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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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 &...
Hauke Reddmann's user avatar
16 votes
2 answers
887 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 ...
Eric's user avatar
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22 votes
3 answers
2k 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 ...
Eric's user avatar
  • 6,498
3 votes
2 answers
290 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 ...
Oray's user avatar
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7 votes
6 answers
623 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 ...
DrD's user avatar
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1 vote
1 answer
200 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?
Tamás Sengel's user avatar
3 votes
1 answer
292 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 ...
Auribouros's user avatar
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2 votes
4 answers
565 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 ...
bob's user avatar
  • 511
5 votes
1 answer
283 views

Circle inscribed in triangle problem [closed]

You need to find the angle BEC knowing that the side BC is tangent to the circumference.
user78580's user avatar
3 votes
1 answer
415 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 ...
Dmitry Kamenetsky's user avatar
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 ...
DrD's user avatar
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12 votes
2 answers
830 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 ...
DrD's user avatar
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2 votes
1 answer
200 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 ...
Dmitry Kamenetsky's user avatar
4 votes
2 answers
432 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 ...
Dmitry Kamenetsky's user avatar
2 votes
1 answer
616 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 ...
Daniel Mathias's user avatar
6 votes
2 answers
557 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? ...
Allure's user avatar
  • 163
2 votes
1 answer
238 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 ...
François Jurain's user avatar
5 votes
1 answer
381 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 ...
Dmitry Kamenetsky's user avatar
8 votes
1 answer
276 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 ...
Penguino's user avatar
  • 13.9k
7 votes
2 answers
2k 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 ...
ThomasL's user avatar
  • 12.2k
11 votes
2 answers
252 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) ...
Parcly Taxel's user avatar
  • 7,678
9 votes
2 answers
531 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 ...
Eric's user avatar
  • 6,498
15 votes
2 answers
932 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 ...
Eric's user avatar
  • 6,498
15 votes
1 answer
698 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 ...
Florian F's user avatar
  • 29.8k
14 votes
2 answers
874 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 ...
ThomasL's user avatar
  • 12.2k
12 votes
0 answers
859 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 ...
Rodolfo Kurchan's user avatar
7 votes
2 answers
662 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 ...
user929304's user avatar
-3 votes
5 answers
849 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 ...
DrD's user avatar
  • 39.2k
9 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 ...
Dmitry Kamenetsky's user avatar
2 votes
1 answer
128 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 $n\geq2$?
ThomasL's user avatar
  • 12.2k
2 votes
1 answer
119 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 ...
user avatar
2 votes
3 answers
586 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 ...
Rodolfo Kurchan's user avatar
5 votes
1 answer
178 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 $a_1$...
Eric's user avatar
  • 6,498
9 votes
3 answers
842 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 ...
ThomasL's user avatar
  • 12.2k
11 votes
2 answers
492 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'...
loopy walt's user avatar
  • 21.2k
8 votes
1 answer
422 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 ...
Anon's user avatar
  • 5,340
9 votes
2 answers
435 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 ...
Penguino's user avatar
  • 13.9k
54 votes
3 answers
16k 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 ...
Dmitry Kamenetsky's user avatar
0 votes
1 answer
232 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?
gyancey's user avatar
  • 519
11 votes
2 answers
539 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 ...
Farcher's user avatar
  • 213
7 votes
2 answers
314 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-...
Penguino's user avatar
  • 13.9k
8 votes
0 answers
276 views

Doubling the cube with rational Meccano strips

In three monographs published in 2006, 2008 and 2014 Gerard 't Hooft considered "Meccano mathematics": how to construct specified distances and regular polygons by a rigid system of ideal ...
Parcly Taxel's user avatar
  • 7,678
8 votes
1 answer
382 views

Taping a cardboard square is getting me fired!

I was mucking around in the storeroom today when I should have been working. I had a square $20\times20$ cm piece of cardboard, and a roll of masking tape I'd been using (I can't remember the tape's ...
Anon's user avatar
  • 5,340
7 votes
1 answer
303 views

Lost Bearings: A Puzzle That I've Designed

I'm Benjamin Curran. I'm a 14 year old puzzle lover and I come to you with this puzzle that I've made up myself. Hopefully you won't find it tricky. Two tourists have gotten themselves lost in the ...
Benjamin Curran's user avatar
0 votes
1 answer
149 views

Catching the Vegetable-Goose [closed]

After an altercation involving giant bees, you have been sentence to some hard labour, specifically harvesting barnacle-lambs: a gourd-like fruit that grows geese in its shell While working, you get ...
Ichthys King's user avatar
  • 1,447
-8 votes
1 answer
152 views

Puzzling rectangles [closed]

Can you find two unequal rectangles whose areas add to $5\sqrt5$? No computer solutions please.
Vassilis Parassidis's user avatar
3 votes
2 answers
799 views

Most points on a circle

What is the most number of integer lattice points that lie on the circumference of a single circle whose radius is 80 or less? Please no computer computations.
Dmitry Kamenetsky's user avatar
4 votes
2 answers
533 views

Catch the angel in less than 7 units of time

The devil has trapped the angel in a regular hexagram of firewalls. The perimeter of the hexagram is 12. The devil starts at the apex of the hexagram. can move at speed $1$ to leave a trajectory of ...
Eric's user avatar
  • 6,498
9 votes
7 answers
6k views

Prove that sin(x) ≥ x/2, but without calculus!

Important Note: After when this puzzle was posted, many people pointed out errors and improvements that could be made. I also noticed many flaws, so the post once had gone through drastic changes. ...
EsoJihun's user avatar
  • 507
5 votes
1 answer
720 views

Guess the algorithm

Here you are seeing part of a pattern I created with my computer. Can you either reproduce it or describe the algorithm I used?
loopy walt's user avatar
  • 21.2k

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