Refering to http://www.mathpuzzle.com/AmbiguousTowns.gif I made a mistake, which I didn't realize until after posting. Now that I understand the problem better, let me try rephrasing it. Actually, there are two different problems, one asked by Stan Wagon, Between four towns are roads of length 3, 5, 6, 7, ~3.10, ~5.44 . There are two distinct town configurations. (See picture in link) Q1. Are there 6 integer road lengths that lead to distinct town configurations? Q2. Are there 10 road lengths that lead to distinct town configurations? Ed Pegg Jr Fred lunnon <fred.lunnon@gmail.com> wrote: I simply could not understand the point of this puzzle, as it stands. The "cities" should be at the vertices of a quadrilateral, of course --- what else is a solver supposed to deduce? The distances given are presumably (very) approximate --- if they were exact, the landscape would have to occupy hyperbolic space. [The enclosed "tetrahedron" --- which should be flat --- actually has imaginary volume approximately \iota.] If you're setting a puzzle based on this idea, why not put some work in and base it on a matrix of exact integer distances in the Euclidean plane? A candidate with small lengths, which avoids any simple parallelograms, would be 8 4 4 6 7 8 Perhaps the final distance might be omitted, and the solver challenged to find it. [Some side condition would be needed to exclude the other, shorter, possible solution.] Even then, elementary dexterity with a pair of compasses would in practice reveal the answer in a few seconds! Fred Lunnon On 11/26/06, Ed Pegg Jr wrote:
In my latest large update (http://www.mathpuzzle.com/ ), I give a problem based on ambiguously placed cities. You are given straight line distances between cities A, B, C, and D. Can you make a map of where the 4 cities are, in relation to each other?
D C B A | 3.10 7 5 B | 5.44 3 C | 6
I show a diagram at my site, along with 16 other recreational math items that have come to my attention.
Ed Pegg Jr http://www.mathpuzzle.com/
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