We prove NP-completeness of deciding whether a given loop of colored right isosceles triangles, hinged together at edges, can be folded into a specified rectangular three-color pattern. By contrast, the same problem becomes polynomially solvable with one color or when the target shape is a tree-shaped polyomino.

We show that jigsaw puzzles, edge-matching puzzles, and polyomino packing puzzles are all NP-complete. Furthermore, we show direct equivalences between these three types of puzzles: any puzzle of one type can be converted into an equivalent puzzle of any other type.

A monotonous polyomino is formed by all lattice unit squares met the graph of some fixed continuous function $$f:[a,b] \to \mathbb{R}$$ with $$f(k) \notin \mathbb{Z}$$ whenever $$k \in . Our main result says that least cardinality a covering $$(m \times n)$$ -rectangle polyominoes $$\big\lceil \frac{2}{3}(m+n-\sqrt{m^2+n^2-mn})\big\rceil.$$ The paper motivated problem on arrangements straight l...

We show that the complexity of a cutting word u in a regular tiling by a polyomino Q is equal to Pn(u) = (p+q−1)n+1 for all n ≥ 0, where Pn(u) counts the number of distinct factors of length n in the infinite word u and where the boundary of Q is constructed by 2p horizontal and 2q vertical unit segments.

In this paper, we propose a new notion of rep-cube which is a net of a cube, and it can be divided into some polygons such that each of them can be folded to a cube. This notion is inspired by polyomino and rep-tile; both are introduced by Solomon W. Golomb, and well investigated in the recreational mathematics society. We prove that there are infintely many distinct rep-cubes. We also extend t...

A permutomino is a polyomino uniquely defined by a pair of permutations. Recently permutominoes, and in particular convex permutominoes have been studied by several authors concerning their analytical and bijective enumeration, tomographical reconstruction, and the algebraic characterization of the associated permutations [2, 3]. On the other side, Beauquier and Nivat [5] introduced and gave a ...

Given N distinct memory modules, the elements of an (infinite) array in storage are distributed such that any set of N elements arranged according to a given data template T can be accessed rapidly in parallel. Array embeddings that allow for this are called skewing schemes and have been studied in connection with vector processing and SIMD machines. In 1975 Shapiro (IEEE Trans. Comput. C-27 (1...

A set partition of the set [n] = {1,. .. , n} is a collection of disjoint blocks B 1 , B 2 ,. .. , B d whose union is [n]. We choose the ordering of the blocks so that they satisfy min B 1 < min B 2 < · · · < min B d. We represent such a set partition by a canonical sequence π 1 , π 2 ,. .. , πn, with π i = j if i ∈ B j. We say that a partition π contains a partition σ if the canonical sequence...