# 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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### How many cylinders in a cube [closed]

Could someone please help me with this math question: How many cylinders with a diameter of $7.44 cm$ and a height of $4.9 cm$ can fit in a cube of $1m \times 1m \times 1m$?
214 views

### Professor Halfbrain and the wonderful rectangles

Professor Halfbrain calls a rectangle wonderful, if it is similar to the rectangle with side lengths $1$ and $2-\sqrt[3]{5}$. The professor claims to have a proof for the following theorem: ...
871 views

### Arrange matchsticks so that

Your challenge is to arrange matchsticks to make a pattern. Matchsticks are all the same length and cannot be broken. These are geometric arrangements: Make 4 triangles using 6 sticks. Make 3 X's ...
393 views

Today I have drawn a regular $18$-gon on a piece of paper. My drawing shows the $18$ vertices of the polygon labeled as $P_1,P_2,\ldots,P_{18}$ in clockwise order, and it also shows all $135$ ...
775 views

### Join six five-link chains to form a circular chain

Join six five-link chains to form a circular chain. To join two chains, you must cut, and then re-weld, a link. The final number of links in the circular chain will be 30. What is the minimum ...
719 views

### A triangular Quidditch field

A Quidditch field is usually in the shape of an oval, a hundred and eighty feet wide and five hundred feet long. But today the Gryffindor house team is training on a smaller field in the shape of a ...
146 views

### Professor Knowfair's square dissection assignment

While at the public library with professor Knowfair, he also told me his plans for this week's assignment. He said: Dissecting a triangle is all fine and dandy, but squares are the real deal. For ...
151 views

### Professor Knowfair's triangle dissection assignment

Last Sunday I met with Professor Knowfair at the public library. We caught up and he told me about the most recent assignment he gave his students. He said: We had been studying medians and how ...
177 views

### How many times?

You have a high-precision watch where all the hands start at noon all pointing exactly to the 12 at the top. All the hands move in atomic ticks where there's no intermediary position in between ticks. ...
378 views

### An Exercise in Spatial Geometry

You're on a treasure hunt for a mysterious object, and the clue you need is hidden somewhere in a giant library. You have the following clue: uxq pt as well as the following piece of paper: ...
181 views

### Can you crawl between the ground and under the middle of the cable? [closed]

Two vertical posts are in level ground at 150 feet apart. A bendable cable, but one that does not stretch from its original length, is 150 feet and 1 inch in length. The cable is fastened in ...
646 views

### Geometry puzzle [closed]

The lengths of the triangle sides satisfy $x < y < z$. Lines that look parallel indeed are parallel. Triangles that look similar indeed are similar. The three red lines are concurrent (meet at a ...
247 views

Cheherazad, the carpet salesman, has bought a rectangular piece of carpet. Unfortunately, Cheherazad's has lost his tape measure and he has no other measuring instruments. However, he finds that if he ...
3k views

### A chess board and a coin! [closed]

You have a standard chessboard, consisting of $64$ squares ($32$ white and $32$ black) each of dimensions $4\times4$ cm, and you have a standard coin of $2$ cm diameter with tail and head. You flip ...
432 views

### Ernie and the Revisitation of the Aunts

I was hanging around at home last week, a little lost for something to do, when I got a phone call from Ernie. He was in a bit of a panic because a guest was going to drop in later in the day. He had ...
3k views

### Gunfire at dawn

A group of ten bandits stand in a flat desert, with no pair the same distance apart. Tensions grow, and at the crack of dawn each bandit fires a single bullet at the bandit closest to him. All have ...
241 views

### Bigger board with the least squares

What is the minimum number of squares to be used to draw a 8x8 board which is divided into 64 unit squares. The squares you are going to use can be of any size you want (see example below). There ...
235 views

### Squaring a cross one more time

This is the lateral-thinking fifth part to this question. In this cross shape, all twelve sides are the same length and all angles are right angles. What is the smallest number of cuts that can ...
2k views

### Tiling a rectangle with nine squares

A rectangle is tiled by nine squares with side lengths $2,5,7,9,16,25,28,33$ and $36$ (without overlapping and without gaps). What are the side lengths of the rectangle? What does the tiling look ...
3k views

### Dividing the pentagon

It is easy to divide an equilateral triangle into three equal, though not equilateral, triangles. It is even simpler to divide a square into four equal squares. The difficult part is, whether you ...
1k views

### Squaring a cross

In this cross shape, all twelve sides are the same length and all angles are right angles. How many cuts does it take to divide the cross into pieces that can be rearranged to form each of the ...
5k views

### The Non-Pythagorean Theorem

Everyone knows the Pythagorean theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The ...
445 views

### Bearing due, due and back

[A variation of : What color was the bear?$/$Where is your rescue partner?] From which latitudes on a theoretically spherical Earth can you proceed as follows? •  Set ...
768 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 ...
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 ...
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 ...
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 ...
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 ...
682 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 ...
572 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 ...
289 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 ...
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 ...
478 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?
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 ...
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. ...
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 ...
512 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 ...
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 ...
495 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 ...
33k 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?
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 ...
181 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 ...
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?
623 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, ...
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 ...
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 ...
443 views

### It needs some thinking but solvable and nice

Replace the question mark with your solution
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 ...
How does one dissect a $10\times2$ rectangle into four pieces that can be reassembled to form a square?