Varying box-counts

Suppose that I follow our definition of box-counts from the text quite closely:

N_{\varepsilon}(E) is the smallest number of \varepsilon-mesh cubes whose union contains E.

My wife (who doesn’t like to miss any detail, however fine) uses the following definition:

N_{\varepsilon}(E) is the number of closed \varepsilon-mesh cubes whose intersection with E is non-empty.

1. Suppose that my wife and I both compute N_{1/2}(S), where S is the closed unit square in \mathbb R^2. What values do we each obtain?
2. Show that, in general, my wife’s count might exceed mine by at most the factor five.
3. Show that, regardless, we both obtain the same value for the box-counting dimension of a set in the plane.