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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1answer
146 views

Polygons circumscribed by lines [closed]

You have a plenty of segments of the same length at your disposal. Put a segment and put another to meet at their ends. Set the counterclockwise angle between two segments at 180/5 degrees. If you ...
17
votes
3answers
1k views

Pythagorean walk

It's a hot day. You are in the middle of a flat sandy plane that stretches as far as you can see. Your hands are shackled and you are surrounded by soldiers of the mighty Pythagorean Brotherhood. Ten ...
4
votes
2answers
207 views

A stroll in the park

Professor Erasmus has returned from his saturday walk in the park. He has counted the number of trees in the park and also the number of lines formed by these trees. Professor Erasmus claims that ...
5
votes
1answer
735 views

Now help me choose a new lock pattern for my phone

Well, thanks to Len, I managed to unlock my phone again, but now everyone knows my pattern, so I need to choose a new one. For brevity, I won't include the explanation of how an Android pattern lock ...
16
votes
2answers
9k views

Help! I've forgotten my phone's lock pattern!

I have an Android phone with a pattern lock screen, which if you've never seen one allows you to unlock your phone using a pattern drawn on the screen over a 3x3 grid of dots. So for example, if we ...
31
votes
1answer
3k views

Ernie and the Island of Stability

Unfortunately, it appears that I may have misled you a little in this puzzle (as you probably know - my memory of events isn't always perfect). When I was writing it, I re-checked the Kzijekistanian ...
12
votes
2answers
557 views

Polyominoes on a checkerboard

Professor Halfbrain has spent his entire weekend by cutting lots of wooden $50\times50$ checkerboards into lots of polyominoes. He looked at various pattern polyominoes with area $49$, and always ...
12
votes
5answers
1k 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 ...
13
votes
7answers
3k views

Fair n-sided dice

I have recently bought a fair $10$-sided dice for generating equally distributed random numbers from the range $1,2,\ldots,10$. Let me describe this dice in some more detail. Consider a regular 5-gon $...
10
votes
2answers
737 views

Polygon wrapping a cube

A polygon is folded to perfectly wrap a cube, covering all of its surface area with no overlap. Show that the polygon had at least two equal angles.
7
votes
1answer
617 views

Smallest number of matchsticks for a 3D structure with 6 matchsticks at each vertex

Make an as small as possible three dimensional structure of matchsticks, all of which have equal length, such that the end of each matchstick meets exactly five other ends. A matchstick is not allowed ...
8
votes
1answer
273 views

The Erasmus tedrahedron

Professor Erasmus has constructed a special tetrahedron that he modestly calls the "Professor-Erasmus-tetrahedron". The professor claims that all four faces of his tedrahedron are right-angled and ...
23
votes
6answers
3k views

Fair share of a square watermelon?

One hot day, Stan, Kyle, and Kenny were sitting outside with a square watermelon (actually it was a cube like the picture below). Stan says "Let's cut the watermelon into 3 equal slices (like the ...
8
votes
2answers
851 views

Ant on a rectangular box

An ant walks in constant steps from corner to corner over a rectangular box, always chosing the shortest path. From any given corner, the ant observes the shortest paths to the other seven corners all ...
3
votes
2answers
279 views

Find seven points so any three contain two of distance one

Choose 7 distinct points in the Euclidean plane so that among any 3 of those points, there are (at least) 2 that are a distance of exactly 1 apart.
10
votes
3answers
686 views

The Erasmus isosceles triangle

Professor Erasmus has constructed a special isosceles triangle that he modestly calls the "Professor-Erasmus-triangle". The professor claims that he can cut his triangle into three smaller triangles, ...
15
votes
1answer
1k views

Touching matchsticks with compass and straightedge

The question Touching Matchsticks asked for the smallest matchstick graph where every node is connected to four distinct edges. The Harborth Graph is the smallest such, with 104 edges connecting 52 ...
4
votes
2answers
739 views

Puzzle pieces, each in contact with 5 others

You are asked to create puzzle pieces by joining together identical squares. This needs to be done such that the puzzle pieces can be arranged in a pattern with each piece being in contact with ...
14
votes
3answers
1k 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 ...
8
votes
4answers
2k views

Find the width of the river

You find yourself with a friend on a completely deserted (devoid of objects or people) and flat plane. The only thing of note is a wide river; you can just manage to see the other side of it. There is ...
9
votes
2answers
834 views

Dissecting a square

You are asked to dissect an $N \times N$ square into polyomino pieces such that each piece shares portion of its boundary with exactly $D$ other pieces, and no piece has area exceeding $N$. This can ...
12
votes
2answers
1k 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. ...
12
votes
2answers
903 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 ...
5
votes
1answer
242 views

Optimal size of n circles to fit an area

Let us say that I have a rectangular area that has to always look "filled" with circles. (the void spaces with the given number of circles should be minimal)(Goal) Let us assume that, I am also told ...
1
vote
3answers
345 views

