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Topic: Catcher in the fog (Read 674 times) |
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alien
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3 kids are standing in the plain shrouded in a dense fog. There are four flags and each points to a direction be it north, south, west or east. There are 2 crevices in the shape of a cube with a side 100 meters long 50 meters in front of them but the kids don't know if the crevices are placed north and south or west and east of them. If you could draw a 100 meters straight line passing through two flags of opposite directions it would cut the edge of crevices closest to them in two parts where one part is 60 meters long and the other 40 meters long, and you could also draw two more lines parallel to the first line between them connecting the corners of the same two edges, and all of the three lines would be of the same length. Two kids need to be blindfolded before they start to run at the same pace with their backs always turned to each other and they need to run for half an hour. They are both autistic in that they can run in a straight line for more than an hour without deviating from it not even for a millimeter, they are infallible when orienting in space and they can infallibly count seconds. Holden is the oldest and smartest kid and he needs to give them advice in what directions will they run. What is the best advice he can give them?
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| « Last Edit: May 5th, 2005, 10:37am by alien » |
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Grimbal
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Re: Catcher in the fog
« Reply #1 on: May 5th, 2005, 9:59am » |
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If I understand:
- 3 kids are lost in the fog.
- 2 of them have to run in opposite directions.
- there are 2 huge cubic holes nearby.
- they don't want to fall in the holes.
- some geometric rule tells something about the hole's placement.
Questions:
- The flags, are they planted 50m away in the 4 directions, or is it more like a signpost indicating these directions where the kids are?
- How far do they see in the fog? Can they see the one crevice 50m away? If they can not see anything in the fog, why blindfold them?
- Can they communicated? Yell "I ran 100m and am still alive!".
- Can they change directions? Can they stop? If no, I don't see the point of counting seconds. If yes, I'd advise them to stop immediately or run in a close circle.
- What is the goal of the whole game after all?
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rmsgrey
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Re: Catcher in the fog
« Reply #2 on: May 5th, 2005, 10:32am » |
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To check whether I've understood it correctly, the problem is equivalent to:
Pick an angle 0<=A<90.
Half the time set B to A, the other half to A+90 (at random)
Then, taking 3 consecutive squares of a chessboard (BWB), locate a point 1/10 of the side length above the center of the white (central) square. Produce a straight line (in both directions) at angle B from the vertical. The aim is to minimise the number of times the line intersects the two black squares.
What value should you pick for A?
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alien
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Quote:| The flags, are they planted 50m away in the 4 directions, or is it more like a signpost indicating these directions where the kids are? |
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The latter. The closest two edges of crevices are located exactly 50m away from the autistic kids, so the distance between these two edges is exactly 100 meters.
Quote:| Can they see the one crevice 50m away? |
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No.
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No.
Quote:| Can they change directions? |
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After they start running no.
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They can stop only after 30 minutes.
Quote:| What is the goal of the whole game after all? |
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Not to fall in the hole. 4, 5, how to stay alive? 6, 7, go to hell or go to heaven!
rmsgrey, it is easy to say the solution, that might be considered the best solution, without fancy geometry.
Quote:| What value should you pick for A? |
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I suck at this.
Quote:| some geometric rule tells something about the hole's placement. |
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That's right. I could draw a picture but this should be clear. Keep in mind that crevice can have 8 potential exact positions, but also keep in mind that there are two crevices and together they can have 4 potential exact positions.
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| « Last Edit: May 6th, 2005, 6:58am by alien » |
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Deedlit
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Posts: 476
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Re: Catcher in the fog
« Reply #4 on: May 5th, 2005, 4:20pm » |
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There are 8 angle intervals in which you will survive for three of the four possible crevasse placements, rather than the usual two. One of them is at between arctan (4/5) and arctan (5/6), and you can obtain the other 7 through the symmetry of the square. In that interval, you will avoid both north/south placements, and you will also avoid the southern east/west placement.
