Questions tagged [geometry]
A puzzle related to shapes, geometric objects (polygons, circles, solids, etc.) of any number of dimensions, the relative position of figures, and the properties of space. Use with [mathematics]
1,321 questions
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Where should the bridges be built to minimize the length of the path between two towns?
Two towns are separated by two rivers, as in the diagram.
The banks of each river are parallel lines. Where should bridges that cross the river perpendicular to the banks be built so that the distance ...
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Can a 10*10 square be paved with 1*4 rectangular stone plates?
Can a 10 * 10 square be paved with 1*4 rectangular stone plates?
I seek a very intuitive and simple answer to this puzzle.
P.S. Will post the source later. The source contains the answer but it is not ...
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Crossing a square pond with beams
Here is a classical mathematical puzzle:
In the middle of a square pond lies a square island (whose sides are aligned with the sides of the pond). On the island is a tower with a princess locked ...
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Regular pentagram inscribed in a parabola
The diagram shows a regular pentagram inscribed in a parabola.
Can another red pentagon (congruent to the one shown) fit in the gray region?
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Three-cushion billiards challenge.
World champion three-cushion billiards player can perfectly make a billiard ball end where asked for (if possible), on a standard rectangular shaped competition table.
The champion now, for ...
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A "crappy" pentagon puzzle
Lately, we've had plenty of puzzles based on the regular pentagon and its geometric properties. So I propose one that literally brings it all together.
Use eleven copies of the larger (left) piece ...
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Sliding coins down an envelope
The envelope is a regular pentagon with two diagonals. The red coins are two vertically aligned and touching congruent circles, the top one passing through the apex of the pentagon, and the bottom ...
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Two semicircles in a parallelogram
Two congruent semicircles (blue) are inscribed in the two triangles formed by the short diagonal (black) of a parallelogram (also black) as shown below. The short diagonal has the same length as the ...
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Heptagons trompe-l'œil ?
Seven outer blue, equal sized, regular heptagons touch the sides of a given, also equal sized, inner blue regular heptagon.
The green triangle has black vertices: the south and the, up to one, most ...
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A single magical ant wandering in a vast garden of cubes
The following image depicts a small part of a vast "cube garden":
This garden started as a single 1x1x1 cube, which was grown into the complex shape you see here by performing the following ...
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Can another circle fit in this diagram?
Two vertically-aligned and touching congruent (blue) circles are inscribed in a pendant (the black quadrilateral) in a regular star as shown below. Can you fit another circle of the same size that’s ...
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Circles in a pendant in a star
Four vertically-aligned touching circles of equal radius are inscribed in a pendant (the black quadrilateral) inside a regular star as shown in the figure below. Show that the centre of the green ...
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Two semicircles in a regular pentagon
Two semicircles, pink and blue, are inscribed inside a regular pentagon as shown in the figure. Which semicircle is bigger?
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Two triangles in a light bulb
Here’s a light bulb made of a regular dodecagon (yellow), two equilateral triangles (light gray), and a square (dark gray). Show that the red and blue triangles have the same area.
Of course, you can ...
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Show that the area of the region inside this square and regular pentagon is greater than 3/4
A square and a regular pentagon, each of area 1, are coplanar and concentric. Show that the area of the region inside both shapes is greater than 3/4.
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Is there a perfect squared square made entirely of squares of prime edge?
There are infinitely many sets of distinct primes whose squares add up to a square number and, presumably, sets of any size (https://mathoverflow.net/questions/501745/primes-whose-squares-add-up-to-...
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A tree-cutting problem [closed]
Dans un parc il y a 10000
arbres (vus comme des points), placés en un quadrillage carré de 100
lignes et 100
colonnes. Déterminer le nombre maximal d'arbres que l'on peut abattre de sorte que, quelle ...
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Show that the triangle has a 60° angle
On square ABCD, points E and F lie on sides BC and CD, respectively, such that BE=CF.
Line BD intersects lines AE and AF at G and H, respectively.
Using pure geometry, prove that a triangle with side ...
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Angle between lines on pentagons
The two regular pentagons share a vertex and an edge. What’s the angle between the red and blue lines?
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Which triangle has the larger area?
In the diagram, AFH, ABGF, ABCDE are all regular polygons.
Which triangle has larger area: red or blue? Or do they have the same area?
Note: Geogebra can be used to help, but an answer saying "...
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Triangles on regular pentagons
The two regular pentagons share a vertex and an edge. The side of the larger pentagon is twice that of the smaller one. Which triangle has larger area, red or blue? Or do they have the same area?
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The "e" in apple
I cut a thin slice from a spherical apple, cutting along a plane.
Then I realized, there is an "e" in apple.
Why is there an "e" in apple?
Hint:
EDIT: Puzzling Meta post about ...
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Sliding block puzzle: Get square to bottom centre
Consider these 10 blocks here, each with a different colour. One block has 4 squares, five have 2 squares and four have 1 square (the white area is unoccupied, and is not a block):
Text version:
<...
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Most symmetrical tangram
We discovered a new toy store near our house, and my daughter and I were very excited to check it out. We found a curious puzzle there and brought it home with us. After googling about it a bit, I ...
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Design a better magazine page layout. Or not.
