A graph of a puzzle is obtained by associating each possible position with a vertex and by inserting edges between vertices if and only if the corresponding positions can be obtained from each other in one move. Computational methods for finding the vertices at maximum distance \(\delta\) from a vertex associated with a goal position are presented. Solutions are given for small sliding block puzzles, and methods for obtaining upper and lower bounds on \(\delta\) for large puzzles are considered. Old results are surveyed, and a new upper bound for the 24-puzzle is obtained: \(\delta \leq 210\).

The total domination number \(\gamma_t(G)\) of graph \(G = (V, E)\) is the cardinality of a smallest subset \(S\) of \(V\) such that every vertex of \(V\) has a neighbor in \(S\). It is known that, if \(G\) is a connected graph with \(n\) vertices, \(\gamma_t(G) \leq \left\lfloor{2n}/{3}\right\rfloor\). Graphs achieving this bound are characterized.

In this paper we define the imbalance of equi-replicate incomplete block designs. We prove that the imbalance measure of an equi-replicate incomplete block design has a lower bound, and this bound is attained if and only if the design is a 2-concurrence design. This result allows one to formulate the construction of 2-concurrence designs as an optimization problem.

Until quite recently, very few weakly completable critical sets were known. The purpose of this note is to prove the existence of at least one Latin square of each order greater than four in which a weakly completable set exists. This is done by actual construction of such a square. Non-existence of weakly completable sets in Latin squares of orders 2, 3, and 4 is already known.

In our recent paper Necessary and sufficient conditions for some two variable orthogonal designs in order 44, Koukouvinos, Mitrouli and Seberry leave 7 cases unresolved. Using a new algorithm given in our paper A new algorithm for computer searches for orthogonal designs by the present four authors we are able to finally resolve all these cases.

This note records that the necessary conditions for the existence of two variable designs constructed using four circulant matrices are sufficient. In particular, of 484 potential cases, 404 cases have been found, 68 cases do not exist, and 12 cases cannot be constructed using four circulant matrices.

We determine the number of non-isomorphic triple systems with bipoints in those cases for which the total number of triples does not exceed 20.

Temporal load-balancing – “spreading out” the executions of tasks over time — is desirable in many applications. A form of temporal load-balancing is introduced: scheduling to maximize minimum inter-completion time (MICT-scheduling). It is shown that MICT-scheduling is, in general, NP-hard. A number of restricted classes of task systems are identified, which can be efficiently MICT-scheduled.

Given a finite-dimensional vector space \(V\) over a finite field \(F\) of odd characteristic, and equipping \(V\) with an orthogonal (symplectic, unitary) geometry, the following two questions are considered:

1. Given some linearly independent vectors \(w_1, w_2, \ldots, w_k \in V\) and the \(k \times k\) matrix \(A = (\langle w_i, w_j\rangle)\), and given scalars \(\alpha_1, \alpha_2, \ldots, \alpha_k, \beta \in F\), how many vectors \(v \in V\), not in the linear span of \(w_1, w_2, \ldots, w_k\), satisfy \(\langle w_i, v\rangle = \alpha_i\) (\(i = 1, 2, \ldots, k\)) and \(\langle v, v\rangle = \beta\)?
2. Given a \(k \times k\) matrix \(A = (\lambda_{ij})\) with entries from \(F\), how many \(k\)-tuples \((v_1, v_2, \ldots, v_k)\) of linearly independent vectors from \(V\) satisfy \(\langle v_i, v_j\rangle = \lambda_{ij}\) (\(i, j= 1, 2, \ldots k\))?

An exact answer to the first question is derived. Here there are two cases to consider, depending on whether or not the column vector \((\alpha_i)\) is in the column space of \(A\). This result can then be applied iteratively to address the second question.

Let \(G\) be a finite graph and let \(\mu\) be an eigenvalue of \(G\) of multiplicity \(k\). A star set for \(\mu\) may be characterized as a set \(X\) of \(k\) vertices of \(G\) such that \(\mu\) is not an eigenvalue of \(G – X\). It is shown that if \(G\) is regular then \(G\) is determined by \(\mu\) and \(G – X\) in some cases. The results include characterizations of the Clebsch graph and the Higman-Sims graph.