Similar dissection of sets

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 $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_k$ be injective continuous mappings. Does there exist a set $X$ such that $D=X\cup f_1(X)\cup \mathrm{.}\mathrm{.}\mathrm{.}\cup f_k(X)$ is satisfied with a non-overlapping union? We prove that such a set $X$ exists for certain choices of $D$ and $\{f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_k\}$. 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 $a\ge (3+\sqrt{5})\mathrm{/}2$.