Consider two sets in the Euclidean plane, each consisting of 4 points.
First set: A(0, 0), B(3, 4), C(12, 4), E(4, -1)
Second set: F(0, -8), G(12, -4), H(9, -8), K(4, -9)
It is easy to check that the set of all pairwise distances between the points of each of the given sets (6 numbers for each set ) are the same. At the same time it is obvious that there is no any isometry of the Euclidean plane carrying one set of points to another.
Pairwise distances for each of the sets: {5, 4*sqrt(10), sqrt(17), 9, sqrt(26), sqrt(89)}
The author of this example is Yog-urt from Mathematical Forum of Moscow University. On the following link you can
see the details http://www.mathforum.ru/forum/read/1/61206/
It is interesting that similar examples exist for any n>3 , where n is the number of the points . They show that it is impossible to prove the isometry of two finite point sets by comparing the multisets of pairwise distances.
The similar example for n= 5 points:
