How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square?
By "nonoverlapping" I mean: not sharing an interior point.
By "touch" I mean: sharing a boundary point.

It seems the answer for a square in $\mathbb{R}^2$ should be $8$,
and for a cube in $\mathbb{R}^3$, $26$.

But a 1999 paper by Larman and Zong,

"On the Kissing Numbers of Some Special Convex Bodies."

Discrete Comput Geom21:233–242 (1999). (Springer link)

says

"In this note we determine the kissing numbers of octahedra, rhombic dodecahedra and elongated octahedra. In fact, besides balls and cylinders, they are the only convex bodies whose kissing numbers are exactly known."

In that paper, they were interested in the translative kissing number and the lattice kissing number, whereas I want to consider arbitrary orientations of each square/cube. Despite the quote above, it seems this should be known...?

**Update**(

*30Dec12*) The following explains (I believe) the $0.82$ in Henry Cohn's comment, leading to his proof for $\le 9$ in $\mathbb{R}^2$:

overlapone unit square? – Erel Segal-Halevi Aug 13 at 6:29