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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8 votes
3 answers
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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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25 votes
5 answers
2k views

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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17 votes
4 answers
19k views

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 ...
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11 votes
2 answers
915 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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6 votes
7 answers
4k views

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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9 votes
5 answers
1k views

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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  • 1,008
6 votes
3 answers
381 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 ...
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5 votes
1 answer
290 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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139 votes
8 answers
41k views

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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14 votes
2 answers
611 views

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 ...
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8 votes
2 answers
272 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 ...
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7 votes
6 answers
566 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 ...
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6 votes
2 answers
275 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 ...
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3 votes
3 answers
234 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 ...
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57 votes
8 answers
5k 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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40 votes
5 answers
5k views

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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  • 792
26 votes
4 answers
17k views

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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  • 10.9k
3 votes
2 answers
2k views

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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  • 14.2k
12 votes
6 answers
7k 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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28 votes
4 answers
9k views

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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  • 3,012
22 votes
1 answer
962 views

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 ...
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  • 1,517
16 votes
4 answers
1k views

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. ...
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8 votes
1 answer
2k views

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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8 votes
6 answers
3k views

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 ...
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  • 197
4 votes
1 answer
640 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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109 votes
2 answers
10k views

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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12 votes
7 answers
3k views

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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  • 15.8k
19 votes
3 answers
1k views

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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  • 4,883
19 votes
10 answers
72k 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 ...
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  • 12.7k
14 votes
2 answers
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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  • 4,242
11 votes
2 answers
659 views

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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36 votes
7 answers
2k views

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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  • 38.8k
11 votes
1 answer
479 views

The adventitious 18-gon

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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  • 44.7k
5 votes
2 answers
249 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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5 votes
2 answers
426 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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2 votes
2 answers
177 views

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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17 votes
4 answers
1k views

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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  • 1,517
11 votes
4 answers
518 views

Largest and smallest hexadecagon with sides $1, 2, 3, \dots,16$

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? ...
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9 votes
3 answers
413 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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  • 21.4k
9 votes
2 answers
940 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 ...
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  • 1,193
4 votes
1 answer
558 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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  • 21.4k
173 votes
1 answer
10k 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 ...
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  • 1,963
51 votes
11 answers
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 ...
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  • 5,275
30 votes
5 answers
4k views

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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  • 7,678
39 votes
7 answers
5k views

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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  • 31.4k
19 votes
7 answers
2k views

Triangle in a circle

Suppose three points are chosen at random in a circle. A triangle is made with these three points as vertices. What's the probability that the triangle contains the origin of the circle? (Although I ...
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  • 341
25 votes
4 answers
4k views

A COVID-19 puzzle: How large a class do you need to fit 30 pupils?

Some countries are proposing to reopen high schools soon. To ensure safety, they want to make sure that all pupils in a class are at least 2 m apart. To help them find the smallest room that can ...
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  • 5,298
22 votes
7 answers
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 ...
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  • 533
14 votes
3 answers
2k views

Touching Matchsticks

You are asked to place matchsticks on a flat surface such that each matchstick end meets three others, and no matches cross. It is easy to achieve this for patterns that extend indefintely: The ...
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  • 4,242
13 votes
2 answers
975 views

Find smallest rectangle divided into figures so each figure has 5 neighbours

The following 3x4 rectangle can be cut into pieces along grid lines, so that each piece has exactly three neighbors: Problem: Find the smallest rectangle on the integer grid that can be cut into ...
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  • 1,008

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