# 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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### Tiling rectangles with F pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with N pentomino plus rectangles Tiling rectangles with T pentomino plus rectangles ...
429 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? ...
417 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 ...
951 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 ...
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 ...
117 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 ...
149 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, ...
171 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 ...
221 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 ...
505 views

### Squaring the new year

Happy new year! You are asked to cut a rectangular strip 2015 times longer than its width into pieces that can be reassembled into a square with area equal to that of the original strip. How many ...
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 ...
433 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 ...
121 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 ...
218 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-...
685 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 ...
608 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 ...
88 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 ...
122 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 ...
208 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 $f$. ...
637 views

### Can't figure this one out.. What is the missing box?

I've been stuck on this for ages, and can't figure this out. What is the missing box, and the logic behind the answer? This was taken from a Korn Ferry Leadership Assessment practice trial.
114 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 ...
174 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 ...
62k views

### Cutting a cake into 8 pieces

Say, you are given a cake which you must share with 7 others. So, you must cut the cake into 8 pieces. But, you are only allowed to make 3 straight cuts. You cannot move pieces of the cake after the ...
106 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 ...
278 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. ...
193 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 ...
409 views

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

There are 9 equally sized matchsticks, move 2 to make 3 equal triangles
159 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.
138 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 ...
132 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 ...
131 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, ...
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 ...
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 ...
646 views

### The farmer and the olive trees

A farmer has a rectangular ground of 100 m by 50 m, he wants to plant olive trees, in sufficiently spaced ways (to avoid exhaustion by the roots) at least 10 meters from each other. How much can one ...
785 views

### Bouncing ball on a billiard board

Consider a unconventional billiard board in the shape of an equilateral triangle (depicted below). An incredibly small ball (size in picture is increased for the sake of visibility on your screen) is ...
2k views

### Blue. Orange. Green. Magenta. What does this strange picture represent?

Is it text? Is it a face? Is it code? What is it?
180 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 ...
231 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 $12$? ...
119 views

### 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 ...
198 views

### 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 ...
631 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 ...
131 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 ...
209 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 ...
2k views

### Is there a simple algorithm for solving Kami 2 puzzles?

I'm finding my life has been consumed by Kami 2, partly because I seem to have achieved some "insight" and am able to solve the puzzles reliably, almost always on the first try. The rules are simple: ...
183 views

### 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 ...
3k views

### Variant of lion and 100 zebras

Note: This problem remains unsolved, as of 19 April 2020, so do try it out. 400 rep bounty guaranteed for a correct answer This a variation of this question by @Gamow Suppose there are $100$ lions ...
602 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 ...
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?