Spivak A.V.

Given a cube with an edge length of $n$ cm. We have a long piece of insulating tape 1 cm wide. It is required to cover the cube with the tape, with the tape being able to freely cross over an edge to another face, travel in a straight line parallel to the edge on a face, and not hang off the face. Into how many pieces does the tape need to be cut to cover the cube?
$2 n$ pieces are sufficient: we cover four faces of the cube, forming the lateral surface of a rectangular prism, with $n$ rows of tape, each row 1 unit wide, and then similarly cover another set of four faces (two of which have already been covered once, and the other two are being covered for the first time).

Let's prove that it is impossible to do with fewer pieces. Consider two faces that share a common edge, and divide them into $n$ strips $1 \times 2 n$, perpendicular to the common edge of these faces. If there are pieces of tape running along these strips and covering their entire width (though not necessarily their entire length), then there must be at least $n$ such pieces. Then there are still two faces completely free from these pieces. To cover one of them, at least $n$ more pieces are required, as each piece covers no more than $\frac{1}{n}$ of the area of this face. If, however, there is a strip that is free from pieces running along it for its entire length, then to cover this strip with pieces running across it, at least $2 n$ pieces of tape are needed.

## Answer

$2 n$ pieces.