### No. 10 ### Extremal {p,q}-Animals

An animal is a planar shape formed by attaching congruent regular polygons, known as tiles, along their edges. In this paper, we study extremal animals defined on regular tessellations of the plane.
In 1976, Harary and Harborth studied animals in the Euclidean cases, finding extremal values for their vertices, edges, and tiles, when any one of these parameters is fixed. Here, we generalize their results to hyperbolic animals. We exhibit a sequence of spiral animals and prove that they attain the minimal numbers of edges and vertices within the class of animals with n tiles. In their conclusions, Harary and Harborth also proposed the question of enumerating extremal animals with a fixed number of tiles. This question has previously only been considered for Euclidean animals. We find special sequences of extremal animals that are unique extremal animals, in the sense that any animal with the same number of tiles which is distinct up to isometries can't be extremal.

### No. 09 ### High-dimensional Holeyominoes

What is the maximum number of holes enclosed by a d-dimensional polyomino built of n tiles? Represent this number by fd(n). Recent results show that f2(n)/n converges to 1/2. We prove that for all d≥2 we have fd(n)/n→(d−1)/d as n goes to infinity. We also construct polyominoes in d-dimensional tori with the maximal possible number of holes per tile. In our proofs, we use metaphors from error-correcting codes and dynamical systems.

### No. 08 ### Topology and local geometry of the Eden model

The Eden cell growth model is a simple discrete stochastic process which produces a "blob" in |R^d: start with one cube in the regular grid, and at each time step add a neighboring cube uniformly at random. This process has been used as a model for the growth of aggregations, tumors, and bacterial colonies and the healing of wounds, among other natural processes. Here, we study the topology and local geometry of the resulting structure, establishing asymptotic bounds for Betti numbers. Our main result is that the Betti numbers grow at a rate between the conjectured rate of growth of the site perimeter and the actual rate of growth of the site perimeter. We also present the results of computational experiments on finer aspects of the geometry and topology, such as persistent homology and the distribution of shapes of holes.

### No. 07 ### Topology of random 2-dimensional cubical complexes

We study a natural model of random 2-dimensional cubical com-plex which is a subcom-plex of an n-dimensional cube, and where every possible square 2-face is included independently with probability p. Our main result is to exhibit a sharp threshold p= 1/2 for homology vanishing as n→ ∞. This is a 2-dimensional analogue of the Burtin and Erdös–Spencer theorems characterizing the connectivity threshold for random cubical graphs. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random 2-dimensional simplicial complexes. However, the models exhibit strikingly different behaviors. We show that if p >1−√(1/2) ≈0.2929, then with high probability the fundamental group is a free group with one generator for every maximal 1-dimensional face.

### No. 06 ### Isotopy Graphs of Latin Tableaux

Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000s in a paper of Chow, Fan, Goemans, and Vondrák. Here, we extend the notion of isotopy, a per-mutation group action, from Latin squares to Latin tableaux. We define isotopy graphs for Latin tableaux, which encode the structure of orbits under the isotopy action, and investigate the relation-ship between the shape of a Latin tableau and the structure of its isotopy graph. Our main result shows that for any positive integer d, there is a Latin tableau whose isotopy graph is a d-dimensional cube. We show that most isotopy graphs are triangle-free, and we give a characterization of all the Latin tableaux for which the isotopy graph contains a triangle. We also give a formula for the degree of a vertex of each component of an isotopy graph which depends on both the shape of the Latin tableaux and the filling.

### No. 05 ### Art Gallery Problem with Rook and Queen Vision

How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can ‘‘see’’ whichever squares the rook or queen attacks. We show that n/2 rooks or n/3 queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

### No. 04 ### Polyiamonds Attaining Extremal Topological Properties, Part II

We consider two optimization questions with respect to polyiamonds. What is the maximum number of holes that a polyiamond with n tiles can enclose, and what is the minimum number of tiles required to construct a polyiamond with h holes? These numbers will be given by the sequences f(n) and g(h), respectively. In this paper, we construct a sequence of polyiamonds with hk= (3/2)(k^2−k) holes and nk= (1/2)(9k^2+3k−4) tiles such that g(nk) =hk. Furthermore, these polyiamonds all attain a specific set of efficient geometric and topological properties.

