# 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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### 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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984 views

### 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 $0^{\circ}$...
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### 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 ...
317 views

### Find maximum circular array sum [closed]

Take this 10 by 10 grid of numbers. ...
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1 vote
227 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 ...
360 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 ...
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182 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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530 views

### Find the optimal dividing line

Consider the following grid of numbers: In machine readable form: ...
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172 views

### 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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996 views

### 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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3k views

### 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 ...
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### 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 ...
410 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....
2k views

### 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
225 views

### 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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310 views

### 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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195 views

### 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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338 views

### 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 ...
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342 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 ...
410 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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141k views

### 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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### Connect dots on a grid with one continuous line (optimization)

(This question is the third puzzle of the Connect dots puzzle series. You can find the first two puzzles here and here, respectively. The original question and photos originate from webadventurer. ...
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### Connect dots on a grid with one continuous line 2.0

This is the same question posted by webadventurer with one extra condition (number 5) that disallows the previous solution posted here. (All credits go to webadventurer, including all images.) Rules: ...
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612 views

### Connect dots on a grid with one continuous line

Rules: Line must be straight. Line must be continuous. Line must not intersect itself. Line is allowed to take 45 degree turns to itself, for example: The dots:
314 views

### Logic and geometry problem #4: are these games functionality equivalent?

Two games, Crossway and Mincut, are believed to be functionally equivalent. That is, a win by Crossway rules will necessarily lead to a win by Mincut rules. And a win in Mincut will be a win in ...
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1 vote
139 views

### Regular polygons meeting at a point

How many ways can regular (convex) polygons meet at a point (vertex), so there are no gaps or overlaps? Here's an example with a square, hexagon, and dodecagon.
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### Game of the glasses on the windowsill

The windowsill above the sink is where my wife and I place our dirty wine glasses. And while both of us love each other, neither of us love loading the dishwasher. As a result, these dirty glasses ...
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### 9 trees in 7 rows with 3 trees in each row

The following puzzle is a variant of a puzzle published in the May 8, 1926 issue of THE WINNIPEG TRIBUNE MAGAZINE: In the picture below there are nine trees arranged in two rows with five trees in ...
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157 views

### Logic and geometry problem #3: are cycles possible in Scattercut with added "maximum" rule?

My question is whether or not cycles can occur in the game of Scattercut. That is, you kill some of mine, I kill some of yours, you kill some of mine... Endless cycle of turns. Game never finishes. ...
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96 views

### Assemble n^3 cubes into n different n×n×n larger cubes [duplicate]

You might have seen this question before: Goal: Paint 27 cubes using three colors (for example, red, yellow, and blue), so that you can form a 3x3x3 cube with all surfaces in red (for example), a ...
441 views

### Tiling a dodecahedron

The surface of a dodecahedron is tiled with 6 of the shown tiles, each tile covering two faces of the dodecahedron. In how many essentially different ways this can be done? Two tiled dodecahedrons are ...
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417 views

### Ship collisions

Four ships are sailing on a 2D planet. Each ships traverses a straight line at constant speed. No two ships are traveling parallel to each other. Their journeys started at some time in the distant ...
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### Capture a laser beam

Design a mirror box that can capture a laser beam, so that it will keep reflecting forever. The setup looks like in the following image: The goal is to design a box in a way, that the light beam will ...
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858 views

### Fewest polyominoes adjacent to 3 copies

What is the smallest positive number of polyominoes P, such that You can place grid aligned copies of P without any overlap; and Each polyomino is adjacent to exactly 3 other polyominoes. ...
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733 views

### Smallest polyomino adjacent to 3 copies

What is the smallest polyomino P in number of cells, such that You can place grid aligned copies of P without any overlap; and Each polyomino is adjacent to exactly 3 other polyominoes. Polyominoes ...
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### An immortal ant on a gridded, beveled cube divided into 3458 regions

This puzzle takes place on the surface of the following gridded, beveled cube: The surface of this cube is divided into 3458 small regions separated by black lines. Of these regions, 3450 of them are ...
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### Making regions with 6 joined lines

What is the most number of enclosed regions^ you can produce by drawing 6 straight lines that are all joined end to end in sequence? ^ an enclosed region is the maximal region whose perimeter is ...
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### 12 piece cube packing puzzle

Consider the following hexacube (made from 6 unit cubes): GOAL: Pack a 3 x 3 x 3 cube using three of these hexacubes plus nine unit cubes. This puzzle comes from: https://puzzlewillbeplayed.com/333/...
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### Can you pack these pentacubes to form a rectangular block with at least one odd side length other the side whose length must be a multiple of 5

This puzzle is part of the Monthly Topic Challenge #11: Now in 3D. Consider the following pentacube (made from 5 unit cubes): It is possible to pack four of these pentacubes to form a 2x2x5 ...
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1 vote
181 views

### Create a 3D object to demonstrate the pyramid volume equation [closed]

Under the assumption that every triangle area is given by the equation "S = constant X Base X height", with a simple drawing it is demonstrated that the constant is 1/2. Assuming that a ...
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