# 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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### Build a slanted pyramid with ten L-shaped blocks

Consider the following L-shaped 3-dimensional object made up of three unit cubes joined at their faces: Use 10 of the above L-shaped pieces to make the following shape:
• 9,876
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### Make a square table top with the minimal needed amount of straight cuts

inspired by : Make a square table top with six (or fewer) pieces A carpenter has three pieces of beautiful wood, measuring 12 inches, 15 inches, and 16 inches square, respectively. They want to use ...
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### Make a square table top with six (or fewer) pieces

A man had three pieces of beautiful wood, measuring 12 inches, 15 inches, and 16 inches square respectively. He wanted to cut these into the fewest pieces possible that would fit together and form a ...
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635 views

### Circles crossing every cell of an NxN grid

What is the least number of circles you need to draw, such that every cell of an NxN grid is crossed? A circle crosses a grid cell if one of the points on its circumference lies completely inside the ...
1 vote
162 views

### Recursive rhombic dodecahedron tiling

Say you have a single rhombic dodecahedron, call it "layer p0". If you then tile identical dodecahedrons around it so that it gets completely covered, the number of dodecahedrons on the ...
521 views

### Is it possible to fill an arbitrarily large hex grid completely given these rules? #2

Based off of this. Lets say you have two players, Red and Blue, that alternate filling an arbitrarily large hexagonal grid of tessellated hexagons with pieces of their color. Hexagons can either be ...
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### Is it possible to fill an arbitrarily large hex grid completely given these rules?

Lets say you have two players, Red and Blue, that alternate filling an arbitrarily large hexagonal grid of tessellated hexagons with pieces of their color. Hexagons can either be filled or empty. A ...
2k views

### Circles crossing every cell of an 8x8 grid

What is the least number of circles you need to draw, such that every cell of an 8x8 grid is crossed? A circle crosses a grid cell if one of the points on its circumference lies completely inside the ...
255 views

### April Fools Origami Update

Me (Sunny Lu) and Ωmega_3301 have made a special April Fools edition of origami puzzles. The objective is to fold a shape into a rectangle with uniform thickness. The thickness will be given to you. ...
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### origami WAVE t2

Fold the shape into a rectangle such that the rectangle is 2 layers thick everywhere. The grey part is not included in the puzzle; the purple part is the puzzle. Although solutions are not required ...
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### Logic and Geometry Problem #6

Is it possible for a deadlock to occur in Necklace? Can there be a square on which neither Red nor Blue can place a stone? If it is possible, I need to see an example of that. If a deadlock is not ...
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### origami USHAPE t3

Fold the shape into a rectangle such that the rectangle is 3 layers thick everywhere. Although solutions are not required to be applicable in real life, using an actual sheet of paper is a great ...
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### ORTHOGONAL origami FISH t8

Since the last puzzle got cheesed, we're adding a new restriction (in italics) Fold the shape into a rectangle such that the rectangle is 8 layers thick everywhere. All folds must be orthogonal ...
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### origami FISH thickness 8

Fold the shape into a rectangle such that the rectangle is 8 layers thick everywhere. A puzzle for this type of orgami puzzle game made by Ωmega_3301, similar to this question.
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### Can 42 1x2x4 cuboids be packed into a 7x7x7 cube?

Can 42 1x2x4 cuboids be packed into a 7x7x7 cube without cutting any of them? Assume that all cuboids have their axes parallel to the axes of the big cube. I tried using https://www.jaapsch.net/...
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### Anna tries to make triangles from broken sticks

Anna and Boris play a game with a red stick, a white stick and a blue stick, each of which is 1 meter long. Anna starts by breaking the red stick into three pieces. Then Boris breaks the white stick ...
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### How do I constrain a puzzle and keep a singular solution?

I am tinkering with a puzzle framework that has the following rules: In a 6x6 grid of squares, arrange 8 strips of connected squares such that there exists exactly one strip of every length (i.e. a ...
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### A pizza dilemma

You are a waiter at a restaurant. The restaurant is known for its signature dish: the Donut Pizza. The Donut Pizza is a 5-inch square pizza with a 1-inch square hole in the middle. After several ...
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### Pleasant Cuboids

A rectangular prism (or cuboid) made up of xyz identical unit cubes (x along its width, y along its length, and z along its height). Some of those cubes are internal, while the rest are external. Such ...
417 views

### The shady puzzle that will keep you in the dark

The image below is the horizontal cross section of a room. The bulb shows the position of the single light source. When the light is switched on, one wall (marked in brown) remains completely in ...
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### Join six cities with roads

Warmup question: Each of five cities is connected to the others by four roads. Show that it is possible for the roads to intersect only once with exactly two roads crossing over at that single ...
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### Walking in a random direction

I walk $\pi$ km in one direction followed by $\pi$ km in another direction. In expectation how far am I now from my starting location? Both directions are chosen uniformly at random between $0^{\circ}$...
3k views

### What is the total area of the two quarter circles?

Puzzle by Catriona Agg. The yellow circle has radius 4. What’s the total area of the two quarter circles?
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### icosahedral net [closed]

