  According to Courant's theorem, an eigenfunction as\-sociated with the $n$-th
eigenvalue $\lambda\_n$ has at most $n$ nodal domains. A footnote in the book
of Courant and Hilbert, states that the same assertion is true for any linear
combination of eigenfunctions associated with eigenvalues less than or equal to
$\lambda\_n$. We call this assertion the \emph{Extended Courant
Property}.\smallskipIn this paper, we propose simple and explicit examples for
which the extended Courant property is false: convex domains in $\R^n$
(hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on
the round sphere $\mathbb{S}^2$, and on a flat torus $\mathbb{T}^2$.
