# 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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### 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 ...
493 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 ...
137 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 ...
1k views

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 ...
233 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-...
457 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? ...
761 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 ...
617 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 ...
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 ...
91 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 ...
131 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 ...
215 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 $f$. ...
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### 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 ...
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 ...
185 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 ...
282 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. ...
217 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 ...
167 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.
419 views

### Move just 2 matchsticks to make three equally sized triangles

There are 9 equally sized matchsticks, move 2 to make 3 equal triangles
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### 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 ...
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 ...
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, ...
183 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 ...
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 ...
183 views

### Cutting a Rectangular Board

There is an $m \times n$ rectangular board drawn on a graph paper. You need to cut it into $mn$ $1 \times 1$ squares by straight cuts along the grid lines. You are allowed to stack several pieces ...
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### How can I cut a cube so that all its vertices except for two mutually opposite vertices are equally distanced from the plane of the cut?

A friend of mine has been struggling with a solid geometry problem and, knowing my imagination skills developed by playing gomokunarabe and renju, has asked me to help her, but the problem has proved ...
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### Pythagorea Toughie

I recently happened upon a game called pythagorea. The idea is that you're given a 6x6 grid. You may click at any intersection to create a point and you may join any two points to create a line (that ...
237 views

### An $n$-sided polygon with area $n$

Here is a $10$-sided polygon which area is $23$ (i.e. it contains exactly 23 unit squares). Can you draw a polygon with: $6$ sides and area $6$? $8$ sides and area $8$? $12$ sides and area $12$? ...
133 views

### When is a robotic arm able to reach any point (closer than the length of the outstretched arm)?

In a plane, there is a robotic arm consisting of $n \ge 2$ segments of length 1, like this: The first segment is fastened to a single point ("origin"), but it can rotate freely around that point. All ...
219 views

### Interplanetary blips and bleeps

Things were quite different in 3000 AD. We'd discovered other planets with sentient life for instance. Five to be precise. Adam, Bill, Carl, Dave, and Eric we called them, and we used the lately ...
641 views

### Perfect Golomb Circles

A Golomb ruler of order $n$ is a straight line with $n$ marks (at integer locations) such that no two pairs of marks are the same distance apart. We can extend the concept to circles. Place $n$ marks ...
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### The death prism

One day, you are caught by a evil wizard. He presents you with a prism, and says, "You can ask me to turn this prism to any $n$-angled right prism. Then you shall fill in $1$ to $3n$ with no ...
607 views

### A rectangle in a rectangular hole

I have a carpet of 240 inches by 120 inches, but my floor, which it needs to cover, is 180 inches by 160 inches. How can I do this by cutting the carpet into exactly two pieces? Source: Rational ...
117 views

### Can we visit each block of 4*4 exactly once? [duplicate]

A person is standing at a corner of $4\times4$ square, he would like to travel each block exactly once before exiting from the opposite corner. Is there a way?
293 views

### Goldman's Transformation Puzzle

In an old book (*) I found an advert with a puzzle I had never seen before. Unfortunately the book is very rare and it is hard to make out in the scanned photocopy above. I have redrawn it: It looks ...
319 views

### Ernie and the Chocolate Bomb

Last week when I visited Ernie he complained that his trousers had shrunk in the wash. I suggested an alternate possibility – he was putting on weight. So he decided to go on a diet and to support him,...
2k views

### Two Cannons - A Beginner's Physics Puzzle

Standoff Let's say we have two cannons aimed directly at one another ( as my horrible attempt shown in the image ). The Angles The cannon on top is aiming down towards the one on bottom right, and ...
2k views

### Cut this shape into 3 pieces and fit them together to form a square

A shape is drawn on a sheet of squared paper as shown in the picture below. The shape is then cut from the sheet and given to you. You are asked to first make a straight cut across the shape and then ...
154 views

### Squared, what is in the post-scriptum?

Just received an invitation from a fellow mathematician for a party. The invitation is quite clear: where I need to be at what time is clearly said in the invitation, but at the bottom of the ...
384 views

### How to obtain the 6- and 12-gon with 6 equal rectangles?

Let's say you have 6 equal business cards. How can you place them on a table to create these shapes: regular hexagon regular dodecagon Edit. Feel free to propose an answer for options: a) you have ...
259 views

### Picture-Puzzle. Find the number!

Make sense out of this picture below and find the two-digit number! Note: Pay attention to everything except the rounded rectangles
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### You find a piece of paper in your bag

In your bag you find a piece of paper with a size of 5 x 5. You want to make it a 6 x 4 but you may only make 1 continuous cut and reposition the pieces. You are not allowed to bend or twist the ...
506 views

### Ten squares inside a rectangle

You are given 10 squares with side length of 1 to 10 units each. You want to put them in a rectangle such that there is no overlapping and no piece of a square is outside the rectange. The sides of ...
137 views

### Icosahedron-Wrapping Monstrosity

The following monstrosity of a shape: ... can be wrapped onto the surface of an icosahedron in a way that completely covers the entire icosahedron with no gaps and no overlaps. How can it be done?
351 views

The following tri-bladed boomerang shape: ... can be folded onto the surface of an octahedron in a way that perfectly covers the entire octahedron with no gaps and no overlaps. How can it be done?
198 views

### Mark Two Points Which Have a Distance of $\sqrt{3.6}$

Here is a grid graph with $7$ horizontal and $7$ vertical lines which are $1$ unit apart. It is trivial to mark two points which have a distance of $\sqrt{36}$. Drawing at most two extra lines as ...
2k views

### A Slightly Moth-Eaten Integral Sign

The following slightly moth-eaten integral sign: ... can be folded onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps. How can it be done? (You ...
339 views

### Two shapes that cover a 4x4 grid with any 1x2 missing

Can you find two geometrical shapes with the following property: If you remove any 1x2 rectangle from a 4x4 grid, then the remaining area can be exactly covered with the two shapes. What do these two ...