The net of $20$ triangles shown to the right can be folded to form a regular icosahedron. Inside each of the triangular faces, write a number from $1$ to $20$ with each number used exactly once. Any ...
328 views

### Find maximum circular array sum [closed]

Take this 10 by 10 grid of numbers. ...
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1 vote
229 views

### Covering a Square Floor with Square Rugs [closed]

You are given a finite collection of axis-aligned square rugs. (You do not choose the collection of rugs that you receive and the rugs are not necessarily all the same size.) Your objective is to move ...
365 views

### Longest cycle on a cube

What is the length of the longest straight path on the surface of a unit cube, such that it starts and ends at the same point? The path can cross itself and must be straight on every edge and face ...
183 views

### Find the optimal partition in this matrix

Given a particular matrix of integers, the challenge is to draw a boundary line through the cells so that the sum of the numbers on the boundary line or above is as large as possible. In this case &...
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### Find the optimal dividing line

Consider the following grid of numbers: In machine readable form: ...
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### Deciding whether a set of points on a 2D plane has axial symmetry [closed]

The problem to solve: Let's say we have a set of $n$ points on the 2D plane. Determine whether it has axial symmetry. My attempt so far: For n=2 the answer is trivially "yes". For n=3 ...
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### A Sierpiński Carpet ratio

This math problem popped into my head and I wanted to share it with you: We have the Sierpiński carpet, which is a fractal built like this: Draw a square. Divide it into 9 equal subsquares arranged ...
268 views

### Nimber mnemonic combinatorial puzzle

Please see my previous question for more background. The following represents an unfolded version of PG(3,2) with 1 as the center point: Given that each number must be an end point of a line which ...
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### Mishustin's circle problem

This problem was given to high school students by the Russian prime minister Mishustin. We have a circle. We are given some point on the circle and its diameter, as shown below. We are given a ...
231 views

### Nimber Mnemonics

Note I originally tried to ask a variation of this question on math.stack; however 1 commenter pointed out that math.stack is not a puzzle site, which made me think maybe the fine folks of puzzling ...
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### Rearrange words to make a sentence

The following puzzle is from the October 1961 issue of the Eureka journal (published by The Cambridge University Mathematical Society): Rearrange the order of the following so as to make a true ...
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### A Prime Ant's Excursion in the Cartesian Plane

An ant resides at the origin of the Cartesian plane. One morning she sets out on a long excursion of its first quadrant and pledges to walk a different prime number of units every day starting with 2, ...
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### Put infinitely many equilateral triangles of equal size on the plane

...such that There's no overlapping No more such triangles can be added without overlapping. Let $r$ be, on average, the ratio of the area covered by triangles with respect to the area which is not. ...
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### Construction of non-rhombus but still paralellogram non-square-non-rectangle non-kite via Pythagorean triplet

Is it possible to construct a non-rhombus but parallelogram and quadrilateral non-square, Non-rectangle by putting four 3-4-5 (Pythagorean triplet) triangles together and making the 90 degree angle ...
453 views

### nice places on earth

We call a place on earth nice if you go 1 mile north, 1 mile west, 1 mile south, 1 mile east and then you end up exactly at the same place you started but you didn't visit any location more than twice....
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### Colouring a rug

I have a rectangular rug in my living room composed of coloured patches (shown below). For convenience I have labelled each distinct colour from 1 to 6. Let's suppose that it was created by starting ...
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### 27 spaces, 27 curious places

Grid 1: Grid 2: Grid 3: Find the meaning behind these images. Hint 1: Hint 2: Hint 3:
1 vote
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### Logic and Geometry Problem #5: does Savage Go have cycles?

My question is whether or not a cycle can occur in the game of Savage Go. That is, you kill some of mine, I kill some of yours, you kill some of mine... Endless cycle of turns. Game never finishes. No ...
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### Waffleing my Egg

This morning I had a waffle and a fried egg for breakfast. The fried egg was cooked with a mold, so it was perfectly round and three inches in diameter. The waffle was a 3-inch by 4-inch grid of one ...
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### Continue the squence of hexagons

This is from c't, a German IT magazine (link will download file). You see seven hexagons, each with seven green and blue dots, following a specific rule. Find out what colors the circles must have to ...
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### Cutting the points evenly

I draw an even number of points on a piece of paper. Is it possible to cut the paper into two pieces with a single straight cut, such that: Each piece gets the same number of points The cut does not ...
345 views

The works and titles of this painter were recently unveiled at an exhibition. What ten letters did the Germans use to describe his collective of artists? errata: the final image should be this:
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### An engineer, a mechanic and an athlete walk into a bar

...to exchange puzzles about the thing they talked about the other day and have a drink. The engineer showed these numbers: 2, 8, 14, 2, 99 2, 4, 0, 0, 1 The ...
416 views

### Closed path on a dodecahedron

Your task is to draw lines between edges on a regular pentagon such that if you tile a dodecahedron with 12 identical copies of that pentagon you get a single closed line which does not intersect ...
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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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