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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8x8 Grid with no parallels

In the 8x8 grid graph shown below; you can put points to the edge of grid as shown below (blue dots). The example above has 4 points and you construct a line between two points as shown below; so ...
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0 votes
2 answers
126 views

Two triangles in a circle

This puzzle is inspired by this great puzzle. You are given a circle. You can draw two non-overlapping triangles of any size and shape inside that circle. What is the highest percentage of the circle ...
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3 votes
1 answer
258 views

A donut, a piece of string and a pair of spectacles

This is a simplified version of this physical puzzle. I believe it captures the essence at much reduced complexity. Please, forgive my poor drawing skills. The goal is disentangling the orange torus ...
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3 votes
1 answer
175 views

Complete sets of pictures by replacing blanks

I made a couple of new ones. I liked making the first one the most. The first one seems to be the easiest for me but I can see someone getting stuck on it if they don't figure out the idea. The second ...
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0 votes
2 answers
129 views

Fill the blank with an image that fits into the given set

I tried to make them not too boring. The intended solution is very hard so I tried to slightly hint at it by implementing multiple solutions that lead to the same answer. I also tried to make more ...
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13 votes
2 answers
433 views

Put three pieces of cake into a round box

You're about to cut three pieces from a large cake to put in a round box of radius 1. If the pieces must be congruent triangles, and cannot overlap, what shape gives you the maximum amount of cake?
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4 votes
1 answer
207 views

Each snowflake is beautiful but some are "pretty"

Let's define "snowflakey" pattern as Regular polygon surrounded by other regular polygons number of surrounding polygons equals to a number of angles of polygon in the middle Here are two ...
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2 votes
1 answer
95 views

How to fill up the numbers in a set of empty discs drawing a pentagon? The target sum is 10 [duplicate]

Can anyone explain to me the math behind the problem? I want to convert the mathematical solution into an efficient algorithm. The target sum can be any given number. For reference please check the ...
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1 vote
1 answer
260 views

Can you escape from two lions?

You're at the center of a circular arena. A pair of lions are at the border, planning to catch you. One of them moves as fast as you, but the other moves slower than you. The three of you are confined ...
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3 votes
1 answer
193 views

Geometric game on a n*n chessboard

You can get famous (OK, Warhol-15 minutes-famous :-)! First a few definitions. Of course, two rooks of the same colors don't attack, but since two colors are needed, "attacking" here means &...
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15 votes
2 answers
728 views

Can the lion protect the sheep from the wolves?

In a closed arena, three wolves are on the vertices of an equilateral triangle at the border. The sheep and his lion friend are at the center. The wolf eats the sheep if their distance is $0$, and ...
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22 votes
3 answers
1k views

Dividing a piece of land

Alice and Bob try to divide a piece of land $D$, shaped in a perfect closed disk of radius 1. Alice moves first to mark some finite (at least one) number of points in $D$. Bob then draws any number of ...
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3 votes
2 answers
254 views

Finding the treasure on a square island

Some treasure is hidden underground in a small square-shaped island of area $64 km^2$. You have no idea where the treasure is exactly, and no time to dig the whole island anyway. But, luckily, you do ...
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7 votes
6 answers
595 views

Breaking the Heart geometrically

The King of Geometro nation has 2 very smart wives. On the Geometro Wives day he gets a nice heart shaped cake made. It has a number of icing flowers on it. The King wants to split the cake in half so ...
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1 vote
1 answer
155 views

What are the exact times an analog clock with two identical hands and its vertically mirrored image show the same time?

Suppose you have a clock with two identical hands (there is no second hand). What are the exact times when this clock and its vertically mirrored image are identical?
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3 votes
1 answer
264 views

The Bouncer of the Last Circle

Following the success of your last paper, you received an invitation to The Last Circle, a private bar for mathematicians and logicians. But the bouncer in front of the only entrance won't let you in ...
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2 votes
4 answers
537 views

Beyond The Edge?

