Questions tagged [geometry]

A puzzle related to shapes, geometric objects (polygons, circles, solids, etc.) of any number of dimensions, relative position of figures, and the properties of space.

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11
votes
1answer
763 views

Professor Halfbrain and the dissection of a rectangle

Professor Halfbrain has spent his entire weekend with cutting rectangles into smaller rectangles. In particular, he proved the following deep theorem on such dissections. Professor Halfbrain's ...
6
votes
1answer
277 views

Good sets on small circles

Let's call a set of points good if every pair of points in the set are an integer distance apart. Let's call a circle small if its radius is less than 7. A good set lies on the boundary of a small ...
4
votes
1answer
342 views

A mysterious grid

Upon reaching the door to the office, instead of a normal keypad, you are presented with a strange grid. Tacked to the wall there is a punch-card with five columns, each column containing the numbers ...
8
votes
7answers
2k views

Two Blindfolded and disoriented near the Great Wall of China

Following on from the recent blindfolded near the great Wall of China puzzle. This time, two wise guys are blindfolded and disoriented and standing together exactly 1 mile from the Great Wall of ...
14
votes
4answers
2k views

Touching coins flat on a table

On an infinite table are $n$ identical circular coins lying flat. Each coin touches exactly $k$ other coins, and any two coins are connected by a path of touching coins. Determine all possible pairs ...
8
votes
2answers
675 views

Bouncing ball on a billiard board

Consider a unconventional billiard board in the shape of an equilateral triangle (depicted below). An incredibly small ball (size in picture is increased for the sake of visibility on your screen) is ...
11
votes
1answer
569 views

Blindfolded and disoriented near a space station

Following on from the recent blindfolded near the great Wall of China puzzle. Suppose you are floating in space a mile away from a huge space station (the death star perhaps). Assume that the death ...
3
votes
1answer
287 views

The lost drone at the Great Wall of China

Inspired by Blindfolded and disoriented near the Great Wall of China A drone is stationary at a spatial point about 1 m from the Great Wall, which is a vertical plane rectangle with height 5 m and ...
38
votes
5answers
5k views

Blindfolded and disoriented near the Great Wall of China

You are blindfolded and disoriented, standing exactly 1 mile from the Great Wall of China. How far must you walk to find the wall? Assume the earth is flat and the Great Wall is infinitely long and ...
6
votes
1answer
475 views

Points on the boundary of a circle

Does there exist a circle whose boundary contains 6 points whose 15 pairwise distances are distinct integers?
-1
votes
2answers
244 views

Robbers - The ultimate compass challenge [closed]

Please read the Cops post for all details. Points are scored on the basis of how long a challenge remains unsolved before you solve. You may solve any number of challenges. You cannot solve your own ...
0
votes
2answers
354 views

Cops - The ultimate compass challenge [closed]

Based on The square and the compass This is a new kind of challenge proposed on Meta Puzzling SE. Any discussion about the general type of puzzle (rather than this particular one) can be done there. ...
2
votes
2answers
215 views

The square and the compass II - Midpoints

Based on The square and the compass The rules are almost the same. (The only difference is the actual task, marked in bold.) You have a compass and a pencil but no scale/straightedge. Your job is ...
11
votes
3answers
508 views

Neighboring circles

If we join two circles on a plane, each will have exactly one neighbor. Given three or more circles, we can build a chain where each circle has exactly two neighbors. There are also arrangements ...
-1
votes
1answer
196 views

Trianglify the Shapes

For each of the following shapes, draw extra lines to divide the shape into the smallest number of triangles that can completely fill the shape. Example: Solution: Shapes (a correct answer answers ...
6
votes
1answer
491 views

The square and the compass

(I don't mean $x^2$ or $N\cdot S\cdot E\cdot W$) You have a compass and a pencil but no scale/straightedge. Your job is to mark four points on a plane paper that would form a square if joined. Your ...
127
votes
8answers
32k views

How can 64 = 65?

