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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13
votes
2answers
917 views

Prove why this mechanical linkage for a triangle centroid works

I saw on Twitter this cool mechanical linkage for which the red dot corresponds to the centroid of the triangle defined by the blue dots: Can you prove why this linkage works?
11
votes
0answers
692 views

Hidden message in rap text II: “Ya, a ray 'ave july joy!”

This puzzle is inspired by this one by @LukasRotter. Hopefully you don't mind me using similar (well, basically the same) format as yours, Mr. Rotter! ;) The same rapper released another teaser for ...
12
votes
1answer
865 views

Four non-right angled triangles passing through every dot of a 5x5 grid

This puzzle was suggested by jwezorek in Three triangles passing through every dot of a 5x5 grid 25 dots are drawn as a 5x5 regular square grid. Can you draw 4 non-right angled triangles that pass ...
18
votes
1answer
1k views

Four triangles passing through every dot of a 7x7 grid

49 dots are drawn as a 7x7 regular square grid. Can you draw 4 triangles that pass through every dot? The corners of the triangles must lie on the dots, ie., they cannot lie outside the grid. The 7x7 ...
15
votes
3answers
3k views

Three triangles passing through every dot of a 5x5 grid

25 dots are drawn as a 5x5 regular square grid. Can you draw 3 triangles that pass through every dot? The corners of the triangles must lie on the dots, ie., they cannot lie outside the grid.
-5
votes
2answers
150 views

Inscribing a cylinder within a sphere [closed]

Let's have a sphere with R=3. What is the trick to inscribe in this sphere a cylinder almost half its volume? The circumferences of the two bases of the cylinder have to lie on the surface of the ...
8
votes
2answers
486 views

Largest rectangle from 20 Lego bricks

You have twenty 2x4 Lego bricks, like the one shown below What is the area of the largest rectangle you can make satisfying the following conditions: All bricks must be connected in a single ...
9
votes
3answers
736 views

Lines and Squares

This 'puzzle' is from the New York Times website, from its puzzles. I decided it was fun and so I would share it with you. Question Here are 10 straight lines and 17 squares. Here are 9 straight ...
21
votes
7answers
4k views

Can you irrigate your lawn with 23 sprinklers?

You have a perfectly circular lawn with radius exactly 4 metres. Lately the grass has been turning yellow and quite rough, so you go to Stiv's Diabolical Instruments and describe your problem. "...
5
votes
2answers
399 views

My special animal

The answer to this puzzle is an image not hosted on Imgur. The image host and the ID of the image on that host should become clear when solving the puzzle. (SVG source of the above) Exact coordinates ...
7
votes
2answers
121 views

Vertices of a regular $13$-gon and $14$-gon on a circle with center angle $< 1°$

All vertices of a regular $13$-gon and all vertices of a regular $14$-gon lie on a circle and divide it into $27$ circular arcs. Is there always an arc, which corresponding center angle is less than $...
13
votes
3answers
981 views

A line not intersecting points in the plane

Is it possible to draw a finite or infinite set S of points in the plane, such that any line drawn in this plane neither intersects with exactly one point in S or an infinite number of points in S?
9
votes
1answer
269 views

Survive the infinite zombie attack II

Zombies are back again. Same as last time, You're at the origin, and zombies occupy the points $(100𝑖,100𝑗)$ for all integers $𝑖,𝑗$ except the origin, as shown below: The ratio of your speed to ...
13
votes
2answers
689 views

Paint Eight Squares

Inspired by this question Given a $5 \times 5$ grid of white squares, can you paint $8$ of the squares black so that each white square is orthogonally adjacent to exactly one black square?
1
vote
1answer
243 views

1 lion, with a zebra and a fixed enclosure

Background See the puzzle Variant of lion and 100 zebras from @ghosts-in-the-code which remains unsolved years after it was posted. Several times in the last couple of years I've started to write up ...
-1
votes
2answers
247 views

Filling up squares! ⬜ ⬛ [closed]

You have a white grid of size a) 9x9 b) 10x10 and you are asked to paint some squares black. Your generous friend has given you a challenge: Can you paint them so that each white square has only one ...
-2
votes
1answer
136 views

