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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13
votes
2answers
1k views

The Erasmus polyhedron

Professor Erasmus has constructed a special convex polyhedron from perfectly homogeneous material, which he modestly calls the "Professor-Erasmus-polyhedron". The professor claims that he can put the ...
2
votes
1answer
176 views

Round plates on a round table [duplicate]

(I did not make this one up, but it's one of my favorites and I didn't see it on here!) After a long shift at the restaurant, your fellow waiter Jeremiah proposes the following game: Start with an ...
-2
votes
2answers
965 views

What is the minimum number of line segments that need to be made to cross all points on a $n \times m$ grid?

Given a grid of $n \times m$ points, on a sheet of paper, what's the minimum amount of lines (which can extend infinitely) required to pass through each point, that you can draw without lifting a ...
22
votes
5answers
3k views

Worm and an Apple

Johnny has a perfectly spherical apple, with a diameter of 70 mm. While he looks away, a worm burrows through (entering and leaving) the apple, forming a single tunnel of length 69 mm and negligible ...
3
votes
2answers
421 views

Professor Halfbrain's quadrilateral theorem

Professor Halfbrain has recently made a fascinating discovery on quadrilaterals in the plane. Halfbrain's quadrilateral theorem: Let $ABCD$ be a plane quadrilateral that possesses an incircle and ...
8
votes
2answers
689 views

Termite eating through a large cube composed of 27 smaller cubes while not moving diagonally

The is a large cube formed by gluing together 27 smaller cubes of uniform size (see figure). A termite starts at the center of a face of any of the outside cubes and bores a path that takes him once ...
25
votes
1answer
1k views

Careless smokers

Cosmo complains: "At the party yesterday at our place, some guys were smoking in our living room. This morning we detected that these careless smokers had burnt four holes into the carpet." Fredo: "...
1
vote
1answer
182 views

Bee and lizard in a room [closed]

There is a bee and a lizard at the corner of an $l\times b\times h$ cuboid. The bee can fly. The lizard can walk on walls, the ceiling or the floor. Both must reach the diagonally opposite corner. ...
5
votes
2answers
957 views

Arrange ten coins in a bowling formation. Fewest pennies to remove so no three pennies that remain have centers that form an equilateral triangle?

Arrange ten coins in the familiar ten-pin bowling formation (see figure). What is the smallest number of pennies you must remove so that no equilateral triangle, of any size, will have its three ...
5
votes
5answers
2k views

Find the simple 3D solid associated with these views [closed]

Find the simple 3D solid associated with these views. It is given that there are no hidden lines (lines which are not visible from the side or top view).
3
votes
3answers
622 views

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 ...
4
votes
1answer
266 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 ...
5
votes
3answers
750 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 ...
2
votes
3answers
531 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 ...
57
votes
12answers
13k 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 ...
10
votes
2answers
774 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). ...
1
vote
1answer
207 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 ...
5
votes
3answers
372 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.
18
votes
5answers
3k 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 $120^{\circ}$ from each other? It's a simple question, but I thought it was a fun ...
18
votes
6answers
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 ...
7
votes
3answers
854 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 ...
6
votes
3answers
2k 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 ...
-4
votes
4answers
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 ...
15
votes
1answer
911 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 ...
7
votes
3answers
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 ...
6
votes
1answer
448 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 ...
0
votes
3answers
272 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 ...
-2
votes
2answers
155 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 ...
6
votes
1answer
332 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?
32
votes
2answers
3k 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 ...
13
votes
3answers
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 ...
27
votes
1answer
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, ...
13
votes
4answers
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 ...
10
votes
2answers
724 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 ...
11
votes
1answer
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 ...
8
votes
3answers
535 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 ...
4
votes
1answer
161 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 ...
18
votes
6answers
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, ...
12
votes
1answer
836 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 ...
1
vote
2answers
754 views

Quadrilateral inside a square

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 (...
2
votes
8answers
765 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 ...
3
votes
1answer
239 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 ...
21
votes
10answers
112k 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 ...
60
votes
5answers
10k 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 ...
7
votes
5answers
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 ...
8
votes
1answer
740 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 been ...
49
votes
11answers
9k 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 ...
22
votes
7answers
2k 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 ...
31
votes
3answers
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:
18
votes
4answers
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 ...