Here is a puzzle:

Find 5 identical coins. Can you arrange them so that every coin is touching every other coin?

The solution is here. The hint is: use the third dimension.

My questions are based on this puzzle.

- Can you prove that six coins can not be arranged to pairwisely touch each other?
- How much coins can be arranged to pairwisely touch each other in dimension $n$?

~~Here, "coins" in higher dimensions refer to $2$-dimensional unit disks (or unit $k$-balls for your favorite $k<n$), but the relative interiors of the disks / balls are disjoint.~~

Here, "coins" may refer to "thin" unit disks, i.e. $B_1^2\times B_\epsilon^{n-2}$ for $\epsilon$ arbitrarily small, or $B_1^k\times B_\epsilon^{n-k}$ for your favorite $k<n$.