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4 points in a grid 4x4x such that 6 pairwise distances of them are distinct

Source: Spanish Mathematical Olympiad 1993 P5

August 1, 2018
combinatoricsgridsquare grid

Problem Statement

Given a 4×4 grid of points, the points at two opposite corners are denoted AA and DD. We need to choose two other points B B and CC such that the six pairwise distances of these four points are all distinct. (a) How many such quadruples of points are there? (b) How many such quadruples of points are non-congruent? (c) If each point is assigned a pair of coordinates (xi,yi)(x_i,y_i), prove that the sum of the expressions ∣xi−xj∣+∣yi−yj∣|x_i-x_j |+|y_i-y_j| over all six pairs of points in a quadruple is constant.
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