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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Can you Avoid the Spear-Wielding Gladiator?
You are trapped in a circular coliseum, and a gladiator with a spear is chasing you. You can't defend yourself, but you can run faster than the gladiator.
You run at 11 feet per second, and the ...
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What is a Good Pasta Number™?
This puzzle is inspired by JLee's What is a Word/Phrase™ series and the subsequent "Number" variants. (Actually, I'd originally tried to create a more original puzzle using the same idea, ...
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Double chess stalemate with 60 units
Inspired by Symmetrical Chess Position With No Legal Moves
Can you arrange on a chessboard 18 kings, 6 rooks and 6 bishops of each colour (i.e. 60 units, leaving 4 empty squares) so that neither side ...
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A pentagon that can measure the first 7 integer distances
A pentagon can be used to measure 10 distances - one distance between each pair of its 5 vertices. Can you find a pentagon that can measure every integer distance from 1 to 7, inclusive?
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Cut a square piece of paper
You are given a square piece of paper shown below:
Can you cut this paper in a way that:
The shortest distance from A to B is double the shortest distance from A to C; and
The shortest distance from ...
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Six loops of threads
The puzzle:
Put six loops of threads together, in such a way that they cannot be separated from each other, but if any one of the loops is cut, then all threads can be separated from each other.
As ...
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What is the measure of $\angle{BAE}$ and the length of $BE$ and $ED$ in a square?
The puzzle is as follows:
Suppose that you have a square whose sides measure 1 inch. Let each vertex be $A$, $B$, $C$ and $D$. Now, pick a point $E$ on the interior of this square so that $\angle{EDA}...
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Turn two cubes into one!
Here are two identical cubes:
Your challenge:
Start with two cubes of exactly the same size.
Cut the surface of each of these two cubes along its edges and unfold the surface into a 2D shape. (So ...
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Tetromino in a Pentomino Lair
Inspired by this question:
Can you fit twelve pentominoes (not necessarily distinct) and one tetromino inside a 10 x 10 grid such that they do not overlap or touch each other orthogonally (...
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Flipping through the faces of a cube?
Let's place a cube on a table and flip it around a bit. In fact, flip it according to the following instructions:
Flip forward twice.
Flip left twice.
Flip backward twice.
Flip right twice.
Assuming ...
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Flipping Platonic solids
A cube is flipping on a table along its edges without sliding.
If the cube flips two steps forward, two steps to the left, two steps backward, two steps to the right, then the cube is back to its ...
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Fitting pentominoes inside a 10x10 grid
What is the most number of pentominoes that you can fit inside a 10x10 grid, such that they do not overlap or touch each other orthogonally (horizontally or vertically)?
Bonus: what is the most number ...
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Ten tetrominoes inside an 8x8 grid
Can you place ten tetrominoes inside an 8x8 grid, such that they do not overlap or touch each other orthogonally (horizontally or vertically) ?
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Ernie and the Menacing Monopoles
While driving home from a fishing trip Ernie and I saw a road-side sign pointing down a gravel track that announced ‘MYSTERIOUS SOUTH SLOPING TREES’. It was getting well on into the afternoon, and I ...
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Twenty-four trees in eighteen rows of four
A very old puzzle, #146 from American Agriculturist, April 1865:
How may twenty-four trees be planted in exactly eighteen rows, with four trees in each row? A row consists of a number of trees in a ...
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Can you refold a hyper plus sign into a cube?
If you take a cube, and grow a new cube out from each of its six faces, you will get a "hyper plus sign":
This 3D solid has an interesting property. It can be sliced along its edges and ...
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How high does the ladder reach up the wall?
A ladder of length $l$ rests against a vertical wall. Suppose that there is a rung on the ladder which has the same distance $d$ from both the wall and the (horizontal) ground. Find explicitly, in ...
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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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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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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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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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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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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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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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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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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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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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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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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.
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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 ...
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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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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 ...
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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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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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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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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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