Question 32
Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\lfloor \log_2 d \rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2n$ distinct values.
*Note:* For any $d \gt 0$, $\lfloor \log_2 d \rfloor$ is the unique integer $k$ such that $2^k \le d \lt 2^{k+1}$.
*Note:* For any $d \gt 0$, $\lfloor \log_2 d \rfloor$ is the unique integer $k$ such that $2^k \le d \lt 2^{k+1}$.