# 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]

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### Line of destruction

There is a line of infinite length (about zero thickness) in a 3D space. It can rotate or move across any plane at any speed. Every point (flagged or not) that has already been visited by the line ...
272 views

### A Hollow Sphere

Imagine a sphere with a hole that has been drilled clean through its center (i.e. a cylindrical piece of the sphere is now missing). This new shape, with the core missing, has height of 6 when ...
772 views

### Tricoastal, quadricoastal, and hexacoastal countries

Let’s define a distinct coastline as a coastline you can theoretically walk completely without going through the territory of another country. For example, apart from islands, Mexico has two ...
602 views

### Rapunzel and the Prince

This is a simple mathematical puzzle, which I decided to improve a bit one year after posting. Some of the answers below consider slightly different, but equivalent setting of the problem. Rapunzel ...
15k views

### Is it always possible to balance a 4-legged table?

A perfectly symmetrical small 4-legged table is standing in a large room with a continuous but uneven floor. Is it always possible to position the table in such a way that it doesn't wobble, i.e. all ...
1k views

### Various cross-sections of platonic solids

We're going to take the 5 platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron) and suspend them in various ways (we'll assume that they are solid and of uniform density). ...
229 views

### Another watch puzzle

You are given a rectangular watch, as provided in the picture below. How many times during a period of 12 hours, starting at 12:01 AM and ending at 12:01 PM, do the hour and minute hands divide ...
387 views

### Square inscribed in a circle (with a ruler only)

Inscribe a square in a given circle by following the rules of construction with ruler and compass but... without using the compass. The center point of the circle is given too.
4k views

### All clock hands at equal degrees from each other

My dad once asked me: At what time will the second hand, minute hand, and hour hand on an analog clock all be 120° from each other? It's a simple question, but I thought it was a fun one to figure ...
4k views

### Seven overlapping circles

The area of a circle of radius 1 is completely covered by seven smaller circles, all with the same radius as each other. (The circles can overlap - indeed they must!). What is the smallest radius the ...
877 views

### When is the area between the hour and minutes hands equal?

The watch in the picture below contains a square, around which the hours are marked. At 3:00 the area enclosed between the hour hand, the minute hand, and the square sides is ¼ of the total area of ...
3k views

### A dozen into six rows?

You were given 12 coins by your friend. He bet that if you could arrange these dozen coins into 6 rows of 4 coins such that it makes two similar shapes, he will give you 12 more coins. How will you do ...
2k views

### Turning a dog (Part 2) [closed]

Here is Mr. Dog, friend of Mrs. Goat. Move only two matchsticks such that Mr. Dog's head faces the opposite direction, i.e. to the right side. This is Part 2 of turning the animal series. See Part 1 -...
1k views

### Ernie and the Artificial Emmental

I dropped in on Ernie last week to help him prepare for his annual hosting of the Wine and Cheese Club party. This was a special occasion because the invited guest was the famous oenophile and ...
2k views

### Measure the diagonal of a brick

You are given three identical bricks (cuboids) and a ruler (scale) as shown in the figure. You have to find the length of the brick's diagonal without using any formula and by using the ruler only ...
486 views

### 2 of each Tetris Puzzle

In Is this Tetris puzzle solvable? we established that it is not possible to form a rectangle with an uneven number of each Tetris piece. But if we had a solution for 2 of each piece, we would now ...
277 views

### Win the relay competition by selecting the longer path for competitors [closed]

Two teams relay race: members of a team of three take turns running from the point P to a points on the circle; A for the first, B for the second, and C for the third, starting and returning at point ...
161 views

### Walking around in circles [closed]

Imagine you are an ant walking on a circle. You start your journey from the blue point. Your progress is given in percentages - 100% being a full round trip. In the figure, you've made a progress of ...
357 views

### Tetrahedrons and Octahedrons

Is it possible to cut a regular tetrahedron with edge length 100 into regular tetrahedrons with edges of length less than 1 and regular octahedrons with edges of length less than 1?
4k views

### Fairly Sharing a Frosted Cake

You are serving a cake to $10$ children. The the cake is shaped like a box, whose top face is square. The top and sides are covered with a thin layer of frosting. $\qquad\qquad\qquad\qquad$ Every ...
1k views

### Black and White

OK guys, I think this is my best puzzle yet. Hope you enjoy it, the solution is neat and simple. A boy draws 2015 unit squares on a piece of paper, all oriented the same way. The squares can overlap....
2k views

### Want to See My Crossword? Too Bad!

I had a wonderful crossword all prepared for you. But... you suck. >:(   So I'm just going to tell you about it. It was an 8-by-8 American-style grid with white and black squares, ...
1k views

### Aquaman's Revenge!

After you posted mean things online about Aquaman, he found out where you lived and kidnapped you (using his awesome aqua-powers), taking you to his super-secret aqua-lair. He has placed you in a room ...
756 views

### Sly Cooper and the Space Police

Last Space Police question... Sly Cooper has landed on a deserted planet with 2 policemen. He is moving around the planet, painting a line along his path, claiming any land which is surrounded by ...
1k views

### Star-Lord and the Space Police

Star-Lord is on a deserted planet along with two space policemen. If he runs into either of them, he will get immediately arrested. Fortunately for Star-Lord, somewhere on two opposite ends of the ...
592 views

