Points moving towards the nearest point, where will they meet?

Solution 1:

Solution for $n = 3$ (that can be extended partially)

Label the points $A,B$ and $C$ such that $\overline{AB} < \overline{AC} < \overline{BC}$ and move them such that $A = (0,0)$. Transform the coordinates for $B = (b_x,b_y) \mapsto (0,1)$

$$T = \pmatrix{b_y & b_x \\ -b_x & b_y} \qquad T^{-1} \cdot C = \pmatrix{x_0 \\ y_0}$$

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$A$ and $B$ will meet at $(0,0.5)$. According to this $C$ will have moved to the following coordinates

$$c_x = x_0 \sqrt{\frac{1}{k} \operatorname{W} \left( k \exp{\left( k - \frac{2}{r_0 - y_0} \right)} \right)} \qquad k = \frac{r_0 + y_0}{r_0 - y_0} \quad r_0 = \sqrt{x_0^2 + y_0^2}$$ $$c_y = \frac{1}{4} \left( (y_0 + r_0) \left( \frac{c_x}{x_0} \right)^2 + 2 \, (y_0 - r_0) \ln{\left( \frac{c_x}{x_0} \right)} + 3y_0 - r_0 \right)$$

We then transform back to obtain the coordinates of the final point

$$T \cdot \left[ \frac{1}{2} \pmatrix{c_x \\ c_y + 0.5} \right] = \pmatrix{x \\ y}$$

This also applies to $n = 4$ if there are two pairs of points that don't influence each other

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