Related papers: The n-queens solution count Q(n) is divisible by 4
We solve unconditionally the class number one problem for the $2$-parameter family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ with square-free discriminant $d=(an)^2+4a$ for positive odd integers $a$ and $n$.
The counterfeit coin problem requires us to find all false coins from a given bunch of coins using a balance scale. We assume that the balance scale gives us only ``balanced'' or ``tilted'' information and that we know the number k of false…
For given posets $P$ and $Q$ and an integer $n$, the generalized Tur\'an problem for posets, asks for the maximum number of copies of $Q$ in a $P$-free subset of the $n$-dimensional Boolean lattice, $2^{[n]}$. In this paper, among other…
Let $N_q$ be the number of solutions to the equation $$ (a_1^{}x_1^{m_1}+\dots+a_n^{}x_n^{m_n})^k=bx_1^{k_1}\cdots x_n^{k_n} $$ over the finite field $\mathbb F_q=\mathbb F_{p^s}$. Carlitz found formulas for~$N_q$ when…
A cake has to be divided fairly among $n$ agents. When all agents have equal entitlements, it is known that such a division can be implemented with $n-1$ cuts. When agents may have different entitlements, the paper shows that at least $2 n…
The elliptic curve y^2= x^3-Nx where N=m^4+n^4 has rank at least 2 over Q(m,n). When N can be written in two different ways as sum of two fourth powers, then we prove that the rank is at least 4.
A placement of chess pieces on a chessboard is called dominating, if each free square of the chessboard is under attack by at least one piece. In this contribution we compute the number of dominating arrangements of $k$ rooks on an $n\times…
For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane…
It is pointed out that separability problem for arbitrary multi-partite states can be fully solved by a finite size, elementary recursive algorithm. In the worse case scenario, the underlying numerical procedure, may grow doubly…
A generalized `$15$ puzzle' consists of an $n \times n$ numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' `15 puzzle problem'…
We give an example of two $n\times n$ chess positions, $A$ and $B$, such that (1) there is a sequence $\sigma$ of legal chess moves leading from $A$ to $B$; (2) the length of $\sigma$ cannot be less than $\exp \Theta(n)$.
The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…
We consider the problem of resolving the envy of a given initial allocation by adding elements from a pool of goods. We give a characterization of the instances where envy can be resolved by adding an arbitrary number of copies of the items…
The famous combinatorial problem of Euler concerns an arrangement of $36$ officers from six different regiments in a $6 \times 6$ square array. Each regiment consists of six officers each belonging to one of six ranks. The problem,…
For given integers a,b, and j at least 1 we determine the set of integers n for which a^n-b^n is divisible by n^j. For j=1,2, this set is usually infinite; we find explicitly the exceptional cases for which a,b the set is finite. For j=2,…
Sequential allocation is a simple and widely studied mechanism to allocate indivisible items in turns to agents according to a pre-specified picking sequence of agents. At each turn, the current agent in the picking sequence picks its most…
For any odd integer $n\geq3$ a board (of size $n$) is a square array of $n\times n$ positions with a simple rule of how to move between positions. The goal of the game we introduce is to find a path from the upper left corner of a board to…
It is well-known that for $p=1, 2, 3, 7, 11, 19, 43, 67, 163$, the class number of $\mathbb{Q}(\sqrt{-p})$ is one. We use this fact to determine all the solutions of $x^2+p^m=4y^n$ in non-negative integers $x, y, m$ and $n$.
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of…
We use a combinatorial approach to compute the number of non-isomorphic choices on four elements that can be explained by models of bounded rationality.