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. General geometry questions are considered off-topic but can be asked on Mathematics Stack Exchange with the geometry tags.

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1answer
111 views

Heptagon, nonagon

What is the trick to constructing a heptagon and a nonagon which have all their sides equal? The length of the side has to be a natural number. Your answer should include a drawing of the two polygons....
8
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2answers
256 views

Cutting a shape into two equal area shapes

Given the following shape - an hexagon ABCDEF of which a parallelogram CDGH is cut out. With a single cut divide the shape into two equal area shapes by means of an unmarked . You may draw lines and ...
3
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2answers
77 views

A triangle inside a triangle

All sides of a triangle T1 are shorter than the shortest side of a triangle T2. Is it always possible to put triangle T1 completely inside triangle T2?
3
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2answers
435 views

How to fill 1/3 of the cylinder?

You are given 3 containers - pictured in this order below: a box with side 2, height 1, and a cone with base radius 1 and height 1 in the middle. a box with side 2, height 1, and a half sphere with ...
7
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4answers
579 views

Divided by Pie Squared. Aaahhh

I have a machine that can divide a square pie into 9 equal square pieces using 4 blades: The blades can be moved, but there is only one control - which defines the width of the blades in both ...
2
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2answers
287 views

Four touching circles

Three identical circles are placed such that they touch each other. A larger circle is drawn around the smaller circles such that it touches them, as shown in the diagram. Can you find the ratio ...
20
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3answers
1k views

An ant's walk in the Cartesian Plane

An ant lives in the origin of the Cartesian Plane. Every morning, at 6 am, it sets out on a 16-hour walk which gets her back home precisely at 10 pm. In the first hour the ant walks exactly one unit, ...
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3answers
151 views

Maximum number of triangles formed in a pentagon with equal area

All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and 10 triangles. Find the maximum number of triangles with the same area that may exist in the division. The best I ...
0
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1answer
75 views

Fill big box with smaller boxes

Let's say we have a big box with inner edges with the lengths 2m, 1.5m, 1.4m. Can we fill this with smaller boxes with the edge lengths of 3dm, 5dm and 1m, without any gaps?
0
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0answers
69 views

Pinocchio and the board [duplicate]

Pinocchio has driven two nails into the board to secure them, so that if one of them falls, the other nail will prevent him from falling.(Maybe he wanted to stand on the board)The fox claims that he ...
1
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1answer
80 views

Painting a plane!

Paint the points on a plane with three colors, so that the points on each line are a maximum of 2 colors, and all three colors are used. (Math Festival 1990)
7
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3answers
459 views

Put line segment the way they cover end points

Is it possible to place 1000 line segments on the page so that the two ends of each line segment are on the inner points of other line segments? (By the inner point of the line segment, we mean a ...
0
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2answers
124 views

How do you find the perimeter of a set of odd looking squares and triangles?

The problem is as follows: The alternatives given in my book are: 76 cm 80 cm 92 cm 100 cm Upon the first inspection. I'm getting the idea that I have to make a system of equations. Assuming that ...
11
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1answer
691 views

How many circles needed to pass through each of 5x5 lattice points?

You are given a 5x5 set of lattice points. What is the minimum number of circles, which pass through each of the 25 points at least once?
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1answer
101 views

Splitting figures on the Cartesian plane

What is the minimum number of lines to separate the sets? a) 2 b) 3 c) 4 d) 1 e) 5 Observe the graph below, it is possible to separate linearly with a line at least two of the classes? a) Yes, just ...
5
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2answers
204 views

A square in the plane with 4 vertices of the same color

Every point in the plane is colored either red or blue. Is it necessarily the case (i.e., is it true for all such colorings) that there exist some four points of the same color that are the vertices ...
2
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0answers
103 views

Does anybody know the history of this star/triangle puzzle?

Does anybody know the history of the puzzle posted by Mario Bilotti? Who created this puzzle? Who deserves the credit? Find 10 triangles in a five pointed star using two straight lines
39
votes
6answers
2k views

What fraction of the larger semicircle is filled?