### Separating the Pieces of a Rubiks Cube

I recall coming across this problem in one of my classes senior year of high school. (We only did the first part). Easy (and probably somewhere on the internet) Taking a standard 3x3x3 Rubik's ...
168 views

### Riddick and the Space Police [duplicate]

The next problem may be way too easy, but will post it anyway. Riddick, trying to escape from the space police, lands on a deserted planet. How many cops should the space police send to the ...
4k views

### How Can I Safely Double The Length of A Ladder?

I'm posting this from the roof of a building because I can't figure out a safe way down. I've scoured the roof for anything useful, and all I found was a coping saw. I also found a 100 foot ladder, ...
939 views

### Ernie and the Bubbles of Lucretia

"So, what do you know about milking spiders?", asked Ernie as he passed a fresh cup of coffee to me. Good grief!, I thought and gave an involuntary shudder as I glanced at the milk jug sitting ...
1k views

There is a square with a side length of 1. Inside this square there is a quadrilateral. Each vertex of the quadrilateral is on a side of the square. The area of the quadrilateral is bigger than half (...
775 views

### Optimal Rope Burning [closed]

You are given a single long rope of length L (ft) with radius R (in), a very sharp knife, and 2 matches. Your task is to completely burn the rope in the minimum amount of time. Calculate the total ...
262 views

### Painting a large sphere

An artist duo (Artist A and Artist B) were hired to make a large blue sphere resembling the earth. They decided to make it out of three sections, two spherical caps and one middle ring with the same ...
133k views

### Five Angles in a Star

In a regular pentagram (5-pointed star), the angle in each point is 36 degrees, so the angles in all five points sum to 180 degrees: What about an irregular pentagram, such as the following? Now the ...
11k views

### The Jeweller's Dilemma

You are well known as the best jeweller in Puzzovania; your shop is always well stocked and your pockets are always bulging. One day, the local 'godfather' of Puzzovania's organised crime comes into ...
2k views

### Cut up a cheese cube into $6$ equal parts with $5$ slices

Bob has a cheese cube and wants to cut it up into exactly $6$ equal pieces for his guests. These guests are professional cheese connoisseurs who will complain if they get less than anyone else. Bob ...
758 views

### Pretzel recipe with a twist

Browsing the geometry section in the local library you come across a flimsy book, seemingly misplaced: "The Pretzel Cookbook: Five More Patterns Using The Spoke Method" What's more, it's ...
10k views

### Turning a goat?

This is a goat made up of 5 sticks. You have to move (change position) any one stick of them such that its head turns to the right side (above the right leg). Notice currently its head is on the left ...
3k views

### How much water do you need to cross the desert?

This question is inspired by Terry Pratchett's "Small Gods," in which an army crosses a vast desert by making multiple trips and caching water along the way. 1. Provide an answer. 2. I doubt I'm the ...
6k views

### Draw 4 straight lines to create 10 equal squares in this image

You can draw up to 4 straight lines in order to create 10 equal squares in the following figure:
1k views

### The 2 million, er, 20 dollar problem

Alfred is a guy who really likes solid shapes. He also really likes to keep his money in his wallet. What makes him the happiest? Getting a boxful of shapes at the Cheap Solids Store! The store has ...
5k views

### A clock where the hour and minute hands are the same length

Your buddy Frankie sold you a shoddy clock: it keeps good time, but the minute and hour hands look exactly the same! Both of these hands move continuously, and there is no second hand. How many times ...
862 views

### Ghost Ship Collisions

There are five ghost pirate ships drifting on an infinite ocean in the spiritual realm. Each ship moves at a constant speed and never changes direction. No two ships are traveling in parallel paths. ...
938 views

### The Origins of a Confusing Maze

Behold a maze of fuses (the black dotted lines) laid out on a grid:         The fuses can be lit, causing sparks to burn along them at a perfectly uniform rate: one segment per ...
4k views

### Connecting blue dots to red dots

Suppose $n$ red dots and $n$ blue dots are arranged in the plane so that no three dots lie on a single line. Show that you can connect each red dot to a blue dot using a line segment so that no pair ...
2k views

### Fitting rectangles into square (optimal/perfect rectangle packing)

I gave the puzzle you can see on the image below to a friend of mine for christmas last year. I thought it would be fun to dump it out in front of him so he would not know the solution. Unfortunately ...
899 views

### hunting for treasure on an infinite grid

You are hunting for treasure located at some point on an infinite square grid. You have two tools: a pointer, which points toward the treasure; and a shovel, which you can use to dig for it. Here's ...
938 views

### Surviving the Shootout

$101$ gunmen stand in a field. At high noon, everyone shoots the gunman standing closest to him. If there are several gunmen who are equally close, they shoot the tallest one of them. No two gunmen ...
1k views

### Droning On in Circles

In the spirit of wizards and circular prisons: You have been imprisoned by an evil wizard in a perfectly circular prison cell of unknown size. You're shackled to the wall, unable to move about the ...
The vicious wizard Neville has trapped you in the middle of a magical circle of radius $10000$, and you have to find a way out. Every time you want to take a step (of length $1$) in a certain ...