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Buffon-Laplace Needle Problem considers a needle of a length $l$ randomly dropped on a large plane distributed with vertically parallel lines with distances $a$ and $b$ ($a \geqslant b$), respectively. As a classical problem in stochastic…

A star of n (n greater than or equal to 2) line segments (needles) of equal length with common endpoint and constant angular spacing is randomly placed onto a lattice which is the union of two families of equidistant lines in the plane with…

We solve a variant of the classical Buffon Needle problem. More specifically, we inspect the probability that a randomly oriented needle of length $l$ originating in a bounded convex set $X\subset\mathbb{R}^2$ lies entirely within $X$.…

Using the Semple bundle construction, we derive an intersection-theoretic formula for the number of simultaneous contacts of specified orders between members of a generic family of degree $d$ plane curves and finitely many fixed curves. The…

In this paper, we solve Buffon's needle problem for a needle consisting of two line segments connected in a pivot point.

In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a ground made of equispaced parallel…

Consider a line segment placed on a two-dimensional grid of rectangular tiles. This paper addresses the relationship between the length of the segment and the number of tiles it visits (i.e. has intersection with). The square grid is also…

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are…

NOTE: Unfortunately, most of the results mentioned here were already known under the name of "d-separated interval piercing". The result that T_d(m) exists was first proved by Gya\'rfa\'s and Lehel in 1970, see [5]. Later, the result was…

We consider a model of randomness for self-similar Cantor sets of finite and positive $1$-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands $\delta$-close to a Cantor set of this particular…

In this paper we study the problem of locating a given number of hyperplanes minimizing an objective function of the closest distances from a set of points. We propose a general framework for the problem in which norm-based distances…

We study hyperplane covering problems for finite grid-like structures in $\mathbb{R}^d$. We call a set $\mathcal{C}$ of points in $\mathbb{R}^2$ a conical grid if the line $y = a_i$ intersects $\mathcal{C}$ in exactly $i$ points, for some…

In this paper we modify the method of Nazarov, Peres, and Volberg "The power law for the Buffon needle probability of the four-corner Cantor set", arXiv:0801.2942, to get an estimate from above of the Buffon needle probability of the…

The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and…

Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on $n$ vertices and $m$ edges. In the first (edge-independent) model, a random hypergraph $H_1$ is constructed by fixing a…

If you throw a needle or stick at random onto a floor ruled with parallel lines, such as the cracks between floorboards or tiles, from the proportion of times that the stick lands crossing a crack you can estimate $\pi$; can we get $e$ as…

I present a variant of the Buffon Needle method for determination of the value of the mathematical constant, pi. The original method is based on the random casting of a needle of length l onto a planked floor of plank width L. The described…

Consider a problem where we are given a bipartite graph H with vertices arranged on two horizontal lines in the plane, such that the two sets of vertices placed on the two lines form a bipartition of H. We additionally require that H admits…

Let $\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\K_n$ of $\Cant_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit…

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We…