"Tetravex is a widely played one person computer game in which you are given $n^2$ unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a $n$ by $n$ square such that all neighbouring edges are labelled with an identical number. Unfortunately, playing Tetravex is computationally hard. More precisely, we prove that deciding if there is a tiling of the Tetravex board is NP-complete. Deciding where to place the tiles is therefore NP-hard. This may help to explain why Tetravex is a good puzzle. This result compliments a number of similar results for one person games involving tiling. For example, NP-completeness results have been shown for: the offline version of Tetris, KPlumber (which involves rotating tiles containing drawings of pipes to make a connected network), and shortest sliding puzzle problems. It raises a number of open questions. For example, is the infinite version Turing-complete? How do we generate Tetravex problems which are truly puzzling as random NP-complete problems are often surprising easy to solve? Can we observe phase transition behaviour? What about the complexity of the problem when it is guaranteed to have an unique solution? How do we generate puzzles with unique solutions?"

kata for GP

"There’s a charming little brain teaser that’s going around the Interwebs. It’s got various forms, but they all look something like this:…"

"A set of polyominoes is interlocked if no subset can be moved far away from the rest. It was known that polyominoes that are built from four or fewer squares do not interlock. The project of Dhawan and his mentor was to investigate the interlockedness of larger polyominoes. And they totally delivered.

They quickly proved that you can interlock polyominoes with eight or more squares. Then they proved that pentominoes can’t interlock. This left them with a gray area: what happens with polyominoes with six or seven squares? After drawing many beautiful pictures, they finally found the structure presented in our accompanying image. The system consists of 12 hexominoes and 5 pentominoes, and it is rigid. You cannot move a thing. That means that hexominoes can be interlocked and thus the gray area was resolved."

"A one instruction set computer (OISC), sometimes called an ultimate reduced instruction set computer (URISC), is an abstract machine that uses only one instruction – obviating the need for a machine language opcode.[1][2][3] With a judicious choice for the single instruction and given infinite resources, an OISC is capable of being a universal computer in the same manner as traditional computers that have multiple instructions.[2]:55 OISCs have been recommended as aids in teaching computer architecture[1]:327[2]:2 and have been used as computational models in structural computing research.[3]"

"The book was enormously popular, and Mr. Reed and Mr. Brainerd’s diagramming swept through American schools like a refreshing breeze. By the latter half of the 19th century, chalkboards had become increasingly common in classrooms; for students, the impact of watching a sentence take shape on that large surface as a comprehensible, often elegant, and sometimes downright ingenious drawing must have been significant. It’s hard to believe anyone but the most dedicated pedant could have actually enjoyed parsing, but plenty of students — including me — loved diagramming.

A century and a half later, diagramming sentences is even more out of date than writing lessons on a piece of slate. When the book I wrote about it was published in 2006, a couple of hundred people sent me e-mails. One writer accused me of succumbing to Stockholm syndrome because I wrote so benignly about the nun who brainwashed me into thinking diagramming was fun. Another asked me for a date. Two objected to my political attitudes, as they deduced them between the lines. A dozen or so either faulted some of the diagrams or challenged me with a particularly tricky sentence."

"We present two variants of Wythoff's game. The first game is a restriction of Wythoff's game in which removing tokens from the smaller pile is not allowed if the two entries are not equal. The second game is an extension of Wythoff's game obtained by adjoining a move allowing players to remove k tokens from the smaller pile and l tokens from the other pile provided l < k. We show that both games preserve the P-positions of Wythoff's game. This resolves a question raised by Duchene, Fraenkel, Nowakowski and Rigo. We give formulas for those positions which have Sprague-Grundy value 1. We also prove several results on the Sprague-Grundy functions."

"The B36/S125 (or "2x2") cellular automaton is one that takes place on a 2D square lattice much like Conway's Game of Life. Although it exhibits high-level behaviour that is similar to Life, such as chaotic but eventually stable evolution and the existence of a natural diagonal glider, the individual objects that the rule contains generally look very different from their Life counterparts. In this article, a history of notable discoveries in the 2x2 rule is provided, and the fundamental patterns of the automaton are described. Some theoretical results are derived along the way, including a proof that the speed limits for diagonal and orthogonal spaceships in this rule are c/3 and c/2, respectively. A Margolus block cellular automaton that 2x2 emulates is investigated, and in particular a family of oscillators made up entirely of 2 x 2 blocks are analyzed and used to show that there exist oscillators with period 2^m(2^k - 1) for any integers m,k geq 1."

"Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in Z^d. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of the way they are glued together in the image by a substitution. Two problems can arise when defining a substitution in such a way: it can fail to be consistent, and the patterns in an image by the substitution might overlap.

We prove that it is undecidable whether a two-dimensional substitution is consistent or overlapping, and we provide practical algorithms to decide these properties in some particular cases."

"The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the regular tetrahedron and the cube. Higher-order moments of f_tetra and f_cube have not been examined until now. Assume that each polyhedron has edges of unit length. We deduce that the mean square width of the regular tetrahedron is 1/3+(3+sqrt(3))/(3*pi) and the mean square width of the cube is 1+4/pi."

"Consider a large Boolean network with a feed forward structure. Given a probability distribution for the inputs, can one find-possibly small-collections of input nodes that determine the states of most other nodes in the network?…"

"Under neutral reciprocal sign epistasis, two genetic changes are jointly neutral, even though their individual effects are deleterious. By using the widely studied mapping from an RNA sequence to secondary structure, we investigate the effect of this kind of epistasis on neutral spaces corresponding to networks of genotypes that fold to the same secondary structure. Neutral networks for RNA structures with n bonds are typically fragmented into at least 2^n different neutral components that cannot be connected by single point mutations. By exhaustive enumeration of all RNA secondary structures of sequences of length 15 we show that most networks are not dominated by one neutral component, but are rather broken up into multiple large components. Although they generate the same phenotype, components of a single neutral network are heterogeneous, showing wide variations in their robustness and their evolvability. Both properties are correlated with component size, rather than with the size of their underlying neutral network. In particular, sets of accessible phenotypes can vary quite strongly between components. Thus, the potential for future innovation is contingent on which neutral component a population occupies. We further argue that neutral reciprocal sign epistasis may have similar consequences for neutral evolution of other biological systems as well."

"It will be observed that this square when turned upside down is still magic."

"Suppose someone takes a solved 20x20x20 Rubik's cube and makes five moves - can you figure out [from that scrambled state] what those five moves were?" he asks. In other words, can you solve it in five moves? He suspects that you cannot, but has yet to prove it. "We don't know."

"While magic squares have been known and studied for many centuries, it is surprising that for certain types of magic squares we still do not know today which are the smallest possible! In an effort to make progress on these unsolved problems, twelve prizes totaling €8,000 and 12 bottles of champagne are offered for the solutions to twelve enigmas (six main at €1,000 each, six small from €100 to €500 each):…"

"Any two rectilinear figures with equal area can be dissected into a finite number of pieces to form each other. This is the Wallace-Bolyai-Gerwien theorem. For minimal dissections of a triangle, pentagon, and octagon into a square, see Stewart (1987, pp. 169-170) and Ball and Coxeter (1987, pp. 89-91). "