Marker and the cube

If a marker is located in one of the corners of a square cube and we begin to draw according to the following conditions, how many edges of the cube can we cross ? A) You can cross every edge only ...
12
votes
2answers
954 views

Hidden planet area

A solar system contains some number of stationary perfectly-spherical planets of equal radius. Call a point on a planet's surface private if it can't be seen from any other planet. Show that the total ...
31
votes
3answers
9k views

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 ...
3
votes
1answer
515 views

Three 3 x 3 x 3 Polycube Dissections

The Soma cube is perhaps the most well-known polycube dissection. Here are three more dissections to be assembled into 3x3x3 cubes. The best answer will provide: the first correct solution for ...
11
votes
1answer
380 views

Dissect the pixel-heads

Here are two figures, each composed of 54 pixels. Cut each figure into six nonominoes having the same shape and size. Both figures use the same nonomino The nonominoes could be rotated, reflected, ...
18
votes
4answers
799 views

Stitching together two identical planar shapes

A 3 x 1 rectangle has a perimeter of length 8. Two of these rectangles can each be bend in a U-shape and stitched together to yield a cube of unit volume. Surely we can do better than that! What ...
18
votes
1answer
653 views

Dissect the frog

Cut the stylized frog in the picture into six pieces having the same shape and size, possibly mirrored. The white dots are guide points. They help recognizing the shape's geometry. You are not ...
8
votes
2answers
449 views

Squaring the new year

Happy new year! You are asked to cut a rectangular strip 2015 times longer than its width into pieces that can be reassembled into a square with area equal to that of the original strip. How many ...
11
votes
2answers
629 views

A crawling spider and a cautious fly

Inside a rectangular room, measuring 16 feet in length and 8 feet in width and height, a spider resides at a corner. A fly buzzing in the room intends to land at a spot that will take the spider the ...
21
votes
4answers
2k views

Digging a tunnel between random locations

Romeo and Juliet are each placed uniformly at random locations on a spherical planet, meaning that each square meter of surface area of the planet is equally likely. Romeo decides to dig a straight-...
9
votes
1answer
1k 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
28
votes
5answers
3k 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, ...
11
votes
3answers
1k views

Optimal present wrapping with a rectangle

This is a (slightly silly) version of Penguino's Ernie gift-wrapping puzzle. You need to wrap a cubical present with side length 1 meter. As in the linked question, when you wrap a present, you are ...
30
votes
2answers
3k views

Ernie and the Alchemist's Gift

For Ernie, the task of choosing Christmas presents for his friends is always a struggle. I think that is because, while he finds it easy to solve problems relating to machines, mathematics, and ...
18
votes
2answers
1k views

Ernie and the Drill Test

I dropped in to Ernie's place a while back and he asked me to stay and help calibrate his new Drilling Device. It was an impressive looking machine consisting of a large raygun-like thing mounted on ...
4
votes
1answer
534 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 ...
13
votes
6answers
3k views

Can you solve those laser puzzles? (puzzles created by the community) [closed]

So, I was bored and I made a small game for this site! In this game the user who solves the previous puzzle gets to create the next one and they will always be more complex. Website link and string ...
19
votes
2answers
870 views

Staking Out the Integers

Suppose you're given six stakes and an unlimited length of string. Your objective is to plant the stakes in a flat patch of ground in such a way that you can wrap the string around the stakes in ...
2
votes
2answers
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 ...
24
votes
2answers
2k views

The tilted labyrinth - Can you find the fastest path in this 3D-puzzle? (Simulator now included.)

This is a puzzle was inspired by the board game labyrinth, which I very much enjoyed as a kid. It either requires very good 3D-visualization skills in your brain or some paper & scissors work. (...
2
votes
4answers
1k views

What does this shape look like?

I have here a three dimensional, solid object. Its sides are flat planes. In the following views, all lines are shown — there are no hidden edges. From the front, this shape looks like: From ...
3
votes
7answers
2k views

How many faces does this object have? [closed]

The object came with a few clues to help you figure it out. I can be viewed on paper I cannot be viewed in reality by your kind I am made up of what your mind can truly comprehend I resemble ...
29
votes
1answer
2k views

Continue the Pattern of Circles

The following shapes are sequenced from left to right. Describe or create the next five shapes. Bonus points to those who describe how the pattern works. The pattern could continue on forever. The ...
24
votes
5answers
1k 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 ...
5
votes
1answer
965 views

Solutions for generic polyomino puzzles

Inspired by Mosaic with tetris blocks I was wondering if there were any generic algorithms to solve or show there was a solution to these types of problems (i.e. placing polyominos on a 2D board). ...
20
votes
4answers
3k views

Mosaic with tetris blocks

Create the pattern shown in the picture below using the set of standard tetris blocks. This is a rectangular arrangement of 6×5 squares where the first and third squares have been removed from the ...