# 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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### Left and right turns on the surface of a cube

Inspired by A closed path on the Rubik's cube. Let $N$ be a positive integer. Consider an $N\times N\times N$ cube, with each face tiled by $N^2$ squares measuring $1\times 1$. A closed path is drawn ...
11k views

### Divide a circle into three equal parts [closed]

You are at a restaurant. There are a total of three people(including you). Now you order a pizza and the trouble is that all three(including you) want equal parts. You have only a knife. How do you ...
310 views

### Folding a piece of paper

On the table lies a rectangular piece of paper $ABCD$ of area $100$. Cosmo folds the rectangle once along a straight line, so that afterwards corner $C$ lies exactly on top of corner $A$. The ...
987 views

### Cutting a square into seven rectangles

Cosmo has cut a square into seven rectangles, so that the seven lengths $\ell_1,\ldots,\ell_7$ and the seven widths $w_1,\ldots,w_7$ of these rectangles satisfy  \{\ell_1,\ldots,\ell_7\}\cup\{w_1,\...
1k views

### Professor Halfbrain and the tilted cube

When I ran into professor Halfbrain this morning, he told me that he has constructed a cube that can be tilted and balanced on a plane table such that exactly one of the eight corners touches the ...
1k views

### The Erasmus pentagon

Professor Erasmus has constructed a special convex pentagon $ABCDE$ that he modestly calls the "Professor-Erasmus-pentagon". The professor claims that he can cut off a smaller pentagon similar to ...
2k views

### Hikers Meeting in the Middle

Two hikers are separated by a two-dimensional mountain range, like the one shown below. The mountain range alternates between peaks and valleys, connected by straight lines. Both hikers are at sea ...
193 views

### Squares in a dot matrix

We have a matrix made up of $m$ by $n$ dots. Can you give a function that counts the number of squares that can be found by joining any $4$ dots in it?
691 views

### Tiling by trapezoids

An equilateral triangle with sidelength $L$ can be tiled by trapezoids with sidelengths $2,1,1,1$. What are the possible values for $L$?
224 views

### The infinite flea circus

Based on Another curious incident in the flea circus and A curious incident in the flea circus by @Gamow There is a $n$ dimensional cube in an $n$ dimensional world. There is a flea on each vertex of ...
637 views

### Professor Halfbrain and the right-angled triangles

Today I met professor Halfbrain at the tea house. The professor looked very tired, and apparently had not slept for the last couple of days. He told me that he had been spending his time with cutting ...
1k views

### Inside or outside the square?

Enrico draws a square in the plane, and then secretly picks a point $P$ that is either situated inside the square, or outside the square, or on the boundary of the square. Damiano sees the ...
2k views

### Polygonal Pizza

Four friends want to share a pizza. However, the pizza became badly misshapen en route; though it is still flat, it is now shaped like some arbitrary polygon. Prove that it is still possible to ...
375 views

### A big cube and 99 smaller cubes

A big cube is cut into 99 smaller cubes. Exactly 98 of these 99 smaller cubes are unit cubes. Question: What is the volume of the big cube?
1k views

### Find the total number of triangles in the diagram

The title of the question says everything $\ldots$ My attempt: We count $2(1+1+1+2+1+1+2+1+1+1+1)=26$ triangles. (On each side $13$ triangles, and then multiplied by $2$). And then we combine them ...
599 views

### Ant on a hyperbox

Despite strong objections by many here at StackExchange, the mad scientist continues to use ethically questionable methods to study the spatial cognition of ants. Last time, he left ant Juliet at a ...
581 views

### This ant sure is smart. But how fast is it?

A mad scientist uses ethically questionable methods to study the spatial cognition of ants. Last time, he left his lab ant in a cubic room and filled it with painfully intoxicating gas, capable of ...
489 views

### On the shores of circle lake

Eight trees $ABCDEFGH$ are standing (in this order) on the shores of circle lake. The four trees $ACEG$ form a square of area $500m^2$. The four trees $BDFH$ form a rectangle of area $400m^2$. ...
599 views

### Professor Halfbrain's second cutting theorem

Professor Halfbrain has recently made several fascinating discoveries on cutting convex polygons in the plane. Halfbrain's second cutting theorem: Every convex polygon can be cut (by a perfectly ...
754 views

### Professor Halfbrain's first cutting theorem

Professor Halfbrain has recently made several fascinating discoveries on cutting convex polygons in the plane. Halfbrain's first cutting theorem: Every convex polygon can be cut (by a perfectly ...
161 views

### The incredible polyhedron

Based on The Erasmus polyhedron by @Gamow.. Construct a 3D convex polyhedron of any single material that can float in water such that $x\%$ of its volume is below water level and $y\%$ of its ...
1k views

### The Challenge Square

Q: Can you divide this shape into 4 equal parts, and then form a square?
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 ...
180 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 ...
1k 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 ...
4k 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 ...
438 views

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 ...
777 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 ...
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: "...
200 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. ...
1k 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 ...
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).
635 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 ...
268 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 ...
759 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 ...
579 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 ...
14k 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 ...
948 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). ...
221 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 ...
378 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.
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 ...
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 ...
861 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 ...
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 ...
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 -...
966 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 ...
472 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 ...