### No. 03 ### Polyiamonds Attaining Extremal Topological Properties, Part I

We consider two optimization questions with respect to polyiamonds. What is the maximum number of holes that a polyiamond with n tiles can enclose, and what is the minimum number of tiles required to construct a polyiamond with h holes? These numbers will be given by the sequences f(n) and g(h), respectively. In this paper, we construct a sequence of polyiamonds with hk= (3/2)(k^2−k) holes and nk= (1/2)(9k^2+3k−4) tiles such that g(nk) =hk. Furthermore, these polyiamonds all attain a specific set of efficient geometric and topological properties.

### No. 02 ### Extremal topological and geometric problems for polyominoes

We give a complete solution to the extremal topological combinatorial problem of finding the mini-mum number of tiles needed to construct a polyomino with h holes. We denote this number by g(h) and we analyze structural  properties of polyominoes with h holes and g(h) tiles, characterizing their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. For h ≤ 8 the values of g(h) were originally computed by Tomas Olivera e Silva in 2015, and for the sequence hl=(2^(2l)−1)/3 by Kahle and Roldán in 2019, who also showed that asymptotically g(h)≈2h. Here we also prove that the polyominoes constructed by Kahle and Roldan with hl=(2^(2l)−1)/3 holes and g(hl) tiles are in fact unique up to isometry with these fixed parameters; that is, having the minimal number of tiles for hl holes.

### No. 01 ### Polyominoes with maximally many holes

What is the maximum number of holes that a polyomino with n tiles can enclose? Call this number f(n). We show that if nk=(2^(2k+1)+3⋅2^(k+1)+4)/3 and hk=(2^(2k)−1)/3, then f(nk)=hk for k≥1. We also give nearly matching upper and lower bounds for large n, showing as a corollary that f(n)≈n/2.

### No. 02 ### Learning music and math, together as one: Towards a collaborative approach for practicing math skills with music

In this paper we present a study that took place in an elementary school in Mexico. The study aimed to explore the use of a digital application for the design and orchestration of collabora-tive, game-based learning activities for STEAM and to study the impact of group formation with respect to students' background knowledge. The workshops took place in the form of a tournament where groups of students worked together to win sets of music and math rounds. The results indi-cate that homogeneous groups outperformed heterogeneous groups in terms of learning gains but heterogeneous groups achieved better results in terms of game score than homogeneous groups. The former does not confirm related research and it may suggest that the group formation impact on learning gains depends largely on the context. The latter may indicate the need for aligning the game objectives with learning goals in order to ensure that educational games indeed prioritize learning.

### No. 01 ### The Mutando of Insanity

Puzzles based on coloured cubes and other coloured geometrical figures have a long history in the recreational mathematical literature. One of the most commercially famous of these puzzles is the Instant Insanity that consists of four cubes. Their faces are coloured with four different colours in such a way that each colour is present in each one of the four cubes. To solve the puzzle, one needs to stack the cubes in a tower in such a way that each one of the colours appears exactly once in the four long faces of the tower. The main purpose of this paper is to study the combinatorial richness of a mathematical model of this puzzle by analysing all possible ways of colouring the cubes to form a puzzle analogous to the Instant Insanity. We have done this analysis for n cubes and n colours for n=4,5,6. This combinatorial analysis allowed us to design the Mutando of Insanity, a puzzle that we presented at Gathering for Gardner 12 (G4G12).

### No. 02 ### Homology and Geometry of the Plaquette Eden Model

This package includes software to run simulations of the Plaquette Eden Model, which is a discrete 2-dimensional stochastic cell growth process defined on the 3-dimensional regular cubical tessellation of the Euclidian space. The software analyzes the topology (Betti numbers and persistent homology) and local geometry of the structure. The program also has visualization capabilities and can produce a picture of the Plaquette Eden Model or output a .txt file that can be used with Autodesk Maya to produce an interactive 3-dimensional image.

Ripser and GUDHI are used to compute and visulize persistent homology of the the Plaquette Eden Model. If you use this functionality, make sure to cite this library.

### No. 01 ### Topology and Geometry of the Eden Model

This package includes software to run simulations of the Eden Growth Model in Z^d in dimensions 2-5, and analyze the topology (Betti numbers and persistent homology) and local geometry of the structure. The software is also able to read a .txt file with the time at which tiles should be added. This allows analysis of simulations from other stochastic models. Note: the cells in a .txt file should be in chronological order. For graphical representation, the program can create a picture of a two-dimensional growth model and can output a .txt file for 3-dimensional growth modles which can be inputed to MAYA to produce an interactive 3-dimensional image.

GUDHI is used to compute homology and persistent homology in 3D, 4D, and 5D. If you use this functionality, make sure to cite this library.