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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151
votes
1answer
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 ...
127
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8answers
32k 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?
109
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1answer
7k views

How can this shape perfectly cover a cube?

The following shape: can be folded onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps. How can this be done?
60
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5answers
10k views

The Jeweller's Dilemma

You are well known as the best jeweller in Puzzovania; your shop is always well stocked and your pockets are always bulging. One day, the local 'godfather' of Puzzovania's organised crime comes into ...
56
votes
12answers
13k 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 ...
54
votes
2answers
6k views

Can you perfectly wrap a cube with this blocky shape?

The following blocky shape: can be folded onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps. How can it be done?
53
votes
8answers
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 ...
53
votes
4answers
5k views

Folding paper into corners

Here is a piece of paper: Fold it once, and you can get a shape with 9 corners: Starting with a rectangular sheet of paper and folding twice (along any line), what is the largest possible number of ...
51
votes
16answers
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 ...
49
votes
11answers
9k 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 ...
48
votes
2answers
5k views

Six pyramids in a cube

This question is from the German mathematics competition Känguru der Mathematik. In this competition students have to solve 30 mathematical tasks like this in 90 minutes without calculator. Actually ...
48
votes
3answers
4k views

The vicious wizard…and you!

The vicious wizard Neville has trapped you in the middle of a magical circle of radius $10000$, and you have to find a way out. Every time you want to take a step (of length $1$) in a certain ...
42
votes
17answers
8k views

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 ...
39
votes
2answers
4k views

The Non-Pythagorean Theorem

Everyone knows the Pythagorean theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The ...
38
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5answers
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 ...
36
votes
7answers
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, ...
36
votes
3answers
3k views

Which 3D shape can you make out of this?

The above shape can be folded into a closed 3D shape using no more than 14 distinct folds, with no parts overlapping. What is special about the shape that results? Rules and clarifications: Every ...
35
votes
5answers
5k views

Coffee-break Puzzle: Where does the Driver Sit?

Given below is a picture of a car (from top view). Find out on which side is the steering wheel - left or right. Source: Geometry Olympiad in honour of I. F. Sharygin, 2007
35
votes
1answer
4k views

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-...
34
votes
5answers
6k views

Simple geometry. Or is it?

I've got a regular tetrahedron and a square pyramid. Every edge of the two solids has the same length. If I perfectly attach one face of the tetrahedron to one of the triangular faces of the square ...
34
votes
3answers
5k views

Odd-looking circle

A man is told to make a circle He makes this: Where is the man?
34
votes
4answers
3k views

Magnets on a whiteboard

Alice enjoys placing magnets on a magnetized whiteboard. This day, she placed all 16 magnets in her possession on the board in a rectangular fashion. o o o o o o o o o o o o o o o o "...
34
votes
1answer
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:...
34
votes
6answers
3k views

Gunfire at dawn

A group of ten bandits stand in a flat desert, with no pair the same distance apart. Tensions grow, and at the crack of dawn each bandit fires a single bullet at the bandit closest to him. All have ...
33
votes
7answers
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. ...
32
votes
2answers
3k views

Fairly Sharing a Frosted Cake

You are serving a cake to $10$ children. The the cake is shaped like a box, whose top face is square. The top and sides are covered with a thin layer of frosting. $\qquad\qquad\qquad\qquad$ Every ...
32
votes
10answers
3k views

Variant of lion and 100 zebras

Note: This problem remains unsolved, as of 4 Feb 2018, so do try it out This a variation of this question by @Gamow Suppose there are $100$ lions and $100$ zebras. The lions function together as a ...
31
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3answers
6k views

Draw 4 straight lines to create 10 equal squares in this image

You can draw up to 4 straight lines in order to create 10 equal squares in the following figure:
31
votes
6answers
3k views

Find 7 solutions

You have three flat pieces, as shown: Arrange them flat, without overlap, such that the shape formed by the black parts is congruent to the shape formed by the white parts. Rotation and reflection ...
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 ...
31
votes
5answers
2k views

Slicing a rectangle

A friend presented this nice little puzzle to me yesterday. You're given a rectangle which is dissected by one of its diagonals, as well as another line that only meets one of the two remaining ...
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 ...
30
votes
9answers
7k views

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 ...
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 ...
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 ...
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, ...
28
votes
8answers
3k views

A hungry ant on a matchbox

A hungry ant is standing on the top of an empty matchbox cover (without its drawer) and has detected the smell of a drop of honey on the floor of the cover as shown in the figure below. What is the ...
28
votes
3answers
3k views

Can you see out of the forest?

You are standing at the centre of a circular forest of radius 500 metres. The trees of this very regularly planted forest stand in a precise rectangular lattice on the plane, each 10 metres from the ...
27
votes
5answers
729 views

Dissecting the holey octomino into a square

This is a pure dissection problem, with no added twists. Cut the holey octomino (i.e., a square with the middle third removed) into several pieces, and reassemble those pieces into a square with no ...
27
votes
1answer
2k views

Want to See My Crossword? Too Bad!

I had a wonderful crossword all prepared for you. But... you suck. >:(   So I'm just going to tell you about it. It was an 8-by-8 American-style grid with white and black squares, ...
26
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4answers
4k views

Drink a Little Wine, Cut a Little Rug

Consider the following diagram of your prized 9' x 12' Persian rug:                       ...
26
votes
7answers
3k views

What's the perimeter of the hexagon?

Consider a hexagon which is equiangular but not equilateral: all angles equal to 120 degrees but four consecutive sides of length $a,b,c,d$ not necessarily equal. What is the perimeter of the hexagon? ...
25
votes
6answers
7k views

Cut a cake into 3 equal portions with only a knife

You have to determine a way to cut a circular cake into exactly three portions of equal size. The only marking on the cake is a candle in the very center. All you have to work with is a knife that is ...
25
votes
4answers
3k views

What is the most triangles you can make from a capital “H” and 3 straight lines?

So start with an upper case H, and then draw $3$ straight lines. What is the greatest number of closed triangles that you can form? For example: Note that triangles inside of triangles only count ...
25
votes
4answers
1k views

Ernie and the Pirates of the Caribbean

A few weeks ago, I dropped in to see Ernie to ask if he was willing to look after the cat while I was on holiday in the Caribbean. “No problem.” he replied then showed me an old leather-covered book, ...
25
votes
1answer
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: "...
25
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 ...
25
votes
6answers
2k views

Chess at the 2020 Olympic Games

For the 2020 Olympic Games, the World Chess Federation asked the Olympic committee to include chess as official sport. The committee agreed under one condition: The Chess Federation must design a ...
24
votes
8answers
5k views

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 ...
24
votes
4answers
11k 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 ...