Consider these five points in $6$-space.
$\{\{1,2,3,4,5,6\}, \{1,2,3,4,6,5\}, \{1,2,5,3,4,6\}, \{2,1,3,5,6,4\}, \{2,1,6,4,5,3\}\}$
Half the squared distances between pairs of these points are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.
What is the lowest dimension where a set $5$ points has that property? Obviously, $6$ is enough dimensions.
What is the lowest dimension where a set 6 points has squared distances $1$-$15$?
I suspect there are answers in $4$-space and $5$-space. If so .. are there solutions with nice coordinates?
EDIT: $\{\{0,0,0,0\}, \{0,0,0,1\}, \{0,1,1,0\}, \{0,2,2,2\}, \{2,0,0,1\}, \{2,1,3,0\}\}$ -- six points in $4$-space with squared distances $1$-$15$.
$\{\{0,0,0,0\}, \{0,0,0,1\}, \{0,1,1,0\}, \{0,2,2,2\}, \{0,0,4,2\}, \{1,4,1,1\}, \{2,2,2,3\}\}$ -- seven points in $4$-space with squared distances $1$-$21$, almost. The $4$ is missing, and instead there is a $27$. Can squared distances $1$-$21$ be done in $4$-space?