Here is a interesting picture with two arrangements of four shapes. How can they make a different area with the same shapes?
18
votes
1answer
1k views

Ernie and the Cake Cutting

Ernie had spent most of the afternoon at my place, helping me with the tile grouting in my new bathroom. It had been a difficult and messy job, so when he asked if I could do him a "small favour" I ...
5
votes
2answers
180 views

Problem solving, two pieces on a 12-gon

A black and a white piece is in two adjacent corners of a 12-gon. In a move, we get to move any piece to any vacant neighboring corners. If both of the pieces returns to a position which they have ...
8
votes
1answer
379 views

Cutting a 7-by-9 rectangle

Is it possible to dissect a $7\times9$ rectangle into $21$ pieces that are $L$-shaped and that consist of three little squares?
7
votes
1answer
615 views

5 equal area polygons

Given the quadrilateral ABCD (see drawing) with mid diagonals, E of AC and F of BD, you need to create 5 polygons equal in area to 1/4 of the quadrilateral ABCD area by means of a straight-edge, ...
2
votes
2answers
204 views

Geometry problemsolving

The Big and the Small Kingdom are both rectangular islands and divided into rectangular landscape. In each province there is a road that runs along one of the diagonals. On each island exist roads ...
15
votes
5answers
1k views

Tommy's Train Tracks

Tommy just got a new train set. It only came with one type of train track piece, a quarter circle, all of which were the same size. $\hspace{2.5in}$ Prove that, whenever Tommy makes a closed loop ...
4
votes
2answers
442 views

It needs some thinking but solvable and nice

Replace the question mark with your solution
12
votes
1answer
344 views

Fitting the pieces

The following grid has been constructed using the shapes of unfolded cubes. Determine the minimum number of shapes (red and green) needed to cover the blue grid. The shapes may overlap, but must ...
13
votes
4answers
3k views

Cutting a 10-by-2 rectangle

How does one dissect a $10\times2$ rectangle into four pieces that can be reassembled to form a square?
8
votes
3answers
3k views

Chopping a cube into different sized subcubes

You are the author for the magazine Extreme Cuisine Quarterly. Your articles are not concerned with cooking, but rather with the presentation of the meal. Recently, there have been rumors of a ...
11
votes
3answers
744 views

Icosikaitrigons

Original Q: Plot 24 points so that there is only one 23-sided polygon where 23 of those points are its vertices, and the other point is not on any edge of the polygon. A polygon, here, is defined as a ...
6
votes
1answer
173 views

Left and right turns on the surface of a cube

Inspired by A closed path on the Rubik's cube. Let $N$ be a positive integer. Consider an $N\times N\times N$ cube, with each face tiled by $N^2$ squares measuring $1\times 1$. A closed path is drawn ...
0
votes
3answers
10k views

Divide a circle into three equal parts [closed]

You are at a restaurant. There are a total of three people(including you). Now you order a pizza and the trouble is that all three(including you) want equal parts. You have only a knife. How do you ...
6
votes
1answer
302 views

Folding a piece of paper

On the table lies a rectangular piece of paper $ABCD$ of area $100$. Cosmo folds the rectangle once along a straight line, so that afterwards corner $C$ lies exactly on top of corner $A$. The ...
10
votes
4answers
955 views

Cutting a square into seven rectangles

Cosmo has cut a square into seven rectangles, so that the seven lengths $\ell_1,\ldots,\ell_7$ and the seven widths $w_1,\ldots,w_7$ of these rectangles satisfy $$ \{\ell_1,\ldots,\ell_7\}\cup\{w_1,\...
11
votes
4answers
1k views

Professor Halfbrain and the tilted cube

When I ran into professor Halfbrain this morning, he told me that he has constructed a cube that can be tilted and balanced on a plane table such that exactly one of the eight corners touches the ...
13
votes
4answers
1k views

