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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39 views

Dissecting the square

This question is strongly related to this one: Rearranging the square You are given a square piece of paper. You can cut it into pieces and rearrange them to form new shapes. You are allowed to rotate ...
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62 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. ...
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1answer
164 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 ...
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1answer
144 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 ...
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2answers
148 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 ...
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2answers
228 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 ...
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3answers
149 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 ...
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1answer
154 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 ...
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1answer
267 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 ...
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5answers
619 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 ...
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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 ...
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1answer
205 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 ...
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2answers
107 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 ...
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3answers
522 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 $20$.
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1answer
157 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 ...
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2answers
559 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 ...
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0answers
174 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 ...
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2answers
282 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
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1answer
210 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 ...
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1answer
89 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 ...
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3answers
346 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 ...
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2answers
778 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?
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1answer
412 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 ...
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1answer
312 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). ...
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1answer
119 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?
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1answer
390 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?
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1answer
84 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 ...
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1answer
188 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 ...
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2answers
502 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 ...
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2answers
433 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 ...
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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. ...
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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?
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2answers
107 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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2answers
341 views

Pumping triangles

On the inside of the triangle (0,0),(1,0),(0,1), define the "pretty useless map" or pump by the following prescription: Given a point A find its projections x0,y0 to the x and y axes and ...
2
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2answers
99 views

Surrounding an L-shaped tromino

You are given an L-shaped tromino, shown below. Can you surround it by 10 more L-shaped trominoes? The new trominoes can be rotated and must touch the original tromino at an edge or a corner. This ...
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6answers
2k views

5 points on a ball, divide the ball into 2 halves so that one half as exactly 4 points

Suppose that you take a pen and mark five points on a ball. I claim that no matter where you draw those points, I can always slice the ball into two equal halves (two equal and closed hemispheres) ...
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1answer
88 views

Trail passing through squares of a grid

Let's construct a 10x10 grid. 0n the 100 squares you are allowed to place 7 bases (the red dots in the diagram below) in any square on the grid. Then you fill the grid with skinny trominoes. The ...
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1answer
77 views

How to arrange a set of cubes to get a tower of two?

The puzzle is as follows: Assume that you have this peculiar toy. This toy is composed by many plastic pieces which is shown in the figure from below, all of them are cubes. You can use as many as ...
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1answer
91 views

Number of turns a coin makes going around some other coins

The puzzle is as follows: Assume that you have three coins over a rectangular table as it is indicated in the picture from below. These three coins are identical in diameter which is 3 cm and tangent ...
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3answers
313 views

Riding on non-circular wheels

The mathematician Stan Wagon built a tricycle with square wheels that can ride smoothly on a carefully crafted curved surface: Now I have some puzzles for you: Is there a surface that allows one to ...
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3answers
422 views

A circle touches two sides of a triangle and two of its medians

A circle touches two sides of a triangle and two of its medians. Prove that the triangle is isosceles. This problem came from the Mathematical Digest issue 62 (Jan 1986) which in turn cited a Russian ...
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2answers
162 views

How many L-shaped pieces will not be used to make cubes?

The puzzle is as follows: The figure from below belongs to a didactical toy which is comprised of 32 congruent wood pieces as indicated in the figure. Each piece is made up by three cubes whose edges ...
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1answer
138 views

What is the perimeter of a pentomino which can tile this heart-shaped board?

The puzzle is as follows: The figure from below shows a board that is made up of 30 squares whose sides are of 2 centimeters each. Such board must be split in six congruent pieces. These must be made ...
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1answer
101 views

Filling a 26x36 grid with trominoes

Let's have 312 L-trominoes of three different colors, 104 trominoes of each color. Can you fill a 26x36 grid with these trominoes without two of the same colors touching side-to-side anywhere? In ...
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1answer
95 views

How many figures to cover the kitchen floor?

The puzzle is as follows: Figure 1.1 shows a kitchen floor that is made up of 30 squares whose sides measure 1 cm in length and in Figure 1.2 a tile that is made up of 5 squares whose sides also ...
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1answer
71 views

How many figures are needed to cover all the star?

The puzzle is as follows: Figure 1.1 and Figure 1.2 shown are made up of congruent triangles. Assuming you must place tiles congruent to figure 1.2 to cover all the triangles in figure 1.1, without ...
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1answer
116 views

A Clear, Simple, Geometry Problem [closed]

Draw a shape consisting of all the points equidistant from a specific point. Furthermore, draw a segment passing through a side of the shape exactly twice, and draw another segment so that it also ...
0
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2answers
177 views

Nested six-point stars: least number of cuts to dissemble

The puzzle is as follows: The figure from below represents a peculiar structure which consists in congruent triangles whose sides intersect and is made of an iron wire. How many cuts passing through ...
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2answers
259 views

Least cuts to get 44 rods from a metal grid

The puzzle is as follows: Suppose that you have a metal structure made by brass wire. Assuming that you must get 44 rods of the same size each. What is the least cuts to be made using an electric ...
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1answer
124 views

Minimum cuts to make a rectangle into a square, allowing bending

The puzzle is as follows: Mike has a thin sheet of cardboard which is 96 centimeters large by 24 centimeters wide and a guillotine whose maximum cut length is 80 cm. Assuming this guillotine can cut ...

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