Trails on a grid filled with skinny tetrominoes

Let's have a 10x10 grid with 12 empty bases. The rest of the grid is filled with skinny tetrominoes. The 5 regular tetrominoes are marked with a red color and the 2 reflections are marked with a green ...
7
votes
6answers
2k views

Dividing a chocolate frosting cake [closed]

Mrs. Betty made a squared cake with chocolate frosting for his neighbors to the afternoon tea. However, first she sliced a middle piece for her two grandchildren and cut it in half: There was no ...
6
votes
1answer
561 views

Flat share share

This is a sequel to Gentrification in Chessshire. Due to the febrile state of the Chesster housing market you and your flat mates are forced to rent out half your place to another group of sharers. ...
2
votes
1answer
208 views

Rearranging the square

You are given a square piece of paper. You can cut it into pieces and rearrange them to form a new shape. You are allowed to rotate and flip pieces, provided that they are all used. Can you cut the ...
-3
votes
1answer
233 views

Open dice Problem

The following figure is folded to form a box. Choose from the alternatives (A), (B), (C) and (D) the boxes that is similar to the box formed. Source: YouTube Video How to answer these type of ...
2
votes
2answers
159 views

A bear of a different colour [duplicate]

Here is a stunning new version of the famous bear problem. IT IS NOT A DUPLICATE, OR LATERAL THINKING. MATHEMATICS REQUIRED. A photographer stepped out of their tent with a camera and walked: 1 km ...
10
votes
2answers
323 views

Gentrification in Chessshire

You can skip the back story and directly jump to the question. Capitalism has arrived in suburban Chesster the community most famed for The Game, and with a vengeance. Rents have trebled in less than ...
1
vote
3answers
173 views

Test of Pentominoes

These are pentominoes, with letter codes: Create 4 yes/no questions which uniquely classify each pentomino. Examples of such questions are: Does it have rotational symmetry? Does it have reflection ...
1
vote
1answer
183 views

Catch the fugitive

The fugitive is at the origin. He moves at a speed of 100. You have a guard at every integer coordinate except the origin. A guard's speed is 1. The fugitive and your guards move simultaneously and ...
10
votes
1answer
282 views

Wraps and Loops—Which Sequences are Admissible?

If you take a non-intersecting closed loop on a torus (that is to say, a path which ends where it starts and does not cross over itself, drawn inside a square whose edges "wrap" left to ...
7
votes
5answers
638 views

What a coincidence or eighteen is not seventeen take two

Since I botched my first attempt at posing this puzzle, let me try again, this time hopefully closing all loopholes by dropping the packing angle and making this a purely geometric puzzle: The figure ...
6
votes
2answers
2k views

Eighteen is not seventeen

This question is not the same as Adding coins inside a ring of coins From a pile of equal size perfectly round coins take eighteen and make a perfect ring. Show that you can fit at least sixteen more ...
9
votes
1answer
225 views

Connect the dots to form a polygon

a) In each of these two grids of dots, 5 x 7 and 7 x 9, connect all of them so as to form polygons of 35 and 63 sides respectively (two consecutive segments can therefore not be collinear as they ...
-1
votes
2answers
117 views

Tessellation with nonagons and equilateral triangles

What type of convex nonagon is required to tesselate a plane with equilateral triangles and nonagons? All sides of the nonagons are equal. NOTE: Partial tessellation of a plane should accompany your ...
5
votes
3answers
530 views

Another puzzle with area

Squares $ABCD, DCGH, BEFG$ and $ELKM$ are positioned as shown on the picture. Find the area of triangle $DGK$ if you know that the area of square $ABCD$ is $20$.
9
votes
1answer
166 views

Puzzle with the point inside paralelogram and area

We have a paralelogram $ABCD$ and point $P$ inside it. Halflines $BP$ and $DP$ cuts respectively lines $AD$ and $AB$ at $E$ and $F$. Why are the area of $ABPD$ and $CEPF$ the same regardles of the ...
10
votes
2answers
573 views

Adding coins inside a ring of coins

17 identical coins with diameter 1 are lying flat on a table, such that their midpoints build the vertices of a regular 17-gon (regular heptadecagon) and adjacent coins touch each other. What is the ...
3
votes
0answers
206 views

One vs many. Can white force a draw?

