# 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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### Ernie and the Geometric Ginger Cookies

Ernie loves cooking and it's always worth the wait as he follows his curiously precise recipe to make Ginger Crunch Cookies. So I was happy to drop around when he invited me to help test his latest ...
1k views

### Optimization of area

You have a closed cord of length 100 meters which touches point $P$. There is a line $L$ whose closest point is 20 meters from point $P$. The value of a layout of cord is defined by the area ...
2k views

### Cut the disk with a hole in four equal pieces

Cut the shape below in four congruent pieces. The gray area is a hole. In non-mathematical terms, cut the white area in 4 pieces having the same shape, same size, possibly mirrored or rotated. Note:...
838 views

### Dividing the Circle

Here's a neat geometrical puzzle I was once shown Is it possible to divide a circle into a finite number of congruent parts some of which don't touch the center? Clarification: Find a shape for a ...
435 views

### Time for tea: cake division by lines

This 'cake division by lines' problem was used as an example in one of the lectures I attended at business school. I've searched and it doesn't seem to be on the site yet, so here it is: You have ...
4k views

### Connect four towers by roads

Four guard towers are situated in a square formation of side length 1km. A general wants to build roads to connect the towers so that one can walk from any tower to any tower along the roads, possibly ...
5k views

### Slicing a donut 3 ways - what's the most number of pieces?

Let's say you have a donut. You are allowed to slice it 3 times. Each slice must be a perfectly straight cut. What is the highest number of donut pieces you can end up with after 3 slices? Assume ...
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### Maximum number of pieces of equal area you can obtain by cutting a pizza a certain number of times

If I cut a perfectly circular pizza through its center 6 times at 30° angles, I get 12 pieces of equal area. If I don't have to cut through the center, I can cut in a grid shape to divide the pizza ...
784 views

### Cut a bagel into interlocking rings

Let's say you have a standard bagel (one that is NOT pre-sliced). How can you cut this bagel into two interlocking rings? The rings must never be broken.
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### N-dimensional Tic-Tac-Toe variant

Consider the game surface to be an infinite N dimensional Cartesian lattice. The rules are X moves first, but O gets to move ...
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### Drink a Little Wine, Cut a Little Rug

Consider the following diagram of your prized 9' x 12' Persian rug:                       ...
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### Getting Stuck on Purpose

Consider the following building floor plan, with a set of rooms labeled A-G and an Outside:                   ...
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### The Monster Garden

You are the praetor of Tri, a small triangular-shaped nation bordered on the northwest by the Kingdom of Sauria, on the northeast by the Sultanate of Avia, and on the south by the Republic of Cryosta, ...
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### Splitting a Plate into 4 Equal Pieces

You are stuck on an island and have been tasked by the natives with dividing a plate of chocolate into 4 equal pieces, one for each of the island's gods. Each god must have an equal share, or you go ...
529 views

### A Hunter wants to get on a Bus [duplicate]

A hunter wants to get on a bus, but belongings which have larger than 1 meter dimensions is not allowed. So how will he be able to get his rifle (which is 1.5 meters) together with him without ...
11k views

### 7 Trees, 6 Rows, 3 Per Row?

Can you arrange 7 trees so that there are 6 rows of 3 trees? It is entirely possible. Note: A tree can be part of more then one row, for example a if you arranged a 3x3 square (9 trees) the tree in ...
325 views

### Largest No-Capture Setup?

Think of a Chinese Game Board, and how you can make jumps. Now think of actual checkers where you capture pieces. How many marbles can you place without a capture possible? There's one slight rule to ...
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### Ernie's automatic parking valet

I was helping Ernie out in his shed a while back, when his favorite screwdriver suddenly wore out. I offered to go and pick up a new one for him and Ernie thought it was a great opportunity to test-...
2k views

### Baggage Problem: $1.5$-meter-long sword onto a train

This is the problem I came across reading the book The Art and Craft of Problem Solving. When I read this question I wasn't able to figure out the solution and I saw the solution after a while, but ...
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### Can you help the traveller

A traveller decided to explore a square desert. He started from a point on the edge to point B, travelling perpendicular to the edge of the desert he started on. On reaching B, he then decided to take ...
685 views

### Finding treasure on a circular island

Pirates have buried treasure on a very unusual perfectly circular desert island, with no trees or obstacles of any kind. The map takes me to the island but doesn't tell me where to find it on the ...
39k 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 ...
326 views

### Packing an Efficient 9 Pack

This is sorta like How to ship the new Slurm 7-pack efficiently by Matt Malone but instead of 7, it's the 9 pack! It's basically a 2 by 4 pack with a piece sticking out, like in the above link. What's ...
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### A blanket for my baby snake

Mama snake wants to knit a blanket for little baby snake. She is not a dissipater and wants to make the blanket of a minimal size (area). But her baby snake is quite a lively baby and it always ...
216 views

### Paint the Rectangle with least number of Circular Stamp Touches

What is the minimum number of touches for painting at least a $100*100$ rectangle if you have one circular stamp tool that paints a circular area of $1$ unit radius in every touch? This question is ...
3k views

### Can you make a Rectangle from an Odd Number of Triangles?

So, the triangles are the same size, they are equilateral triangles. Can you use an odd number of these, to create a rectangle? If so how many, if possible post an image of your answer! Answer with ...
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### How many equal-sized polygons can be used to cover a soccer ball?

