  In 1994, Martin Gardner stated a set of questions concerning the dissection
of a square or an equilateral triangle in three similar parts. Meanwhile,
Gardner's questions have been generalized and some of them are already solved.
In the present paper, we solve more of his questions and treat them in a much
more general context. Let $D\subset \mathbb{R}^d$ be a given set and let
$f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such
that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a
non-overlapping union? We prove that such a set $X$ exists for certain choices
of $D$ and $\{f_1,...,f_k\}$. The solutions $X$ often turn out to be attractors
of iterated function systems with condensation in the sense of Barnsley. Coming
back to Gardner's setting, we use our theory to prove that an equilateral
triangle can be dissected in three similar copies whose areas have ratio
$1:1:a$ for $a \ge (3+\sqrt{5})/2$.