What fraction of the larger semicircle is filled? The two smaller semicircles are of equal size. This is a puzzle originally set by Catriona Agg, who is a puzzle setting genius.
0
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2answers
71 views

Show that no lines need cross [closed]

There are n red points and n blue points in the plane. Show that you can always join all the red and blue points with straight lines so that no two lines cross. Each point can have exactly one line ...
4
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1answer
218 views

Bisecting a 3D object into two equal volume objects

Given the following object - box of which a rectangular pyramid is removed. By means of unmarked ruler, draw lines on the surface of the object to guide cuts of the object into two objects with the ...
1
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1answer
160 views

Assemble lozenges

You have a large number of 60° rhombi called "lozenges." Each lozenge has its edges marked with four distinct symbols drawn from an infinite alphabet. Lozenges may be rotated by 180° or ...
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1answer
235 views

How many shapes can you form with squares? [closed]

There is a 6 by 6 dot-grid. You will draw two squares by joining the dots. The squares cannot have common dots/points or areas. Rotations or reflections of a drawing are considered distinct. In How ...
9
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3answers
358 views

A theorem about angles in the form of arctan(1/n)

There is a famous classical geometry puzzle about the angles formed by integer coordinates: What is the sum of angle A and B in the following image? Do not use any advanced mathematics such as ...
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1answer
98 views

Construct a Triangle with Triangles

If you have a very large but finite number of equilateral triangles (of any/all sizes) is it then possible to construct an isosceles triangle? By isosceles I mean a non-equilateral isosceles triangle. ...
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1answer
135 views

Self-intersecting polygonal chains in a hexagon [closed]

This is continuation of this Q&A. Given a regular hexagon with center at point O: Question: How many self-intersecting polygonal chains are there that connect 7 points? The self-intersecting ...
0
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6answers
289 views

Move 3 matches to maximize triangles

The problem is as follows: The figure from below shows 9 matches. If only 3 of them are changed from their positions, then what is the most number of triangles that can be made? The alternatives ...
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1answer
95 views

Four equilateral triangles' areas sum to a fith

Let's have five equilateral triangles. From these five triangles one has an area equal to the sum of the other four triangles. All triangles have heights which are rational numbers. Can you find five ...
2
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2answers
108 views

Form three squares of equal area with 8 sticks

This puzzle comes from a Chinese puzzle book. This is the translation of the puzzle: There are 4 sticks 10 cm long and 4 sticks 5 cm long. How to form three squares of equal area with these sticks? ...
10
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1answer
351 views

Equilateral triangle inscribed in a square

A square is drawn on a piece of paper. How can you draw an equilateral triangle such that its vertices lie on the boundary of this square?
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2answers
295 views

How many triangles can you obtain using the 6 vertices and center of a regular hexagon?

Let's say there is a regular hexagon with center at point O. Question 1. How many triangles can you obtain using the 6 vertices and its center? Question 2. What is the largest number of different ...
6
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1answer
159 views

What is the name or author of this tangrams-like packing puzzle?

Does someone knows the name of this packing puzzle, or its inventor? The goal is to repack the shapes into the square with the additional red square. It is an easy puzzle but the construction is ...
3
votes
1answer
152 views

Aluminum Foil Folds and Cut

Your task is to convert a diamond shape monomino that is made from aluminum foil into an x-shape pentomino (see figures). You may fold the monomino and make one straight cut with a pair of scissors. ...
2
votes
2answers
121 views

What’s the big idea, 32.5 does not equal 31.5 [duplicate]

Here’s a visual proof for why 32.5=31.5 Here’s the animation This puzzle came directly from Proofs without words on Mathoverflow. Your Goal: Explain what’s wrong with the proof and where the “missing ...
2
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1answer
101 views

Closing an irrational curve

For any rational number q, a finite number of congruent circular arcs each measuring 2πq radians can be assembled into a continuous (possibly self-intersecting) closed curve. There are many other ...
7
votes
1answer
225 views

Can you fold a paper, make one hole and produce a bear's template?

