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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4 votes
1 answer
121 views

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 ...
5 votes
0 answers
231 views

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, ...
  • 596
6 votes
3 answers
1k views

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 ...
  • 1,197
23 votes
3 answers
2k views

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?
2 votes
2 answers
703 views

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 ...
8 votes
1 answer
232 views

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 ...
  • 7,149
0 votes
2 answers
110 views

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}...
13 votes
1 answer
2k views

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 ...
8 votes
1 answer
193 views

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 (...
  • 125k
2 votes
1 answer
156 views

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 ...
13 votes
1 answer
238 views

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 ...
  • 7,149
10 votes
3 answers
1k views

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 ...
9 votes
3 answers
1k views

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) ?
6 votes
2 answers
198 views

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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20 votes
3 answers
2k views

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 ...
36 votes
2 answers
2k views

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 ...
3 votes
2 answers
164 views

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 ...
  • 5,328
4 votes
3 answers
290 views

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 ...
  • 28.9k
0 votes
2 answers
156 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 ...
3 votes
1 answer
313 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 ...
  • 13.6k
2 votes
1 answer
222 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 ...
0 votes
2 answers
224 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 ...
13 votes
2 answers
467 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?
  • 5,187
4 votes
1 answer
219 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 ...
2 votes
1 answer
104 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 ...
1 vote
1 answer
287 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 ...
  • 5,187
3 votes
1 answer
204 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 &...
15 votes
2 answers
772 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 ...
  • 5,187
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 ...
  • 5,187
3 votes
2 answers
269 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
607 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 ...
  • 38.8k
1 vote
1 answer
164 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?
3 votes
1 answer
278 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 ...
  • 6,234
2 votes
4 answers
553 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 ...
  • 511
5 votes
1 answer
274 views

Circle inscribed in triangle problem [closed]

You need to find the angle BEC knowing that the side BC is tangent to the circumference.
3 votes
1 answer
374 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 ...
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 ...
  • 38.8k
12 votes
2 answers
751 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 ...
  • 38.8k
2 votes
1 answer
168 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 ...
4 votes
2 answers
366 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 ...
2 votes
1 answer
342 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 ...
6 votes
2 answers
307 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? ...
  • 163
2 votes
1 answer
233 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 ...
5 votes
1 answer
285 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 ...
8 votes
1 answer
216 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 ...
  • 12.7k
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 ...
  • 11.6k
10 votes
0 answers
158 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) ...
  • 4,113
8 votes
2 answers
490 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 ...
  • 5,187
9 votes
0 answers
450 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 ...
  • 5,187
13 votes
1 answer
621 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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