# 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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### 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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### 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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### 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:
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### 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
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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
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### 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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### Odd solutions are in 3d

This puzzle is part of the Monthly Topic Challenge #11: Now in 3D. You are given a collection of sticks which are straight lines of length 1. Two such sticks can be attached to each other at their end ...
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### Three clocks problem

Three clocks which show the times for three different time zones are hanging on a wall. The length of their minute hands is the same - but not the lengths of their hour hands. The lengths of the hour ...
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### Three boats problem

Three boats are sailing on the ocean. The three boats form the corners of an equilateral triangle. Boat 1 heads southwards with a constant speed of 10 kn, boat 2 heads westwards with unknown constant ...
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### Can you pack these tetracubes to form a rectangular block with at least one odd side length?

Consider the following tetracube (made from 4 unit cubes): It is easy to pack two of these tetracubes to form a 2x2x2 rectangular block. And from that simple packing it is easy to pack any ...
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### Killer Capybaras

Clues: [contextual images] [2 triangles] [2 trapezoids] [1 rectangle full of letters listed in-order] Instructions: Name That Poison _ _ _ A _ _ _ _ _ _ _ _ Hint:
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### Triangles, Triangles, and More Triangles

Which has more area, Shape A, which is one big triangle, or Shape B, where each 2 triangles have half the side length of the previous two triangles? (Shape B has infinitely many triangles.) Or do they ...
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### Alone in Bronson Canyon

Clues: [contextual images] [boat shape made of more shapes, each with letters inside] [box within a box] [arms on either side] [water] Instructions: Name That Captain _ _ _ _ _ _ _ _ _ _ A _ _ ...
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### F the Planet - Enigmatic Puzzle

Clues: [contextual images] Eye D G C F U K L K A Z W A N C H Q T O U I A E I X L U M N R J M A M V A I R S A E T S Z B F F F F Instructions: Name That Character _ _ _ A _ _ _
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### Cutting a square into integer triangles

You are given a square piece of paper with size 10x10 units. What is the most number of triangles that can be cut from this square, such that: Each triangle has integer sides. Each triangle is ...
1 vote
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### Axis of Symmetry

Give an example of a 3D object with exactly $2$ axes of symmetries? Definition of Axis of Symmetry: Let $l$ be a straight line. If every point $P$ on object $O$ has a corresponding point $P'$ on $O$ ...
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### Drowning Squirrel

On a super foggy day, a squirrel falls from a tree into a swimming pool. Due to fog, it cannot see any bank of the pool, unless it touches the bank. It is aware that the swimming pool is in the shape ...
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### n*n*n Rubik's cube algorithm

Is there a universally working (I mean, regardless of n) algorithm for Rubik's cube n×n×n ? It is acceptable to divide ...
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### Catch the mouse (Easy)

A mouse is at the center of a square. There are 4 cats each on the mid-points of different sides of the square. Mouse is looking to escape to any corner of the square. Mouse and cats have the same ...
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### Attacking Hyenas

$N$ Hyenas are standing on a plane region in a forest. At $t=-1$, they see dead meat nearby. Being selfish, at $t=0$, each Hyena attacks the Hyena which is closest to it. All pairwise distances ...
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### Wizard of subsets

Can you change this into this in three moves? You are the wizard of subsets. With only your mind, you can grab any subset of the 16 stone blocks and move them one unit in any direction (north, ...
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### Lengths of sides of a right-angle triangle [closed]

Let's have a right angle triangle. The sides have lengths equal to rational numbers. The area of the triangle is equal to 39, the length of the hypotenuse is 31.3 and the difference of the other two ...
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### Flipped Einsteins in the Einstein Tiling

The single-tile aperiodic tiling by Goodman-Strauss, Kaplan, Myers and Smith has been all the rage recently: In this tiling a minority of tiles, coloured purple above, are flipped with respect to the ...
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### Average length of chords with a fixed end [closed]

The problem is pretty simple to state. Draw a circumference of radius r, and fix a point on the perimeter. The question is: What is the average length of all the chords defined by that fixed point and ...
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### Most polyominoes in an 8x8 grid

What is the most number of distinct free polyominoes you can form by painting an 8x8 grid in two colours? Here a polyomino is a set of orthogonally adjacent cells of the same colour, so polyominoes of ...
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### World Tour of Planet Rhombicosidodecahedria

This is the planet Rhombicosidodecahedria: This lovely planet has 62 countries, each with its own distinct history and culture. By an amazing coincidence, the countries all happen to coincide ...
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### "Sphere ---> Circle, Ball---> ?"

I am struggling with a puzzle in Swedish. I will translate it to English: Sphere ---> Circle, Ball--->? a) Sphere b) Ring c) Circle d) ? The answer provided is "It is hottest that Ring ...
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### Smallest Magic Hexagon Using Repeat Digits

Consider this image below. Its a magic hexagon using repeated digits to create a magic sum of 10. All rows columns and diagonals, meaning the cells in any straight line through the hexagon in any ...
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### The Joy of Hexasect

This geometric construction challenge is a set of components to be placed in order, \$\begingroup \def \s #1{{ \small\sf #1 }} \def \AB { \overline {\s{AB}} } \def \line #1{{ ...
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### Personal Names Turn Around

This puzzle is part of the Monthly Topic Challenge #8: Cellular Automata. Life is like connecting dots... ACE BTS GKH HE I II ING LOAF ME MOON POND PV QR ROT TUB US Can you identify the themes? ...
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### Cover this disc

Can you place five discs of radius 1, such that they fully cover a disc of radius 2?
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### Multiple Line Lengths around 8 triangles

The image below contains 8 triangles each having side length 6. Inside each triangles are three numbers indicating lengths of 3 lines that are wrapped around the triangle without gaps. Numbers range ...
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The dots around the semicircle are equally spaced. What fraction is shaded? This puzzle is by the amazing Catriona Agg.
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### Just driving around

I drove in a straight line for 10km in one direction then 20km in another direction and then 30km in another direction. Assume the Earth is flat. The three directions are not necessarily distinct from ...