# 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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### 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 ...
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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 ...
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### Polyomino T hexomino and rectangle packing into rectangle

Let's pack some (one or more) T hexominoes together with some (one or more) small $a\times b$ rectangles into some bigger $m\times n$ rectangle without holes and overlapping pieces. For example, I ...
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### Drawing something using one pen stroke

Can you determine if it's possible to draw a geometric figure (made up from shapes like rectangles, triangles, and other regular shapes) with one pen stroke and not drawing the same line twice. I am ...
970 views

### Twelve Labours - #04 Erymanthian Bar

This puzzle is part of the ‘Twelve Labours’ series. Previous instalments can be found here: Prologue | 01 | 02 | 03 Now one crate lighter, Hercules made his way back up the road to the Erymanthian ...
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### Polyomino Z pentomino and rectangle packing into rectangle

See my similar question about T hexomino (Polyomino T hexomino and rectangle packing into rectangle) This is exactly same but with other polyomino - Z pentomino. Let's pack some (one or more) Z ...
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432 views

### Tiling rectangles with X pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ...
• 3,978
318 views

### Tiling rectangles with Hexomino plus rectangle #1

Inspired by Polyomino T hexomino and rectangle packing into rectangle See also series Tiling rectangles with F pentomino plus rectangles and Tiling rectangles with Hexomino plus rectangle #1 Next ...
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### How can 64 = 65?

Here is a interesting picture with two arrangements of four shapes. How can they make a different area with the same shapes?
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### Tiling rectangles with W pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ...
• 3,978
304 views

### Tiling rectangles with T pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ...
• 3,978
609 views

### Tiling rectangles with N pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with T pentomino plus rectangles ...
• 3,978
310 views

### Tiling rectangles with V pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ...
• 3,978
319 views

### Tiling rectangles with U pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle Also in this series: Tiling rectangles with F pentomino plus rectangles Tiling rectangles with N pentomino plus rectangles ...
• 3,978
6k views

### Find a straight tunnel

There is a circular area with radius 1 km. And there is a tunnel, which is just under the surface, but invisible - unless you dig. It is known that the tunnel goes under the area (at least touches it ...
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### Blindfolded and disoriented near the Great Wall of China

You are blindfolded and disoriented, standing exactly 1 mile from the Great Wall of China. How far must you walk to find the wall? Assume the earth is flat and the Great Wall is infinitely long and ...
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### Plant 9 trees in 10 rows of 3

"Tree-planting" puzzles are also known as "points and lines" puzzles. The English puzzle author and mathematician Henry Ernest Dudeney was very fond of them. In 1917, Dudeney published a collection ...
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### How do I arrange pencils so they all touch each other?

How do you arrange 6 pencils so that each one touches the other five? And what about 7 or 8?
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### Peaceful Encampments

This math puzzle is due to Donald Knuth (as far as I know; maybe he got it from someone else) circa 2014. Consider a plain represented by the unit square. On this plain we want to “peacefully encamp” ...
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### My Mother's Dish Collection

From every trip she makes, my mother brings as a souvenir a well-decorated dish to hang in a wall. She now has a collection of 12 dishes, all disks, of radii 1, 2, 3, ..., 12 inches respectively. ...
2k views

### Escaping a hungry lion you can't outrun

You are at the edge of an enormous circular arena. A hungry lion is eying you from the centre of this area. You are both capable of running at the same maximum speed, but constraint within the arena. ...
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### 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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### Martin Gardner - Crazy Cut

You are to make one cut (or draw one line) – of course it needn’t be straight – that will divide the figure into two identical parts. Source
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### How many different non congruent polygons can you make on a 3x3 dot grid?

There is a 3×3 dot grid. How many different non-congruent polygons can you make on the grid? Rules: All vertices of the polygon must be on the grid Only non self intersecting polygons Only polygons ...
• 197
667 views

### Check to see if a Configuration is Possible: prove there's an Hamiltonian path on a connected subset of the square grid graph

Alright, here's a new one for you guys. Instead of solving it, you need to determine if any given configuration is possible to solve, without actually solving it. It needs to be a general method of ...
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### 4D Maze Creation!

