Questions tagged [geometry]

A puzzle related to shapes, geometric objects (polygons, circles, solids, etc.) of any number of dimensions, relative position of figures, and the properties of space.

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23
votes
1answer
1k views

Wrap a squashed, bullet-riddled lowercase lambda around a cube

The following rather squashed and bullet-riddled lowercase lambda: ...can be wrapped onto the surface of a cube in a way that perfectly covers the entire cube, with no gaps and no overlaps. How can ...
10
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4answers
601 views

A packing game!

Amy and Ben are playing a game which is suggested by a genie. Amy first chooses $a,b,c\in\mathbb{R}^+$. Then a empty cuboid box with internal measurements $a+b,b+c,c+a$, and infinite supply of cuboid ...
12
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4answers
1k views

Cover a square with three smaller squares

A square has a side length of 5 units. Is it possible to cover this square with three squares each with a side length of 4 units?
18
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2answers
334 views
+50

Dominoroto-toto

Consider a domino tiling of a plane rectangle of size $n \times m$. (Obviously, at least one of $m$ and $n$ has to be even for that to be possible.) I personally hate those because they tend to look ...
17
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3answers
1k views

Most triangles formed by three triangles

What is the maximum number of triangles you can form by drawing three triangles on a piece of paper? Good luck!
4
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1answer
76 views

Form nine squares from three squares

Can you draw three squares on a piece of paper, such that they form nine distinct squares? Good luck!
7
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1answer
82 views

Minimal 2D pattern for domino tiles where each tile touches three others

Inspired by The five problems of the six domino tiles, where one of the tasks was to place six domino tiles so that each tile touches three others (corner / edge touches don't count). My solution ...
2
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3answers
105 views

How to find the number of ways to go from one point to another in a truncated structure?

I've found this problem in my book "Riddles and reason" and after several attempts I still have no idea how to tackle it. The problem is as follows: The figure from below shows a truncated ...
14
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2answers
830 views

The five problems of the six domino tiles

Here is a set a problems (regarding domino tiles) of a famous Portuguese newspaper weekly magazine. For each problem you have $6$ domino tiles and the goal is always to place them touching each other ...
3
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2answers
143 views

4 Kids vs easter bunny

There is a game played between the easter bunny VERSUS a team of 4 kids. I will fully explain the rules of the game below. However, I'd like to start with the preface. I found this problem as a king ...
4
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3answers
221 views

Cut the square cloth!

I have a square cloth with side length $x$ cm, and I am going to cut it into at least $n$ squares with side length $1$ cm for my customer, and also you cannot cut the cloth to thinner pieces (reminded ...
0
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1answer
98 views

Concerning Tetrahedra

As a pyramid with a triangular base, the volume of a tetrahedron, like all pyramids, is $(1/3)*BH$, where $B$ is the base area and $H$ is the height. If one had $3$ square $45$ degree pyramids (square ...
7
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3answers
270 views

Touching triangles at their vertices

What is the minimum number of non overlapping congruent triangles arranged in the plane, such that each vertex of the triangles coincide with exactly three triangles?
3
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1answer
93 views

An angle between diagonals

The diagonals of a square intersect at a right angle. Is that true in three dimension, i.e. have the two diagonals of a cube, each running from one corner of the cube to its opposite corner and ...
1
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2answers
152 views

Geometric logos 2

Following the idea of Geometric logos. Which famous company/software logos include the following geometric shapes (listed alphabetically by company)? Three congruent parallelograms forming a hexagon ...
14
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3answers
2k views

Ten miles south, east, north and west

I'm standing on the surface of the Earth. I walk ten miles south, ten miles east, ten miles north and ten miles west. I end up exactly where I started. Where on earth am I?
4
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1answer
115 views

Geometric logos

Which famous company logos have the following geometric descriptions (listed alphabetically by company)? Four congruent circles with six points of intersection Regular pentagon containing five ...
15
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1answer
662 views

Arrange ten pawns into ten lines of three

This is not a chess problem! In the following position you can see six pawns that have been arranged into lines of three. Each pawn stands at the intersection of exactly two lines and each line ...
20
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6answers
2k views

Size of a square in a square

Given a unit square (blue in the picture), pick a point on one edge and label it A. Label the distance from the nearest corner to A as x. Pick one of the corners opposite A and label it B. Call the ...
6
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3answers
289 views

Extended face-planes of an Octahedron

When extending the face-planes of a regular Octahedron, how many cells (bounded and unbounded) are formed in space?
23
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5answers
1k views

Painted faces on a cube

Here's another challenge I used to give to my students: Let's begin with a bunch of little white cubes assembled into a big white cube. All the little white cubes are equal. Then I decide to paint ...
8
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4answers
477 views

The broken wheel

In a regular polygon, we can connect six of the vertices to form a convex hexagon. I construct a hexagon by picking six vertices out of a regular $n$-sided polygon. In this hexagon, if you connect the ...
3
votes
1answer
125 views

Fill in numbers on the cube … again!

You are given a cube. You are told to fill in each face randomly with some of the numbers $4, 5, 6, ..., 11$, with no repetition. What is the probability that for each two faces that are connected by ...
4
votes
1answer
158 views

Fill in numbers on the cube!

You are given a cube. You are told to fill in each vertex with the numbers $4,5,6,...,11$, with no repetition. What is the probability that for each two vertices that are connected by a common edge, ...
2
votes
1answer
185 views

An exam question testing your spatial sense. (Is it worded correctly?)

