# 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 &...
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 ...
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### 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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### 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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### 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
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### 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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### 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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### 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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### Circle inscribed in triangle problem [closed]

You need to find the angle BEC knowing that the side BC is tangent to the circumference.
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 ...
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### 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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### 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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### 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 ...
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 ...
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 ...
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### 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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### 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 ...
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 ...
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 ...
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### 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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### 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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### 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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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 ...
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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 ...
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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 ...
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### 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 ...
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 ...
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### 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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### 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 ...
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 $n\geq2$?
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### 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 ...
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 ...
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 $a_1$...
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### 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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### 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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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 ...
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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 ...
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### 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 ...
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?
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### 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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### 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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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Puzzling rectangles [closed]

Can you find two unequal rectangles whose areas add to $5\sqrt5$? No computer solutions please.
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### 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.
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### 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 ...
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### 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. ...
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