100 dot product riddle I was given this riddle by one of my professors, and was wondering if anyone could give some hints on this problem. Say I have 100 vectors, $x_1, x_2, ... x_{100}$. I compute every dot product pair, excluding self-pairs, so a vector is not dotted with itself, i.e $x_1 \cdot x_2$, $x_1 \cdot x_3$, ... $x_1 \cdot x_{100}$, $x_2 \cdot x_3$, $x_2 \cdot x_4$, ... $x_2 \cdot x_{100}$ ... $x_{99} \cdot x_{100}$, and tabulate the result in a list $L_1$. I then create another list $L_2 = -L_1$, $L_2$ is just $L_1$ with the signs flipped. If I only gave you the two lists, could you tell which one is $L_1$ and which one is $L_2$, i.e which list is the true dot product pairs, and which list had it's sign flipped?
 A: Consider a set of vectors $x_k\in \mathbb{R}^{100}:$
$$
\Big\{ x_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \\0 \end{bmatrix}, \; x_2 = \begin{bmatrix} 1 \\ 1 \\ 0 \\ \vdots \\ 0 \\0\end{bmatrix}\; x_3 = \begin{bmatrix} 1 \\ 1 \\ 1 \\ \vdots \\ 0\\0 \end{bmatrix}, \; \dots,\;  x_{99} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ \vdots\\1 \\ 0\end{bmatrix},\;x_{100} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ \vdots  \\ 1\\1\end{bmatrix}\Big\}
$$
Then $L_1=\big\{(1)_{\times 99},(2)_{\times 98},\dots,(99)_{\times 1}\big\}, \; L_2=\big\{(-1)_{\times 99},(-2)_{\times 98},\dots,(-99)_{\times 1}\big\}$.
Now consider a set of vectors $x_k\in \mathbb{R}^{100}:$
$$
\Big\{ x_1 = \begin{bmatrix} -1 \\ 0 \\ 0 \\ \vdots \\ 0 \\0 \end{bmatrix}, \; x_2 = \begin{bmatrix} 1 \\ -3 \\ 0 \\ \vdots \\ 0 \\0\end{bmatrix}\; x_3 = \begin{bmatrix} 1 \\ 1 \\ -5 \\ \vdots \\ 0\\0 \end{bmatrix}, \; \dots,\;  x_{99} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ \vdots\\-197 \\ 0\end{bmatrix},\;x_{100} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ \vdots  \\ 1\\1\end{bmatrix}\Big\}
$$
Then $L_1=\big\{(-1)_{\times 99},(-2)_{\times 98},\dots,(-99)_{\times 1}\big\},\;  L_2=\big\{(1)_{\times 99},(2)_{\times 98},\dots,(99)_{\times 1}\big\}$.
A: First, let's consider all the possible answers this question could have:


*

*It it always possible t otell which list is which

*It is always impossible to tell which list is which

*It is sometimes possible and sometimes impossible to tell, depending on the list.


Since both the dimension and amount of vectors can be arbitrary, we can consider 3 1-dimensional vectors resulting in lists [1,1,1] and [-1,-1,-1]. It is clear that [1,1,1] is the correct one, since you can never have 3 real numbers so that the product of any 2 of them is negative.
On the other hand, consider 2 1-dimensional vectors with lists [1,1] and [-1,-1]. Both are possible from vectors (1), (1) and (1), (-1).
Therefore, the answer 3) is correct. Sometimes it is possible to tell, sometimes it isn't. The fun begins when you try to investigate when it is possible to tell...