I recently got the following question from a friend. This kind of question was asked in one of her exams (it really doesn't matter what kind of exam that was). I after trying to solve said question ...
25
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5answers
2k views

Pythagorean quilts

The King requests Pythagoras to his palace to discuss an important matter. After the usual formal greetings the King asks: - I have been told that you have a marvelous formula about adding squares ...
3
votes
3answers
233 views

Most number of equilateral triangles formed by 13 points

What is the most number of equilateral triangles you can form by drawing 13 points on a piece of paper? Each triangle must have 3 equal sides and pass through 3 points. Only equilateral triangles can ...
7
votes
3answers
489 views

Balance the nails

Someone I know handed me this puzzle, I have seen a couple of solutions for it that follow the instructions and don’t involve bending the nails, etc. Can you figure out how to balance the 6 nails on ...
3
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2answers
128 views

Broken stick riddle

There is a famous mathematical riddle called The Broken Stick Problem. Here’s the extension: If a straight stick is accidentally broken into three pieces, the probability of being able to form a ...
9
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2answers
1k views

Escape from your friend!

I saw this interesting problem in a Mathematics book in Chinese(I translated it): You and your friend is playing a game. There is a square swimming pool, and you are in the middle of it. Your friend ...
6
votes
2answers
228 views

Circle inside circle v2

This question is a kinda follow-up question to: Circle inside Circle You have a large circle with radius $5$ units and you also have a small circle with radius $1$ unit. But this time you have a 10-...
9
votes
4answers
452 views

Largest and smallest hexadecagon with sides $1, 2, 3, \dots,16$

Of all hexadecagons lying in the cartesian plane, all of whose vertices are lattice points, and whose sides are of length $1,2,3,\dots,16$ in some order, which two have the largest and smallest area? ...
13
votes
3answers
755 views

Pythagorean triplets wheat field

A rectangular field has width $a$ and length $a+1$. We cut it into 3 triangles that all have integer side lengths. If all triangles have a different area, then what’s the minimum value of $a$? Please ...
15
votes
3answers
616 views

Largest and smallest dodecagon with sides $1, 2, 3, \dots,12$

Of all dodecagons laying in the cartesian plane, all of whose vertices are lattice points, and whose sides are of length $1, 2, 3, \dots,$ and $12$ in some order, which two have the largest and ...
16
votes
4answers
1k views

The enclosure on a grid

On an infinite 1 by 1 grid, we want to make an enclosure with 20 fences that are each 5 units long. The two ends of each fence has to be on a node of the grid. What is the maximal area of the ...
1
vote
1answer
90 views

A question based on cutting a wire to form a tetrahedron

I am trying exercises of Quantitative Aptitude and I am unable to work out how this problem can be solved: As a tetrahedron has 6 edges, I thought 5 cuts should be required. But that's wrong. Answer ...
3
votes
1answer
128 views

IQ test question (double-arrow with inverted flukes)

I found the following question in the New Zealand Mensa practice test: According to the website, the solution is B: The first symbol is a double-arrow with inverted flukes. The second is the same ...
3
votes
1answer
210 views

Find the equation of this surface - The Snowman

Here is a surface that resembles a snowman: Its equation has the form $f(x,y,z)=0$: if a point $(x,y,z)$ satisfies that equations it is shaded, otherwise it is "left blank". Your goal is to find $f$. ...
1
vote
2answers
115 views

Triangular pool and three swimmers [closed]

Given a triangular pool 100, 120, 140 yards and three swimmers which swims at rates of 3.5, 4.0, 4.5 yards per second - place them on the edges of the pool in such way that when they start swimming at ...
3
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2answers
108 views

Tangential circles

The following figure has two axes of symmetry which define its width and length. The length (horizontal distance) is twice the width (vertical distance). The largest circle has a radius of 2005 and ...
9
votes
2answers
180 views

Cut It Up! (then find a phrase)

This is a "sorry for messing up yesterday's clues" puzzle. For each shape below, divide along grid-lines into identical pieces (rotation and reflection allowed). The first shape has 3 pieces, and the ...
9
votes
1answer
281 views

120 degrees and circles

The bigger circle has a radius of $1$, and it is tangent to the two straight lines that forms an angle of $120$ degrees. The smaller circle is tangent to the two straight lines and the big circle. ...
7
votes
1answer
199 views

Three squares in a triangle

In a triangle, three identical squares of side lengths 2.8 share a common vertex and are each touching two sides of the triangle. If one of the angles in the triangle is 75 degrees and is opposed to a ...
8
votes
2answers
164 views

Intersecting shapes on a flat surface

What is the maximum number of enclosed regions that you can create by drawing two circles and two triangles on a flat surface? Try answering with mathematical arguments.
11
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5answers
413 views

Move just 2 matchsticks to make three equally sized triangles

There are 9 equally sized matchsticks, move 2 to make 3 equal triangles
7
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1answer
139 views

Geometry optimization

Three equilateral triangles with side lengths 28 are placed in the position as shown in the picture above. All the contacts are perfect and a circle passes by exactly one vertex per triangle. What’s ...
9
votes
1answer
133 views

Table covering with tablecloths

In front of me stands a table with the shape of an equilateral triangle with side lengths 1. I can cover the whole surface with five identical circular tablecloths. What is the minimum radius for a ...
5
votes
1answer
132 views

A new Sangaku puzzle

A cyclic hexagon is inscribed inside a circle. The sum of two consecutive sides always equals 149. Then, we triangulate the hexagon into four triangles each containing an incircle, and surprisingly, ...
5
votes
2answers
182 views

The pond of symmetry

There is a $4$m by $4$m square pond. You have $3$ straight planks of wood, each exactly $2$m in length. You need to place the planks so that they go from one corner of the pond to the diagonally ...
24
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4answers
4k views

A COVID-19 puzzle: How large a class do you need to fit 30 pupils?

Some countries are proposing to reopen high schools soon. To ensure safety, they want to make sure that all pupils in a class are at least 2 m apart. To help them find the smallest room that can ...

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