Geometry
Subcategories
Puzzles
- 3d.problemsHere are nine elementary problems in three-dimensional geometry, suitable for advanced high-school students. New, unsolved
- K3,3Can three houses be connected to three utilities without the pipes crossing? Restored
- annulus.chordTwo concentric circles are given. New
- ball.in.cornerA beach ball of radius 1 foot is placed in the corner of a room so it is tangent to the floor and two of the room's walls. New
- bearIf a hunter goes out his front door, goes 50 miles south, then goes 50 miles west, shoots a bear, goes 50...
- bear.square.walkA wildlife photographer goes one mile south, one mile east, one mile north and one mile west, and is back... New
- bisectorProve if two angle bisectors of a triangle are equal, then the triangle is isosceles (more specifically,...
- calendarBuild a calendar from two sets of cubes.
- circle.regionsPut n points on a circle and connect each point to every other point by straight lines so as to form the maximum number of regions. New
- circle.splitting.pointsThere are 2n+3 points in the plane, no three of them collinear and no four of them concyclic. New
- circles.and.trianglesFind the radius of the inscribed and circumscribed circles for a triangle.
- circles.through.gridWhat is the smallest number of circles k that pass through all the points of an n x n square grid of points? New
- coins.to.hexagonArrange 6 coins in two rows of three coins nestled together; what is the minimum number of slide moves to... New
- coins.trianglePut 10 equal coins on a table to make a filled triangle with a side of 4 coins. New
- conicsIs there a formula which describes *all* conic sections, which will generate a particular class of same... New
- cornerA hallway of width A turns through 90 degrees into a hallway of width B.
- cover.earthA thin membrane covers the surface of the (spherical) earth.
- cube.cutsA 3x3x3 cube of cheese can be cut into 27 1x1x1 cubes with 6 cuts (2 along every dimension; like a Rubik's Cube). New
- cycle.polynomialWhat are the cycle polynomials for the Platonic solids?
- duck.and.foxA duck is swimming about in a circular pond. Added material
- earth.bandHow much will a band around the equator rise above the surface if it is made one meter longer? Added material
- fenceA farmer wishes to enclose the maximum possible area with 100 meters of fence.
- four.peopleAlice, Bob and Carol are standing on a straight line. New
- grad.school.triangleIn isosceles triangle PQR, angle QPR = 20 degrees and the other two angles are 80 degrees. New
- grid.linesHow many straight lines do you need to cross every square of a 3x3 board? New
- half.angle.tangentsFind a triangle with integer sides such that the product of the tangents of its half-angles is (2^8 5^2... New
- ham.sandwichConsider a ham sandwich, consisting of two pieces of bread and one of ham.
- hamilton.pathDevise a continuous path that visits each * exactly once: New
- hexagon.in.triangleStart with an equilateral triangle. New
- hikeYou are hiking in a half-planar woods, exactly 1 mile from the edge, when you suddenly trip and lose your sense of direction. Added material
- hole.in.sphereOld Boniface he took his cheer, Then he bored a hole through a solid sphere, Clear through the center,... Added material
- hypercubeHow many vertices, edges, faces, etc.
- inscribed.polygonA polygon of 100 sides is inscribed in a circle. New
- intersecting.cylindersTwo cylinders of unit diameter intersect (100%) at right angles. New
- intersection.free1,000,000 black points and 1,000,000 white points exist on a planar disk.
- kissing.numberHow many n-dimensional unit spheres can be packed around one unit sphere?
- konigsbergCan you draw a line through each edge on the diagram below without crossing any edge twice and without...
- ladder.and.boxA 3' * 3' packing case is placed against a wall. Added material
- laddersTwo ladders form a rough X in an alley.
- lighthouseTwo people walk to the top of a cylindrical lighthouse, one via a straight ladder up the middle and the... New
- manhole.coverWhy is a manhole cover round? Added material
- morleyTake any triangle and trisect its internal angles. New
- mouse.in.penA toy wind-up mouse is put in a circle pen with 1 m radius. New
- nearest.townTake a map and connect each town shown on it to its nearest (as the crow flies) neighbor by means of a straight line segment. New
- nine.dotsDraw 4 straight lines in such a way that all the dots of a 3x3 array are touched. New
- opaque.cubeWhat is the least area surface that can block all lines of sight through a cube? New
- pentagram.ten.trianglesA regular 5-pointed star is composed of five lines, and has 5 non-overlapping triangles in it. New
- pentominoArrange pentominos in 3x20, 4x15, 5x12, 6x10, 2x3x10, 2x5x6 and 3x4x5 forms. Added material
- pentomino.duplicationDivide the 12 pentominoes into two sets of 6 that form two identical shapes (shapes that overlap exactly... New
- photograph.horizonYou have a photograph of a person in front of a sunset over the sea. New
- piano.around.cornerWhat is the longest (or largest) object that can be moved along a corridor of width W and around a... New
- platonic.volumeWhat are the volumes of the regular polyhedra? New
- point.in.triangleGiven a triangle in the xy plane (specified by its vertices) and a point, find a computationally simple... New
- points.in.rectangleGiven a rectangle of sides A and B, two points are picked at random, independently and uniformly with respect to area. New
- points.in.sphereWhat is the expected distance between two random points inside a sphere?
