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h_alpha
Member since Feb-8-06
Feb-08-06, 04:54 PM (EST)

"four travelers alternate/equivalent solution"

 The 4 Travellers problem is posed at CTK in the "beautiful / elegant proofs" part of the manifesto. Taking this problem 'as read': I came up with a solution that I believe is equivalent to the one given. I find the comparison amusing so I'll give my solution here on the chance someone else might find it'so as well. In the rest frame of traveller 1: Travellers T2, T3, and T4 all approach along straight lines we can define in terms of angles (say a2, a3, and a4). Of course once they meet T1 they depart along the same lines with the new headings (a2 + 180 deg) etcetera. We can determine a2 by observing T2 at just about any time (since a2 is constant) so let's choose the moment that T2 meets T3. This gives us T3's angle a3 as well, the same: a2 = a3. Similarly since T2 meets T4 we have a2 = a4. Thus from T1's perspective, T2, T3, and T4 are all travelling along the same line. Given adequate constraints, T2, T3, and T4 must all meet one another. "Adequate contraints" are provided by the conditions "T2 meets T3 and T4" and the condition that no two roads (in the absolute rest frame) are parallel. Demonstrating the constraints are adequate: Not done; just wanted to point out the T1-frame idea.best regards Rob

mr_homm
Member since May-22-05
Feb-08-06, 09:15 PM (EST)

1. "RE: four travelers alternate/equivalent solution"
In response to message #0

alexb
Charter Member
1766 posts
Feb-09-06, 09:52 AM (EST)

2. "RE: four travelers alternate/equivalent solution"
In response to message #0

 It ought to be from THE BOOK. I agree with Stuart: the argument shows that in one frame the four of them are collinear. They remain collinear in any other frame, the original in particular.Many thanks,Alex

alexb
Charter Member
1766 posts
Feb-09-06, 01:56 PM (EST)

3. "RE: four travelers alternate/equivalent solution"
In response to message #0

 >I came up with a solution that I believe is >equivalent to the one given. In a sense, all solutions are equivalent; perhaps some more so than the others. If that idea of equivalence is meaningful then your solution is rather related to Stuart's:https://www.cut-the-knot.org/4travelers/FourTravelers.shtmlHis Lemma 1 is equivalent to saying that in the rest frame of #1, #2 moves along a direction towards #1.His Lemma 2 is equivalent to saying that either a2=a3=a4, or #1 has a position where he meets all the rest without moving.

alexb
Charter Member
1766 posts
Feb-09-06, 02:48 PM (EST)

4. "RE: four travelers alternate/equivalent solution"
In response to message #0

 I just posted your proof at the site proper. Hope you do not object. It makes a good company with other proofs.Thank you,Alex

h_alpha
Member since Feb-8-06
Feb-09-06, 05:19 PM (EST)

5. "RE: four travelers alternate/equivalent solution"
In response to message #4

 Alex: Delighted to hear it, and thanks for building this excellent website.Rob