No holeless tetrad of four congruent pentagons

A tetrad is a set of four plane regions with disjoint interiors, every two of which share a boundary segment of positive length; it is holeless if the complement of their union is connected. Holeless tetrads of four congruent hexagons appear in Martin Gardner's account of the puzzle, one of them found by Scott Kim, and Juris Čerņenoks and Andrejs Cibulis list the least number of sides of such a polygon as an open problem. Four congruent pentagons can form a tetrad around a hole. Walter Trump asked in 2020 whether they can form one without a hole, judging it extremely unlikely, and showed that convex tiles cannot form a tetrad at all. We prove that no four congruent simple pentagons, convex or not and mirror images allowed, form a holeless tetrad, and that triangles and quadrilaterals cannot either; so six is the least number of sides. The proof reduces every such tetrad to one of three plane maps and finitely many corner placements, decides the linear conditions on angles and side lengths in exact arithmetic, and excludes each of the 116 remaining two-parameter families by an exact certificate that the prototile cannot close up as a simple pentagon. A separately written verifier checks the certificates. This work has not yet been reviewed by experts. The proofs, the checking programs and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.