Ꭵn 𝔞ges past when sailors feared the 𝚖onsters 𝔞t The Edge, life was sim⍴lꬲᴦ for me. But ever ƽince Magellan and his cursed vهyage, I’m ever beside myself, ever before, ever after, ever surrounding ...
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5 votes
1 answer
261 views

Circle inscribed in triangle problem [closed]

You need to find the angle BEC knowing that the side BC is tangent to the circumference.
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3 votes
1 answer
369 views

Two points inside a circle

Two points are randomly chosen inside a circle. Is it always possible to draw a straight line through each point, such that they subdivide the circle into 3 regions of equal area? Bonus: can the lines ...
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5 votes
1 answer
1k views

Three lines to get twenty triangles

Shown below are five squares. Starting at any point, draw three straight lines without lifting the pen, and create exactly twenty (20) triangles. It is understood that this will create some other ...
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12 votes
2 answers
725 views

An interesting geometry problem

I found this on the net and tried to solve it with no luck. However there is a tricky way of solving this problem and hence I am posting it here as a puzzle. Give it a try if you have not seen the ...
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2 votes
1 answer
151 views

Five 3:1 rectangles tiling a square

Can you fully tile a square with 5 rectangles such that: Every rectangle has 3:1 ratio, ie., their length is triple their width. No part of any rectangle is outside the square. No two rectangles ...
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4 votes
2 answers
341 views

Seven 2:1 rectangles covering a square

Can you fully cover a square with 7 rectangles such that: Every rectangle has 2:1 ratio, ie., length double its width. No part of any rectangle is outside the square. No two rectangles overlap. Note ...
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2 votes
1 answer
281 views

Thirteen Diagonals of a Nonagon

A regular nonagon has 27 diagonals, and these diagonals intersect in the interior of the nonagon at 126 distinct points. Show that it is possible to select 13 diagonals of a regular nonagon such that ...
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6 votes
2 answers
235 views

Jigsaw puzzle: packing pentominoes into a rectangle

I've got this jigsaw puzzle that I can't figure out. The major problem is that there are no signposts on whether a piece is in the right place. How does one get all the pieces into the 6x10 container? ...
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  • 163
2 votes
1 answer
219 views

A puzzle in tribute to J. J. Sylvester

About the right time of the year I say, since Sylvester's Day is nigh. AFAIK the puzzle is original; comments welcome if you know any better. (Edited) Sylvester's theorem, also Sylvester-Gallai ...
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5 votes
1 answer
252 views

Connecting points to form triangles

$3n$ points are drawn on a flat piece of paper, such that no $3$ points lie on a straight line. Is it always possible to connect triples of points with straight lines, such that you form $n$ triangles ...
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7 votes
1 answer
189 views

Ernie and the Christmas Stars

Although Ernie professes to be an atheistic rationalist, he does love the Christmas season. He thinks long and hard to find appropriate gifts, brushes up on his Christmas Carol repertoire, plans a ...
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7 votes
2 answers
1k views

A rectangle cut into two pieces, which build a square

A rectangle with side length a and b are in ratio $a : b = (n+1)^2 : n^2$, where n is a positive integer. Is it possible to cut each such rectangle into two pieces, which can be put together to build ...
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  • 11.5k
10 votes
0 answers
135 views

Rigid regular nonagon from 21 Meccano strips

You are given 21 Meccano strips, where the distance between adjacent holes is 1 unit: 9 strips of length 10 (hence having 11 holes) 6 strips of length 18 (19 holes) 6 strips of length 19 (20 holes) ...
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  • 3,621
8 votes
2 answers
474 views

Form an equilateral triangle

Alice and Bob take turns to mark points in $\mathbb{R^2}$ (i.e. infinite 2D plane). Alice can only mark $1$ point on her turn, while Bob can mark $4$ points. They're free to mark their points anywhere ...
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9 votes
0 answers
421 views

Can Alice form a unit square?