It seems that mostly either 26 or 32 equal-size polygons are used to cover a spherical soccer ball. What other configurations are possible?
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### rescue operation - where is your partner?

Your partner's space ship has crashed on an uninhabited planet. Only the radio transmitter and his compass were still in operation. He asks you to rescue him, and tells you how to find him by the ...
625 views

### Paint the rectangle with least movement

There is a painting brush tool that can paint a circle area of $1$ unit radius. What is the shortest paint track for painting at least a $100*100$ units rectangle area on wall? How much did the brush ...
514 views

### The robotic vacuum cleaner

An engineer has invented a robotic vacuum cleaner to sweep the floor of his house. The floor plan, shown below, is made up of many square tiles, with walls at certain places, represented by black ...
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### The Laziest Surveyor

The new land surveyor and his boss arrived at the work site, a huge and perfectly flat parking lot. His boss said, "Your first assignment is going to be a tough one. You see this parking lot? It ...
1k views

### How to ship the new Slurm 7-pack efficiently

The six-pack is a thing of the past. Beverages of the future will use the seven-pack format. But how will the mighty spacemen of the future manage to ship the Slurm seven-pack efficiently in ...
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### Find 10 triangles in a five pointed star using two straight lines

This puzzle consists of counting ten triangles (check three sides for each one, remember that there aren't exist triangles with more than three sides :P ) using two straight lines that cross the ...
7k views

### Chaos and Order: a visual puzzle in stained glass

I created a visual puzzle, which my wife then implemented as part of a stained-glass window. I've no idea if it is (a) obvious, (b) stupid or hopefully (c) extremely clever, and hence would love to ...
786 views

### Santa Claus flies to the South Pole

It is a little known fact that Santa Claus is in fact a vampire. Why else would he spend so much time near the North Pole and travel only at night? Yes, he fears daylight! His problem is that ...
8k views

### Divide a rectangle with a rectangular hole into two equal parts

Consider any large rectangle from which a smaller rectangular portion has been removed. The removed rectangular portion may have any orientation i.e. the remaining figure is not necessarily symmetric ...
560 views

### A cube puzzle (Multiple of 6)

If you have a cube, say, $3 \times 3 \times 3$, you know that mathematically it's made of $27$ ($3*3*3$) smaller cubes. Now, just remove one full line of smaller cubes (in this case, 3 smaller cubes). ...
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### Find shortest network connecting four points

Given the figure below, find the shortest network of straight line segments (like a Steiner tree, or like parts of a Delaunay triangulation) that connects the four circled points while staying in the ...
395 views

### Make a wide tower of bridge-shapes

In this question, bridge stands for the illustrated convex quadrilateral, constructed from three equilateral triangles. Letters A, B, C, D, E are assigned to each unit length of periphery as shown in ...
2k views

### Probability of three segments forming a triangle

Consider a line segment of length L. This segment is cut in three segments at arbitrary points. What is the probability that these three segments could be rearranged as sides of a triangle?
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### Tower of tiles revisited

An S-tileset is a collection of n oriented tiles, where no two tiles have the same size, each tile is one unit thick, and its non-zero-integer length and width add up to n+1. (So, an S-tileset has n ...
12k views

### Join all circles together only with 6 lines

In the below image, can you draw 6 straight lines that pass all the circles? As soon as you start drawing lines you can't take your pen up until you draw all six lines. hint: you don't have to keep ...
847 views

### How high a tower of tiles can be made?

An $S$-tileset is a collection of $n$ oriented tiles, where no two tiles have the same size, each tile is one unit thick, and its non-zero-integer length and width add up to $n+1$. (So, an $S$-...
1k views

### Folding a hexagon to a rectangle or square, with uniform overlap

A piece of fabric shaped like a regular hexagon with unit-length sides is given, with vertices A...F and center O. Let a fold correspond to folding some fabric along a straight line, without ...
350 views

### Is there a second solution to points and perpendicular bisectors problem?

There is a problem from M.Gardner book ("Wheels, Life, and other Mathematical Amusements", p. 201): [Hallard T. Croft] asked if there existed a finite set of points on the plane such that the ...
409 views

### Land claim staking

It was traditional during the times of the Oregon Trail and the gold rush that men founding new towns were allowed to stake their claim on land upon arriving to the west coast. Suppose a man is ...
467 views

### Drawing lines on a map

This was a puzzle that appeared in an old math contest as well as an unrelated puzzle book. Suppose we have a two-dimensional map with a lot of cities marked on it. For each city on the map, we ...
830 views

### How long is needed to clear the area of robots?

This question has been written in order to help answer Fastest way to collect an arbitrary army. There is a 1 x 1 square area with corners labelled clockwise A, B, C and D. A finite number of ...