You are given a piece of paper with no marks on it. With this paper (A4 or A5 size), you have to make a bear's template. On the template the sell size is 5 mm, hole is not in scale. You are given no ...
8
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3answers
928 views

Drawing a complete graph of 5 nodes on a torus

A complete graph of $n$ nodes is a graph where every node is connected to every other node. It is known that one cannot draw a complete graph of 5 nodes on a piece of paper (plane) without any ...
8
votes
2answers
409 views

Can a square of size 1000.25 fit a million and one unit squares?

A square with a side length of exactly 1000 can obviously be packed with exactly a million unit squares. If we increase the side length to 1001, then 2001 more squares can fit. But if we increase the ...
1
vote
1answer
103 views

Square forming challenge

Cut a square into 3 pieces Rearrange them anyway you want Cuts must be straight (but can begin/end anywhere). Scoring method 1: For each visible square 1 point For each cut -1 point Scoring method 2: ...
3
votes
4answers
1k views

A rectangle, a circle, and a triangle are drawn on a plane

A rectangle, a circle and a triangle are drawn on a plane. What is the maximum possible number of points of intersection? The sides of the triangle are not collinear with any of the sides of the ...
5
votes
3answers
560 views

Billiard table: how many sides touched?

On a billiard table of 1 by 2 meters lies a billiard (snooker) ball, with a diameter of 52.5 mm, separate from the sides. It is pushed away without effect and then it rolls 3 meters. How many times ...
7
votes
19answers
2k views

Drawing a perfect circle without any tools [closed]

You are given a pen and a large piece of paper. Without using any additional tools (eg., ruler, compass, string etc) how can you draw a perfect circle?
10
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5answers
1k views

Can you fold a square into a square of one-third the area

I do not love origami, but Mitsuko gave me an idea for a extremely hard and (not that?) beautiful puzzle. I'm really curious whether anyone here can solve it. So here's the puzzle. You are given a ...
9
votes
1answer
222 views

Hollow Cube Cuts

The 2x2x2 inches seamless hollow cube with aluminum surfaces can be cut using a box knife. How to cut it into 4 pieces that can be bend to form smaller 1 inch cubes?
3
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3answers
953 views

Most intersections with Olympic rings

The Olympic symbol has 5 rings that intersect at 8 points: What is the most number of intersection points can you achieve by moving the rings?
6
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4answers
1k views

A robot making increasing steps

A robot starts on a cell in an infinite grid. On the first turn it can move 1 cell horizontally or vertically. On the $n$-th turn ($n>1$) it can move $n$ cells horizontally or vertically, but it ...
19
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1answer
1k views

Ernie and the Lock-down Puzzle

During lock-down I was feeling a bit lost for something to do, so one one day I sent Ernie a text reading "Bored". He responded with a text-less message and an attached image (see below). I ...
2
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3answers
263 views

The treasure on a tropical island

My great-great-great-grandpa left this note for my family when he passed away, but no one dared try it out. On the tropical island located at 90.888°N, 123.456°E, there's a gallows where we used to ...
5
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2answers
311 views

General orchard planting problem for circles

My previous puzzle asked for the maximum number of 4-point circles attainable from a configuration of $n=10$ points drawn on a plane. I am now interested in generalizations of this puzzle to arbitrary ...
4
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1answer
135 views

Orchard planting problem for squares

The classic Orchard planting problem asks for the maximum number of 3-point straight lines attainable from a configuration of $n$ points drawn on a plane. Here we are interested in a variant of this ...
6
votes
3answers
451 views

Orchard planting problem for circles

The classic Orchard planting problem asks for the maximum number of 3-point straight lines attainable from a configuration of $n$ points drawn on a plane. Here we are interested in a variant of this ...

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