Alice and Bob take turns to mark points in $\mathbb{R^2}$ (i.e. infinite 2D plane). Alice can only mark $1$ point on her turn, while Bob can mark $N$. They're free to mark their points anywhere as ...
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12 votes
1 answer
613 views

All distances different on a chess board

Here is a simple formulation for, I believe, a quite difficult problem. I have played with it, I don't have the answer yet. The question: How many pawns can you put on a standard 8x8 chess board in ...
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14 votes
2 answers
854 views

Building equilateral triangles by reflecting tokens

Three tokens are placed at the vertices of an equilateral triangle with side length 1. A move is to reflect a token at any other token. After several moves the tokens build again an equilateral ...
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10 votes
0 answers
437 views

Dissecting a figure into 2, 3, 4, and 5 parts but not 6

This figure is divided in 2, 3 and 4 equal parts of same size and shape, but it is not possible to do it in 5 equal parts of same size and shape. Is it possible to find a figure that can be divided ...
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7 votes
2 answers
641 views

Inferrence strategies for hidden pieces on a chessboard

This is a potentially off-topic question, however, the reason I'm asking it here is in the hope of gaining the perspective of puzzlers in tackling such a problem. That is, I am more interested in the ...
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-3 votes
5 answers
743 views

Pouring water from the 10 liter container

The answer to the following "decanting" puzzle Split 10L in half using 4L and 6L jugs was It is impossible to pour out 5 liters from 10 liter jug using 6 and 4 liter jugs Maybe not if you ...
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8 votes
2 answers
3k views

Social distancing in a 5x5 room [duplicate]

I have booked a square meeting room that is 5 by 5 meters. Our Covid-19 policy says that each person must be at least 1.5 meters away from any other person. What is the highest number of people that ...
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2 votes
1 answer
120 views

Triangles with side length a, b and c and $a^n$, $b^n$ and $c^n$

For which triangles with side length a, b and c do exist triangles with side length $a^n$, $b^n$ and $c^n$ for all positive integers $n\geq2$?
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2 votes
1 answer
95 views

Make a topological torus-with-a-hole out of congruent squares that may share an edge or a vertex with other squares

Suppose we arrange, in 3-dimensional space, 8 identical solid cubes in space so they form a square-shaped ring (using a 3x3 arrangement of squares except for the one in the middle). Its surface will ...
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2 votes
3 answers
456 views

Dissecting a figure into three congruent parts in three different ways

Figure 1 is divided in 2 equal parts of same size and shape in 3 different ways Figure 2 is divided in 3 equal parts of same size and shape in 2 different ways Is it possible to find a figure that ...
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5 votes
1 answer
169 views

Fold the plane four times to get the maximum number of cross points

You have a straight line $l$ in an infinite plane. You can fold the plane along any straight line so the line $l$ becomes two rays with a common starting point. In the picture we fold along line $a_1$...
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10 votes
3 answers
724 views

Tipping a tetrahedron in a plane

A regular tetrahedron with one black and three white faces is positioned with its black face at the bottom of a plane. The tetrahedron is tipped several times over an edge and finally reaches the same ...
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  • 11.5k
10 votes
2 answers
467 views

Clash of the Robinsons

"Ridiculous!" you think "What can be the odds? Either I'm hallucinating or the amateur writing this story plunged to new depths of incompetence." Both being equally likely you don'...
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8 votes
1 answer
401 views

Squares in a quadrant: How big is the pool?

The local council is building a new pool complex. A one hectare ($100\times100$m) block of land has ground suitable for pool-digging, but a local ring road runs through it, leaving only an exact ...
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  • 5,253
9 votes
2 answers
325 views

Ernie and the Cuboidal Crystals

When passing Ernie's letterbox this morning, I found a courier bag about the size of a shoe-box, resting inside (along with a smaller unlabelled bag). I immediately guessed that it was the special ...
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51 votes
3 answers
15k views

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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0 votes
1 answer
217 views

When a Cube Loves a Circle [closed]

Align the center of a unit cube to the origin, and one of its long diagonals to the z-axis. In terms of r and h, what proportion of the circle {x^2 + y^2 = r^2, z = h} is inside the cube?
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  • 411
11 votes
2 answers
493 views

Match the colors on the edges of the rectangles - is it possible?

In April 1971 (it says so on the back of the cards) I made the following puzzle which requires one to form a square with adjacent colours being the same. Can this puzzle be solved and, if yes, what ...
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  • 213
7 votes
2 answers
272 views

Ernie and the Equi-area Tetrahedra

I enjoy spending time with Ernie when he is 'pottering' in his workshop. It is a large open-plan area filled with soldering-stations and oscilloscopes, lathes and milling machines, band-saws and drill-...
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