# 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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### 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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### 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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### 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?
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### Is this duplo train track under too much tension?

My kids were making this train track of duplo the other day, and this is what they put together. They are still very young, and for them, this is something big. They were really proud that they ...
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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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### Can you fold a square into a square of one-fifth the area?

I love origami, and it recently gave me an idea for a very hard but beautiful puzzle. I'm really curious whether anyone here can solve it. So here's the puzzle. You are given a large perfectly square ...
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### 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 ...
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### 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 ...
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### 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?
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### 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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### 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 ...
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### 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 ...
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### A new way to cut a pizza

Can you cut a pizza (circle) into 12 congruent pieces, such that half of them have crust (circle boundary), while the other half do not? The pieces must have the same shape and area, but can be ...
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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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### 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 ...
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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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### 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 ...
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### A colorful dodecahedron

Divide a "base" edge of a regular pentagon into three equal parts. Then draw two lines from the base to the center of the other edges such that the lines do not intersect. This splits the ...
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### What percentage is grey?

The evenly spaced lines are drawn parallel to the base of triangle. What percentage of the triangle is grey?
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### Find the perimeter (seemingly unsolvable problem)

It might seem that there is not enough information to solve this problem. But the fact is that there is enough information to find the perimeter.
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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 ...
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### Release the "Q" ball

Hit the "I" ball at such an angle that it creates a chain reaction which ultimately dislodges the "Q" ball. Whenever the ball in motion hits a stationary ball, the ball in motion ...
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### What fraction of the larger semicircle is filled?

What fraction of the larger semicircle is filled? The two smaller semicircles are of equal size. This is a puzzle originally set by Catriona Agg, who is a puzzle setting genius.
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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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### 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 ...
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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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If you take a cube, and grow a new cube out from each of its six faces, you will get a "hyper plus sign": This 3D solid has an interesting property. It can be sliced along its edges and ...
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### Odd-looking circle

A man is told to make a circle He makes this: Where is the man?
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### 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 "...
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### 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 ...
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### Variant of lion and 100 zebras

Note: This problem remains unsolved, as of 19 April 2020, so do try it out. 400 rep bounty guaranteed for a correct answer This a variation of this question by @Gamow Suppose there are $100$ lions ...
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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-...
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### 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 ...
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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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### 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
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### 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:...
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### 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 ...
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### Dissecting the exotic bulbfish

Can you cut the following black shape into exactly three pieces, and then rearrange those pieces into a square?
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### 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:
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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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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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