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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19 votes
4 answers
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How long can you survive at the devil's playground?

The devil has trapped you in his playground. The devil knows that you can't cross over the burning boundary of his circle, so he allows you to choose a position within the circle before he starts to ...
Eric's user avatar
  • 6,498
-1 votes
2 answers
206 views

Rectangles and squares of trominoes filling a grid

Let's have a board 24 squares by 24 squares. This board is to be filled with trominoes of three different colors. There are equal numbers of trominoes of each color. The board is to be filled with ...
Vassilis Parassidis's user avatar
7 votes
1 answer
840 views

Cutting off one's nose to spite one's eyes

Disclaimer: to keep graphic depiction of gratuitous violence to a minimum the face to be spited has been deliberately kept abstract. You are required to further reduce any distress this puzzle may ...
loopy walt's user avatar
  • 21.2k
118 votes
3 answers
12k views

Prove that π > 3

It seems that once upon a time some politicians tried to pass a law fixing the value of π to be exactly 3. The idea being to "make math simpler so that our children can get better at math". ...
Florian F's user avatar
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19 votes
3 answers
1k views

Reunite the Stars

On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of 1 in their ...
Eric's user avatar
  • 6,498
8 votes
1 answer
200 views

Going off on multiple tangents

Given two disjoint circles of diameters D1 and D2 draw all shared tangents. Find the intersection points of all pairs of tangents. Discard those collinear with the two circle centre points. Of the ...
loopy walt's user avatar
  • 21.2k
8 votes
2 answers
575 views

Triangles to diamonds

Given a triangle ABC with sides a=|BC|,b=|CA|,c=|AB| a diamond is circumscribed around the triangle's incircle. The diamond and the triangle share the corner C along with (part of) sides a and b. ...
Albert.Lang's user avatar
  • 5,237
11 votes
4 answers
848 views

Optimal Path between two concentric circle arcs

When travelling along a outer arc between A and B you have two choices, either diverting onto the inner circular arc or carrying on the outer circular arc, as shown below: You start on the outer arc, ...
PuzzlingFerret's user avatar
-7 votes
1 answer
197 views

The three door puzzle [closed]

In a long room are three doors. Behind each door one block is hanging from the ceiling. Behind the first door the block is made of concrete; behind the second door the block is made of hardwood; ...
Vassilis Parassidis's user avatar
12 votes
2 answers
1k 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?
Kikos's user avatar
  • 338
12 votes
0 answers
1k 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 ...
Jerry Dean's user avatar
  • 1,746
11 votes
1 answer
961 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 ...
Dmitry Kamenetsky's user avatar
21 votes
1 answer
2k 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 ...
Dmitry Kamenetsky's user avatar
15 votes
3 answers
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.
Dmitry Kamenetsky's user avatar
-7 votes
2 answers
207 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 ...
Vassilis Parassidis's user avatar
8 votes
2 answers
597 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 ...
Dmitry Kamenetsky's user avatar
11 votes
3 answers
2k 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 ...
Stevo's user avatar
  • 2,613
22 votes
7 answers
5k 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. "...
Parcly Taxel's user avatar
  • 7,678
5 votes
2 answers
469 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 ...
Parcly Taxel's user avatar
  • 7,678
6 votes
2 answers
136 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 $...
ThomasL's user avatar
  • 12.2k
13 votes
3 answers
1k 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?
ThomasL's user avatar
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8 votes
1 answer
319 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 ...
Eric's user avatar
  • 6,498
13 votes
2 answers
772 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?
hexomino's user avatar
  • 136k
2 votes
1 answer
409 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 ...
Steve's user avatar
  • 3,875
-2 votes
2 answers
272 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 ...
Arale's user avatar
  • 135
-2 votes
1 answer
154 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 ...
Vassilis Parassidis's user avatar
7 votes
6 answers
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 ...
Pspl's user avatar
  • 2,263
6 votes
1 answer
606 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. ...
loopy walt's user avatar
  • 21.2k
1 vote
1 answer
334 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 ...
Dmitry Kamenetsky's user avatar
-4 votes
1 answer
625 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 ...
Sourabh's user avatar
  • 15
2 votes
2 answers
185 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 ...
Laska's user avatar
  • 1,919
10 votes
2 answers
393 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 ...
loopy walt's user avatar
  • 21.2k
1 vote
3 answers
327 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 ...
IsaacRoan Sison's user avatar
1 vote
1 answer
234 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 ...
Eric's user avatar
  • 6,498
10 votes
1 answer
326 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 ...
Feryll's user avatar
  • 2,379
8 votes
5 answers
696 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 ...
loopy walt's user avatar
  • 21.2k
6 votes
2 answers
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 ...
loopy walt's user avatar
  • 21.2k
10 votes
1 answer
1k 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 ...
Bernardo Recamán Santos's user avatar
-1 votes
2 answers
198 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 ...
Vassilis Parassidis's user avatar
5 votes
3 answers
546 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 $20$.
nonuser's user avatar
  • 880
9 votes
1 answer
176 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 ...
nonuser's user avatar
  • 880
10 votes
2 answers
632 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 ...
ThomasL's user avatar
  • 12.2k
7 votes
2 answers
518 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 ...
Eric's user avatar
  • 6,498
7 votes
2 answers
376 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 ...
Eric's user avatar
  • 6,498
9 votes
1 answer
355 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 ...
Bass's user avatar
  • 77.4k
2 votes
1 answer
128 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 ...
iBug's user avatar
  • 3,166
12 votes
3 answers
451 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 ...
loopy walt's user avatar
  • 21.2k
7 votes
2 answers
872 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?
ThomasL's user avatar
  • 12.2k
13 votes
1 answer
1k 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 ...
ThomasL's user avatar
  • 12.2k
8 votes
1 answer
496 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). ...
theonetruepath's user avatar

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