- points.on.sphereWhat are the odds that n random points on a sphere lie in the same hemisphere? Added material
- polycubesHow many pentacubes are there really? New
- polyomino.exclusionGiven a finite family of polyominoes (of any sizes), is the set of the polyominoes that do NOT contain any... New, unsolved
- polytope.volumeHow can one calculate the hypervolume of a convex n-dimensional polyhedron given the coordinates of its... New
- puzzle.ringHow can I put together my puzzle rings (where four bands become one)? Added material
- pyramid.trianglesJoin every pair of the 5 vertices and 8 edge midpoints of a square pyramid. New
- regular.polyhedraWhy are there only five regular polyhedra, and why is the solid made of two tetrahedra glued face to face... New
- revolutionsA circle with radius 1 rolls without slipping once around a circle with radius 3.
- rotating.lineA line of unit length is enclosed within a two-dimensional area. New
- rotationWhat is the smallest rotation that returns an object to its original state?
- satellitesYou have a planet that's a perfect sphere. New
- seven.pointsCan you arrange seven distinct points on a plane, so that of any three taken at random, at least two would... New
- shephard.pianoWhat's the maximum area shape that will fit around a right-angle corner?
- shortest.creaseYou have a sheet of paper that is 6 units wide and 25 units long, placed so that the short side is facing you. New
- six.tangent.circlesStart with a circle O of radius 1 and let a and b be arbitrary real numbers. New
- smugglerSomewhere on the high sees smuggler S is attempting, without much luck, to outspeed coast guard G, whose boat can go faster than S's.
- sphere.packingAre face-centred cubic and hexagonal close packing of equal spheres really two different arrangements, or... New
- spider.and.flyA room is 30 feet long, 12 feet wide and 12 feet high. New
- spider.on.treeThere is a tree, circumference of which is 4 ft. New
- spiralHow far must one travel to reach the North Pole if one starts from the equator and always heads northwest? Added material
- spy.planeA spy plane must see the entire surface of the Earth in one flight; what is the shortest flight, starting... New
- square.holeIs it possible to drill a square hole? New
- steiner.treeFour cities stand at the corners of a square. New
- surveyed.triangleA team of surveyors are given the task of finding the size of an equilateral triangular plot. New
- table.in.cornerPut a round table into a (perpendicular) corner so that the table top touches both walls and the feet are firmly on the ground.
- tangent.circlesInside a pipe 3 units in diameter are two pipes, of diameters 2 and 1. New
- ten.lines.6x6Using ten continuous lines, and not lifting your pencil from the paper, go through all the dots of a 6x6... New, unsolved
- tetrahedra.in.cubeHow many regular tetrahedrons with edges of length square root of 2 can be packed into a cube with edges of... New, unsolved
- tetrahedronSuppose you have a sphere of radius R and you have four planes that are all tangent to the sphere such that...
- three.churchesIn George's small home town there are three churches, equidistant from each other as the crow flies - that... New
- three.squaresThree squares are placed side by side: New
- touching.blocksCan six 1x2x4 blocks be arranged so that each block touches n others, for all n?
- touching.cubesGiven a cube, how many cubes (all identical to the original) can you place around it so they all touch it? New
- treasure.mapA treasure map locates the treasure using four landmarks, two of which have vanished. New
- triangle.20.degreesIn isoceles triangle ABC angle A=20, B=80, C=80. Added material
- triangular.regionsShow that when n lines in general position in the real affine plane (no two parallel, no three concurrent)... New
- walkA person walks up a hill starting at 8 pm and arrives at the top at midnight.
- weighted.bisectorsThe perpendicular bisectors of the sides of a triangle meet in a single point. New
- weird.riverA river flows from town A to town B and has the property that any point on either of its banks is no... New
- wooden.blockHow was this wooden block made from two pieces, with the joint showing as a dovetail on two adjacent faces? New
- worm.blanketPerson A has a piece of string exactly one meter long. New
- wrapping.paperYou have a rectangular solid, A x B x C in dimension, and you want to wrap it with a rectangle of wrapping paper, D x E in size. New