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Deedlit
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Re: Catcher in the fog
« Reply #5 on: May 6th, 2005, 2:34pm » |
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on May 6th, 2005, 9:58am, alien wrote:| How many lives can be saved for sure? If you don't say how many lives your solution ensures saving I won't reply even if it's correct. |
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If you're going to be a stickler for a certain answer, it helps to actually ask for that answer. Notice that Grimbal asked you to be more specific about what you wanted, and you gave this cryptic reply:
Quote:| Not to fall in the hole. 4, 5, how to stay alive? 6, 7, go to hell or go to heaven! |
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So nowhere did you say that you wanted to "save as many lives for sure". The strategy above ensures that at most one person falls in a hole, and then only if the crevasses are place east/west. So if the third person then goes straight north, he'll survive if someone died, and at least two lives are saved. Of course, it makes more sense to go in one of the 8 intervals mentioned above - 4 of them are in a north/south direction, so any of those 4 will still save two lives, and also give each person a 3/4 shot at survival.
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| « Last Edit: May 6th, 2005, 5:48pm by Deedlit » |
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alien
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The riddle is about autistic kids: Holden isn't autistic and he won't fall in the crevice. And as you can see from the picture if a kid goes straight north it's impossible that his life will be saved for sure because the crevice might be there. Let's say that east represents an angle of 0° (north 90°, etc.). Between arms of opposite angles autistic kids need to run in order for at least one of them to stay alive for sure. You don't have to specify those angles because that's a bit difficult, but try to describe a straight line on which the kids will run (you could draw a picture if you want) that will guarantee that at least one kid will be saved, and please try to conclude if a) there is a solution that ensures saving one kid, b) there is a solution that ensures saving both kids or c) there is no solution that guarantees saving either kid.
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| « Last Edit: May 7th, 2005, 4:28am by alien » |
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SMQ
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Well, since you put it that way:
Below is a diagram with the four possibilities shown overlapped, each in a different color.
1) Every path from the center crosses at least one potential crevice, so there is no direction which is guaranteed to save both "autistic" kids. (quotes because I've known a few really great autistic people and the whole statement of the riddle strikes me as somewhat offensive)
2) Any path on an angle between tan-1 4/5 and tan-1 5/4, or between tan-1 -4/5 and tan-1 -5/4 intersects at most one square of any given color and so is guaranteed to save at least one kid. It's probably easiest for Holden to pick any of the 45-degree angles.
3) If Holden has half the brains God gave orange marmalade he'll tell the other kids to bloody-well stay put until the fog lifts so they don't risk falling in a crevice! 
--SMQ
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| « Last Edit: May 7th, 2005, 5:35am by SMQ » |
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--SMQ
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alien
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That's right: the so-called diagonal solution ensures saving one life but the other kid will die.
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Deedlit
Senior Riddler
Posts: 476
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Re: Catcher in the fog
« Reply #9 on: May 7th, 2005, 4:23pm » |
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on May 7th, 2005, 3:58am, alien wrote:| And as you can see from the picture if a kid goes straight north it's impossible that his life will be saved for sure because the crevice might be there. |
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I was just giving a simple way to insure two lives are saved; if Holden falls in a crevice, that means the crevices are north/south, so both of the first two kids live. But if Holden never falls in it doesn't matter.
Quote:| 1) Every path from the center crosses at least one potential crevice, so there is no direction which is guaranteed to save both "autistic" kids. (quotes because I've known a few really great autistic people and the whole statement of the riddle strikes me as somewhat offensive) |
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Yes, I agree.
Quote:| 2) Any path on an angle between tan-1 4/5 and tan-1 5/4, or between tan-1 -4/5 and tan-1 -5/4 intersects at most one square of any given color and so is guaranteed to save at least one kid. It's probably easiest for Holden to pick any of the 45-degree angles. |
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If he's nice, though, he should stay between arctan (4/5) and arctan (5/6), or arctan (6/5) and arctan (5/4), to give them a 50% chance of both surviving. It's not nice to let someone die because you like round numbers!
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| « Last Edit: May 7th, 2005, 4:25pm by Deedlit » |
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SMQ
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Re: Catcher in the fog
« Reply #10 on: May 8th, 2005, 5:39am » |
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on May 7th, 2005, 4:23pm, Deedlit wrote:| It's not nice to let someone die because you like round numbers! |
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Ooh, I missed that--you're right, of course!
--SMQ
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--SMQ
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