Related papers: Cubes and Boxes have Rupert's passages in every di…
Given two cubes of equal size, it is possible - against all odds - to bore a hole through one which is large enough to pass the other straight through. This preposterous property of the cube was first noted by Prince Rupert of the Rhine in…
A shape possesses Rupert's property if a hole can be cut through it such that a second identical copy of the shape can cleanly pass straight through the interior of the first. Such a passage proving cubes are Rupert was first shown more…
A polyhedron $\textbf{P} \subset \mathbb{R}^3$ has Rupert's property if a hole can be cut into it, such that a copy of $\textbf{P}$ can pass through this hole. There are several works investigating this property for some specific polyhedra:…
O. H. Keller conjectured in 1930 that in any tiling of $\Bbb R^n$ by unit $n$-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $n\le 6$. We show that for all $n\ge 10$ there exists a…
A rectangular parallelepiped is called a cuboid (standing box). It is called perfect if its edges, face diagonals and body diagonal all have integer length. Euler gave an example where only the body diagonal failed to be an integer (Euler…
Given a chain of $HW$ cubes where each cube is marked "turn $90^\circ$" or "go straight", when can it fold into a $1 \times H \times W$ rectangular box? We prove several variants of this (still) open problem NP-hard: (1) allowing some cubes…
One unsolved mathematical problem remains the perfect cuboid problem. A perfect cuboid is a rectangular parallelepiped whose edges, face diagonals and space diagonal are all expressed as integers. No such cuboid has yet been discovered and…
A convex polyhedron is Rupert if a hole can be cut into it (making its genus $1$) such that an identical copy of the polyhedron can pass through the hole. Resolving a conjecture of Jerrard-Wetzel-Yuan, Steininger and Yurkevich recently…
A perfect cuboid, popularly known as a perfect Euler brick/a perfect box, is a cuboid having integer side lengths, integer face diagonals and an integer space diagonal. Euler provided an example where only the body diagonal became deficient…
The perfect cuboid problem is an old famous unsolved problem in mathematics concerning the existence or non-existence of a rectangular parallelepiped whose edges, face diagonals, and space diagonal are of integer lengths. Recently Walter…
Nobody has discovered any perfect cuboid and there is no formula to deliver all possible Euler bricks. During investigations of famous open problems regarding the perfect cuboid and Euler brick; I have found new important conjectures on…
This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain…
In 1903, noted puzzle-maker Henry Dudeney published The Spider and the Fly puzzle, which asks for the shortest path along the surfaces of a square prism between two points (source and target) located on the square faces, and surprisingly…
The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces.…
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Finding such a cuboid is equivalent to finding a perfect cuboid with all…
In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition,…
In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower…
A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The problem of finding such cuboids or proving their non-existence is not solved thus far. The second…
This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also…
Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each…