Related papers: Multimagic Squares
In this article we prove the impossibility of some disentanglement puzzles, first building mathematical models that reflect the essential characteristics of these puzzles.
We prove that for all n>1 every latin n-dimensional cube of order 5 has transversals. We find all 123 paratopy classes of layer-latin cubes of order 5 with no transversals. For each $n\geq 3$ and $q\geq 3$ we construct a (2q-2)-layer latin…
We study $n$-dimensional matrices with $\{0,1\}$-entries ($n$-cubes) such that all their $2$-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized…
This article explores the limits of geometric construction using various tools, both classical and modern. Starting with ruler and compass constructions, we examine how adding methods such as origami, marked rulers (neusis), conic sections,…
A novel kind of self-referential square matrix is introduced. A certain subset of the matrix entries record the frequencies of occurrence of each distinct number appearing within the entire matrix. Such squares are necessarily elusive. Our…
We study 4-by-4 squares formed by cards from the EvenQuads deck. EvenQuads is a card game with 64 cards where cards have 3 attributes with 4 values in each attribute. A quad is four cards with all attributes the same, all different, or half…
A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots,h_k$ such that $h_1+\dots+h_k = n$ is known as a realization. The existence of realizations is a partially solved problem with a few general results for an…
If our universe has appeared in a result of Big Bang or something like this, whether we have reasons to deny an existence of other universes appearing by the same or similar way? An objection that there is no anything like it, is doubtful,…
In this paper, we derive a formula to express the maximum number of non-intersecting diagonals of arbitrary length that can be drawn in n x n square arrays, where n is a multiple of l+1.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Inspired by the fact that the sum of the cubes of the first $n$ naturals is equal to the square of their sum, we explore, for each $n$, the Diophantine equation representing all non-trivial sets of $n$ integers with this property. We find…
The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $\Lambda_n$ is obtained from $\Gamma_n$ by removing vertices that start and end with 1. We…
Triangular numbers that are multiple of other triangular numbers are investigated. It is known that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers which are triangular numbers. If the…
A procedure that generates parallelograms from any quadrilateral is presented. If the original quadrilateral is itself a parallelogram, then the procedure gives squares. Hence, when applied two times, this procedure generates squares from…
The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we get the explicit expressions of all squares, and then establish the tree structure of the positions of repeated squares…
We derive formulas for the number of polycubes of size $n$ and perimeter $t$ that are proper in $n-1$ and $n-2$ dimensions. These formulas complement computer based enumerations of perimeter polynomials in percolation problems. We…
In this article, we reveal how Benjamin Franklin constructed his second $8 \times 8$ magic square. We also construct two new $8 \times 8$ Franklin squares.
Large set of orthogonal arrays (LOA) were introduced by D. R. Stinson, and it is also used to construct multimagic squares recently. In this paper, multimagic squares based on strong double LOA are further investigated. It is proved that…
We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else…
In this paper, we prove that optimally solving an $n \times n \times n$ Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an $n \times…