I still read print magazines. Whenever I come across an interesting passage, I clip it out and post it on the refrigerator. I am surprised, however, how often a passage, even a relatively short ...
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Some almost smiley face facts ...
Very much inspired by excellent
An angle in a smiley face
While trying to solve it I noticed:
What if two (red) squares ABCD and A'B'CD' sharing C
do not have all 4 vertices AA'BB' on one single (...
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An angle in a smiley face
What's the angle in the following figure?
Clarifications:
The two squares share a vertex.
Each square has two vertices on the circle.
The squares are not of equal size.
Source: Inspired by a recent ...
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How to select P so that the angle APB is as large as possible?
Given a line and two points A and B, which point P on the line forms the largest angle APB?
Bonus question: How should we select P so that the angle APB is as small as it can be?
P.S.
I tried solving ...
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A crown in a polygonal cycloid
Take a regular polygon resting on one of its sides. Pick the left vertex of the side and start rolling the polygon to the right until that vertex touches the ground. Mark the location of that vertex ...
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If 3-D is too easy, go 4-D
It is a well-known puzzle that one can take a cube and make a single planar cut through it so that the intersection of the cutting plane and the cube is exactly a regular hexagon.
One can do an ...
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What is the probability that a random tetrahedron inside a sphere is intersected by the sphere's vertical axis? [closed]
A tetrahedron's vertices are independent uniformly random points in the interior of a sphere.
What is the probability that the tetrahedron is intersected by the sphere's vertical axis?
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A tour through a city of circular roads with no sharp turns
Anita lives in a city with a peculiar road system: every road is a circle (not necessarily of the same radius). The rules of the system are simple: no sharp turns. That is, if you are at a transversal ...
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How to cut a tetrahedron in half using only a pencil and a knife?
You are given a pencil, a knife, and an object, and you are asked to cut the object in half using only the tools provided to you. For example, if the object is a square or a cube, you can mark the cut ...
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What is the limiting probability that the polygon contains the circle?
A red circle is surrounded by a chain of n congruent green circles, with each green circle touching its two neighboring green circles and the red circle. On each green circle, a random (uniform and ...
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Tiling 7×107 rectangle with heptominoes
Is it possible to tile a 7×107 rectangle with the 107 heptominoes that do not have a hole?
Obviously, the heptomino with a hole cannot be used to tile, and there are 107 remaining heptominoes?
Rules:
...
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Polygon whose sides are not completely visible from a point
Can you construct a polygon (not necessarily convex, but definitely not self-intersecting) such that there is an internal point from which no side of the polygon is completely visible? If yes, what is ...
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Dividing both the interior and the boundary of a square into equal parts using polyominos
We want to cover an m×m square with n non-overlapping axis-parallel polyominos such that both the interior and the boundary of the square are divided into n equal areas and n equal lengths, ...
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Fiery tsunami on a sunny day
I was playing around with some ideas for a nice geometry puzzle involving circles on my iPad when my daughter saw it and said it looked like fire. I then added some background elements like the sun, ...
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Can you make a closed loop of interlocking chips with three notches?
Here's a circular chip with three evenly spaced notches. Two such chips can be attached as shown, where the "boundary circle" of each chip goes through the center of the other, and the ...
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Compact arrangement of Qwirkle tiles
Qwirkle tiles are identically-sized squares. Each has two attributes: one of 6 symbols and one of 6 colors. There are 3 tiles for each combination of symbol and color, thus 6×6×3 = 108 tiles. We want ...
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Will the ice cream spill over?
Ever since my daughter discovered ice cream, she’s been obsessed with it. She wants to eat it all the time, and try all flavours every time. So we came up with a plan where she could sample all the ...
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Can you dissect the red part of “12” into any number of congruent pieces that divides 12?
Here’s a diagram that looks like 12.
Can you dissect the red part of the diagram into n congruent pieces for every n that divides 12?
All the angles are multiples of 45° and if two lengths seem equal,...
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Are there other LEGO Duplo track layouts with two trains that trigger all the switches indefinitely?
My daughter has a LEGO Duplo railway set that she loves to play with. Here are some basic track elements in the set (I made the figures myself): the red element is a circular arc of 30°, the green ...
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Which line is more likely to intersect the circle?
A green circle is tangent to a red circle and a black circle. The three circles have equal radii. Their centres are collinear and distinct.
Random point A is chosen on the red circle. Random points B ...
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A disk and a triangle: What is the probability of intersection?
An equilateral triangle has one vertex at the centre of a disk.
One side of the triangle lies completely outside the disk and is colored green.
A red line is drawn through two independent, uniformly ...
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Dissecting a cube of side n into n interlocking congruent polycubes
A polycube is a three dimensional generalisation of polyomino, i.e., it is formed by gluing unit cubes along their faces. Given a cube of side n, one can easily dissect it into n congruent polycubes ...
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How many IKEA train track layouts with self-intersections?
Recently, I got an IKEA LILLABO train set for my daughter. It has twelve curved segments, each is 1/8th of a circle of radius 1, two straight segments of length 1, and a bridge of length 2 (in top ...
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Covering a Rubik's Cube Pattern with Pentominoes
Given a Rubik's Cube pattern:
And using each of the following 12 pentominoes at least once:
Can you cover entirely the Rubik's Cube?
The answer is obviously yes!
However, given that the Rubik's Cube ...
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