The Erasmus pentagon

Professor Erasmus has constructed a special convex pentagon $ABCDE$ that he modestly calls the "Professor-Erasmus-pentagon". The professor claims that he can cut off a smaller pentagon similar to ...
16
votes
5answers
2k views

Hikers Meeting in the Middle

Two hikers are separated by a two-dimensional mountain range, like the one shown below. The mountain range alternates between peaks and valleys, connected by straight lines. Both hikers are at sea ...
2
votes
1answer
182 views

Squares in a dot matrix

We have a matrix made up of $m$ by $n$ dots. Can you give a function that counts the number of squares that can be found by joining any $4$ dots in it?
9
votes
2answers
665 views

Tiling by trapezoids

An equilateral triangle with sidelength $L$ can be tiled by trapezoids with sidelengths $2,1,1,1$. What are the possible values for $L$?
0
votes
1answer
218 views

The infinite flea circus

Based on Another curious incident in the flea circus and A curious incident in the flea circus by @Gamow There is a $n$ dimensional cube in an $n$ dimensional world. There is a flea on each vertex of ...
12
votes
1answer
605 views

Professor Halfbrain and the right-angled triangles

Today I met professor Halfbrain at the tea house. The professor looked very tired, and apparently had not slept for the last couple of days. He told me that he had been spending his time with cutting ...
12
votes
2answers
1k views

Inside or outside the square?

Enrico draws a square in the plane, and then secretly picks a point $P$ that is either situated inside the square, or outside the square, or on the boundary of the square. Damiano sees the ...
10
votes
3answers
2k views

Polygonal Pizza

Four friends want to share a pizza. However, the pizza became badly misshapen en route; though it is still flat, it is now shaped like some arbitrary polygon. Prove that it is still possible to ...
6
votes
1answer
368 views

A big cube and 99 smaller cubes

A big cube is cut into 99 smaller cubes. Exactly 98 of these 99 smaller cubes are unit cubes. Question: What is the volume of the big cube?
3
votes
6answers
964 views

Find the total number of triangles in the diagram

The title of the question says everything $\ldots$ My attempt: We count $2(1+1+1+2+1+1+2+1+1+1+1)=26$ triangles. (On each side $13$ triangles, and then multiplied by $2$). And then we combine them ...
11
votes
4answers
585 views

Ant on a hyperbox

Despite strong objections by many here at StackExchange, the mad scientist continues to use ethically questionable methods to study the spatial cognition of ants. Last time, he left ant Juliet at a ...
7
votes
1answer
565 views

This ant sure is smart. But how fast is it?

A mad scientist uses ethically questionable methods to study the spatial cognition of ants. Last time, he left his lab ant in a cubic room and filled it with painfully intoxicating gas, capable of ...
8
votes
4answers
486 views

On the shores of circle lake

Eight trees $ABCDEFGH$ are standing (in this order) on the shores of circle lake. The four trees $ACEG$ form a square of area $500m^2$. The four trees $BDFH$ form a rectangle of area $400m^2$. ...
11
votes
1answer
582 views

Professor Halfbrain's second cutting theorem

Professor Halfbrain has recently made several fascinating discoveries on cutting convex polygons in the plane. Halfbrain's second cutting theorem: Every convex polygon can be cut (by a perfectly ...
10
votes
1answer
734 views

Professor Halfbrain's first cutting theorem

Professor Halfbrain has recently made several fascinating discoveries on cutting convex polygons in the plane. Halfbrain's first cutting theorem: Every convex polygon can be cut (by a perfectly ...
2
votes
1answer
155 views

The incredible polyhedron

Based on The Erasmus polyhedron by @Gamow.. Construct a 3D convex polyhedron of any single material that can float in water such that $x\%$ of its volume is below water level and $y\%$ of its ...
15
votes
1answer
1k views

The Challenge Square

Q: Can you divide this shape into 4 equal parts, and then form a square?