I have a problem for anybody who cares to try. You're job is to take a 10x10x10x10 size tesseract and design a maze that fits. The maze must be a perfect maze (no loops, one path cannot be followed ...
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### Prove that π > 3

It seems that once upon a time some politicians tried to pass a law fixing the value of π to be exactly 3. The idea being to "make math simpler so that our children can get better at math". ...
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### Create a 3 inch measurement

How will you create a 3 inch (within say plus or minus 0.05 inch) side on a standard piece of paper ( 8.5 x 11 inches) merely by Folding? No marking of any kind allowed. Only one paper available. ...
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### How many cubes can we fit in?

You have a bag with many 3x3 Rubik's cubes . After solving all but one of them, you decide to play a new game. You take the unsolved cube, and start placing solved ones around it. The question: What ...
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### 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 ...
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### Reunite the Stars

On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of 1 in their ...
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### Discrete Peaceful Encampments: 9 queens on a chessboard

Here's a discrete variation of yesterday's puzzle Peaceful Encampments. You have 8 white queens and 8 black queens. Place all these pieces onto a normal 8x8 chessboard in such a way that no white ...
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### Fastest way to collect an arbitrary army

I am looking for solution of this puzzle: There is a kingdom with a square shape with sides of length 1. The castle is located at the center of the square. At the castle the king lives under the ...
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### Largest and smallest hexadecagon with sides 1, 2, 3, ...,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? ...
513 views

Today I have drawn a regular $18$-gon on a piece of paper. My drawing shows the $18$ vertices of the polygon labeled as $P_1,P_2,\ldots,P_{18}$ in clockwise order, and it also shows all $135$ ...
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### Unlucky tiling: Arrange thirteen right isosceles triangles into a square

Link to next puzzle in this series:Five graded difficulty isosceles right triangle into square tilings Two difficult "Seventeen right isosceles triangles into a square" tilings V.hard ...
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### 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 ...
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446 views

### Unreflected infinitely simple polygon reflexivity

(This was retrofitted to more tightly match a surprise solution and to allow for another puzzle with the original intent.)                   &...
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### Cutting a cross made of 5 equal squares by 2 straight cut into 4 figure to together form a square

The figure below consists of 5 equal squares in the form of a Greek cross: Please show how to divide it with two straight cuts into 4 equal pieces which will fit together to form a square. A MSE told ...
275 views

### Five graded difficulty isosceles right triangle into square tilings

Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Two difficult "Seventeen right isosceles triangles into a square" tilings V.hard problem, 20 right ...
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507 views

### 20 right isosceles triangles into a square

Similar: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Five graded difficulty isosceles right triangle into square tilings Two difficult "Seventeen right isosceles ...
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582 views

### Infinitely simple polygon solipsism

Solipsism — The self is all that can be known to exist. Above is a simple polygonal region divided into infinitely many different-sized copies of itself.   Each copy is √2 = 1.414... times as large ...
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### Two difficult "Seventeen right isosceles triangles into a square" tilings

Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square Five graded difficulty isosceles right triangle into square tilings V.hard problem, 20 right isosceles triangles ...
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### 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 ...
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### 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 ...
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### Cover 63 squares of a chess board

Can you find 3 similar geometrical figures (common shape but can be different sizes) A, B and C with the following property: If you remove any square from a 8x8 chess board, then the remaining area ...
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### A Surprising Circle Packing

A grocery store has a long, skinny box, with no top, that it uses to display soda. The box is two soda cans wide and 200 soda cans long. You can neatly fit 400 cans in this box, using two rows of 200, ...
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### What is the minimum number of straight lines to connect all the dots on this grid?

Recently a question was posted with this picture of a 7x7 grid of dots, asking for a possible configuration with 12 lines where you can draw them without lifting a pencil. But is it possible with 11 ...
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### Dissection Puzzle - The Umbrella Stand

You own a square-shaped table. You want to drill a small hole in the center to place an umbrella stand. Unfortunately, you're a little drunk: Alas. Fortunately, not all is lost. You are sober now, ...
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