together all along. Alex suggested this solution only because he knew he and Ben could just leave the remaining three behind to die, thus saving them the trouble of having to serialize three more people by hand and the trouble of being discovered by the police if either of them did return with survivors.

And it's probably not fair for me to answer #2. ]:--8) It's interesting that the forms of the solutions are so similar, as if there's some underlying isomorphism between the two. It must be the same river.

You might want leave a warning that unscreened answers to #2 appear in your previous entry.
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From:baxil
Date:August 16th, 2005 10:42 am (UTC)
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amthrax, correct on #1, and I like your horror movie thinking. ;)

You might want leave a warning that unscreened answers to #2 appear in your previous entry.

Consider this it ... although ... uh ... I don't think the solution has actually been posted there yet! There are certainly spoilers and discussion of the problem, though.
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From:kinkyturtle
Date:August 16th, 2005 04:44 am (UTC)
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Puzzle 1:

Alex and Ben go across.
Alex returns.
Charlie, Dan and Eric cross.
Ben returns.
Alex and Ben go across.


Puzzle 2: I've been thinking and thinking but can't come up with a quicker solution than this:

Runner and fast walker cross (2 min).
Runner returns (1 min).
Runner and slow walker cross (5 min).
Runner returns (1 min).
Runner and limper cross (10 min).
Total time: 19 minutes.

...which, given your hint, must not be the best solution, but I'm stumped.
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From:baxil
Date:August 16th, 2005 10:50 am (UTC)
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kinkyturtle, correct on #1! As for #2 ... Re-read my original post carefully. I wrote two hints about the problem I quoted. (Only one hint is labeled as such.) ;-)
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From:natetg
Date:August 16th, 2005 06:45 am (UTC)

Familiar with the second one.

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Alex and Ben go across the river. Alex takes it back. The other three take the boat accross the river, Ben goes back and picks Alex up.

The second one:
The fast two cross first, one takes the flashlight back to the slow two, and then the other fast one goes back.
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From:baxil
Date:August 16th, 2005 10:51 am (UTC)

Re: Familiar with the second one.

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natetg, right on both counts!
From:lhexa
Date:August 16th, 2005 10:38 am (UTC)
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A, B across
B back
C, D, E across
A back
A, B across

And: the two faster ones cross with the lamp, and the sprinter returns, handing it to one of the others. The sprinter crosses without the lamp, since he knows the route already, then the slow ones. Fourteen minutes. I'm guessing this isn't what you had in mind, though. Would you mind emailing me your solution?
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From:baxil
Date:August 16th, 2005 10:46 am (UTC)
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Won't e-mail it to you, but I'll give you a big hint: Despite how different they look, the two problems virtually share the same solution. :)
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From:packbat
Date:August 16th, 2005 09:50 pm (UTC)
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#1: For convenience, I will choose the alphabetically-earliest option when possible.

Alex and Ben cross. They're already established as safe, and the other three together are safe.

Alex brings the boat back. Ben is safe alone, Alex is safe alone, and the other three are safe in a group of three.

The other three cross. Same thing: Alex safe alone, Ben safe alone, the other three safe together.

Ben brings the boat back. He's safe alone, so is Alex, the group is safe together.

Alex and Ben cross. They're safe together (as established), and the other three are safe together.

All are across.

#2: No idea. I don't see the trick. I'll post again when I do.
(Deleted comment)
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From:baxil
Date:August 16th, 2005 10:42 pm (UTC)
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packbat, correct on both counts!
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From:dreamflow
Date:August 16th, 2005 11:01 pm (UTC)

*amused*

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For #1, the guys should have Dave hold down Dan while they beat him to death. Dave offered a suggestion that would result in nobody else dying and all of the suspects getting back to civilization -- something the serial killer wouldn't do. So his evil twin Dan, who's also travelling with the group, is obviously the serial killer.

Then the remaining five of them can get back to civilization at their leisure.
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From:baxil
Date:August 16th, 2005 11:02 pm (UTC)

Re: *amused*

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Aigh. Fixed.
From:(Anonymous)
Date:August 17th, 2005 09:53 am (UTC)
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#1. Alex and Ben cross first. Then Alex (or Ben) returns. Charlie, Dan and Eric all cross. Then Ben (or Alex) returns. Then Alex and Ben cross again, putting all five people on the far side of the river. And the three potential murderers are kept together. But as long as you were naming characters A, B, C, D, and E in a horror movie shouldn't at least three of them have been female names?

#2. The key is to minimize the time spent bringing the lantern back. So have the runner escort each person across. That will take 19 minutes.
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