On an infinite chessboard there's a single white king and N black kings. The nearest black king must be K moves away from the white king. Given N, white dictates the value of (finite) K, then black ...
7
votes
2answers
303 views

Can the fugitive escape?

A fugitive is surrounded by N police officers, with the nearest one at distance 1 away. The fugitive and the officers move alternatively. In a fugitive move, the fugitive can travel no more than a ...
7
votes
1answer
219 views

The Extraordinary Sky of Saddlestania

The infinite country of Saddlestania has some very interesting geography: its elevation from the mathematically flat sea level exactly follows the equation $$\mathbf{z=x^2-y^2}.$$ After traveling ...
2
votes
1answer
97 views

“Physical” height of an infinitely distant joint point

When driving along long straight roads, this is what you would typically see: Photo by Luke Stackpoole on Unsplash As you're probably aware, the solid lines on the sides of the road are parallel, in ...
11
votes
3answers
361 views

Inhomogeneous circle packing

In the figure, what is the diameter of the smallest circle assuming the two parallel lines are one unit apart? Note: There is at least one elegant, geometric proof. Attribution: Mine, but wouldn't be ...
6
votes
2answers
792 views

A square covering a rectangle

You are given a rectangle with base b and height h with $h>b>0$. What is the minimum side length of a square, which completely covers this rectangle?
13
votes
1answer
427 views

Folding a piece of paper with numbers in sequence

Divide a rectangular sheet of paper with a side length of 2 × 4 into eight 1 x 1 unit squares and label them as shown in the sketch. Then fold the sheet of paper along the boundaries of the square so ...
8
votes
1answer
337 views

Manual tiling with 8 dodecadudes

Here are 8 dodecadudes. (Drawn from the very numerous dodecadrafters, made from 12 half equilateral triangles, dodecadudes are a subset of 770 pieces with sharp points and narrow necks excluded). ...
2
votes
1answer
122 views

Painting cells on the diagonals of a grid rectangle

In a grid rectangle 20210 × 1505, two diagonals are drawn, and all the cells containing segments of diagonals are painted. How many cells are painted?
12
votes
1answer
400 views

Strange tiling pattern

Here is a simple pentagonal shape: Using copies of this shape it seems that you can tile the plane, without even needing to flip over the tile. But can you really?
-6
votes
1answer
87 views

3-Sliced Square [closed]

I remember this one time when my brother told me "with just 3 lines, how many shapes can you form in a square piece of paper?" After 30 minutes, my 7-year-old brain thought that I could only ...
3
votes
1answer
197 views

Infinite beauty

This is a follow-up to Puzzle about 6 infinite cylinders in space Question: Given six identical infinite (no caps) cylinders is there a beautiful arrangement in space such that each touches each ...
4
votes
2answers
515 views

Cubes touching all other cubes

This question is based on this great puzzle: Puzzle about 6 infinite cylinders in space What is the most number of identical cubes that can be placed, such that every cube touches all the other cubes ...
4
votes
2answers
444 views

A geometric puzzle. What is the angle?

This is a simply stated geometry puzzle. What is the angle p in this isosceles triangle? Here's some information about the origin of the puzzle. Following any links therein may spoil the fun if you ...
14
votes
2answers
1k views

Puzzle about 6 infinite cylinders in space

You have 6 infinitely long cylinders (tubes) with the same radius R. Can you arrange them in space in a way that every cylinder touches the other 5? By touching, I mean have a common point or a line. ...
8
votes
5answers
1k views

A pentagon puzzle

Consider the following pattern made of regular pentagons: If the pattern continued, will it form a complete loop or will the pentagons overlap?
-1
votes
2answers
113 views

Surrounding an equilateral triangle

You are given an equilateral triangle. What is the most number of such identical triangles you can place such that they do not overlap, but each one touches the original triangle?

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