# 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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### 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 ...
200 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 ...
91 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: ...
857 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 ...
105 views

### Is there any triangles named with three letters? [closed]

My name is Gabbie and I have a question today. Can any triangles be named with three letters?
545 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 ...
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?
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 ...
216 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?
937 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?
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 ...
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 ...
259 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 ...
306 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 ...
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 ...
443 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 ...
98 views

### Making *9* congruent triangles from the pieces of a triangle dissection

Working on the making 7 congruent triangles from the pieces of a triangle dissection question I realized it's possible to do even better! So here it is for extra points: Use six lines to cut a ...
290 views

### A chessboard tiling with corners removed in 3D

A famous problem asks whether an 8x8 chessboard with two opposite corners deleted can be tiled with dominoes, where a domino is a rectangle congruent to two adjacent squares of the board. Now, let C ...
391 views

### Thick as two short planks

That didn't go to plan. You just wanted to help your friend the artist redecorate. In the process you mananged to make an ugly notch in their favorite table, scratch their wall when moving said table ...
1k views

### How to find half volume of tetrahedron?

Let's say you have a few juice packs that are shaped as regular tetrahedra. Question. Is it possible to measure half of the juice there is in one pack? Edit. You do not have any measuring tools (rules,...
456 views

### Disarray and Organization: a virtual mosaic puzzle

I created this mosaic, which references a country I like to visit in my spare time. What country is that?
205 views

### Escape the Plane

One Sunday morning, you awake to find yourself completely alone on an infinite, flat plane. You don't remember much about the night before, other than that you may have pissed off a wizard. Next to ...
176 views

### Making 7 congruent triangles from the pieces of a triangle dissection

I got this challenging geometrical conundrum from a Russian geometrical magazine. It states: (A. Soifer) Use six lines to cut a triangle into parts such that it is possible to compose seven ...
346 views

### Sharing a field among 4 sons

A wealthy famer has a large estate in the shape of an irregular squarish octogon. In the middle he has a rectangular retention basin for storing water. He is getting old and discusses with his wife ...
288 views

### No algebra please, we are geometers

Given a right triangle with sides $ABC$ make two more right triangles using sides $A$ and $C$ (long side) and a new long side $x$ (same for both new triangles). By Pythagoras the implied third sides ...
6k views

### Can you fold a square into a square of one-fifth the area?

I love origami, and it recently gave me an idea for a very hard but beautiful puzzle. I'm really curious whether anyone here can solve it. So here's the puzzle. You are given a large perfectly square ...
159 views

### What is an opposite face to the letter $T$?

Starting with 27 small cubes connected to each other in a 3x3x3 cube shape, I removed some of the cubes so that all the remaining cubes stayed solidly connected by cube faces. I then used the ...
767 views

### How many right triangles are there with these conditions?

How many right triangles are there with the following conditions: the sides $a$, $b$, and $c$ have an integer length (Pythagorean triplets) the amounts of area and perimeter are the same for each ...
298 views

### Right triangles with polygons

I drew all the regular polygons in a circle of radius one. I decided to take one side of the equilateral triangle, one side of the square and one side of the regular hexagon to form a right triangle. ...
157 views

### Shuffling Magnets

For an experiment, I have to place magnets onto a rectangular board whose dimensions are 3 and 3.5 inches. Between each phase of the experiment, I have to shuffle these magnets around, but if two ...
262 views

### Find the area of the given triangle

Another mathematical puzzle: Find the area of $\triangle FGH$, given that $FG=FH$ and the radii of the circles shown are $2$ and $1$
3k views

### Two chunky pixelated X's locked in mortal combat!

In this dramatic image, we witness two rather chunky pixelated letter X's (having recently fattened themselves up for the approaching winter) locked in mortal combat, fighting to the death for the ...
453 views

### Pretty average --

You are given a tetrahedron $T=ABCD$. Average opposing edges to create a second tetrahedron $T'=A'B'C'D'$ with $\overline{A'B'}=\overline{C'D'}=\frac 1 2[\overline{AB}+\overline{CD}]$ etc. Place $T$ ...
213 views

### Find the radius of the incircle

Here's a small mathematical puzzle I came up with recently: Find the radius of the larger incircle, given that the radius of the smaller incircle is $3-\sqrt{3}$. The hexagon is a regular hexagon, ...
1k views

### Wrap a squashed, bullet-riddled lowercase lambda around a cube

The following rather squashed and bullet-riddled lowercase lambda: ...can be wrapped onto the surface of a cube in a way that perfectly covers the entire cube, with no gaps and no overlaps. How can ...
732 views

### A packing game!

Amy and Ben are playing a game which is suggested by a genie. Amy first chooses $a,b,c\in\mathbb{R}^+$. Then a empty cuboid box with internal measurements $a+b,b+c,c+a$, and infinite supply of cuboid ...
1k views

### Cover a square with three smaller squares

A square has a side length of 5 units. Is it possible to cover this square with three squares each with a side length of 4 units?
485 views

### Dominoroto-toto

Consider a domino tiling of a plane rectangle of size $n \times m$. (Obviously, at least one of $m$ and $n$ has to be even for that to be possible.) I personally hate those because they tend to look ...
1k views

### Most triangles formed by three triangles

What is the maximum number of triangles you can form by drawing three triangles on a piece of paper? Good luck!
86 views

### Form nine squares from three squares

Can you draw three squares on a piece of paper, such that they form nine distinct squares? Good luck!
88 views

### Minimal 2D pattern for domino tiles where each tile touches three others

Inspired by The five problems of the six domino tiles, where one of the tasks was to place six domino tiles so that each tile touches three others (corner / edge touches don't count). My solution ...
118 views

### How to find the number of ways to go from one point to another in a truncated structure?

I've found this problem in my book "Riddles and reason" and after several attempts I still have no idea how to tackle it. The problem is as follows: The figure from below shows a truncated ...
852 views

### The five problems of the six domino tiles

Here is a set a problems (regarding domino tiles) of a famous Portuguese newspaper weekly magazine. For each problem you have $6$ domino tiles and the goal is always to place them touching each other ...
145 views

### 4 Kids vs easter bunny

There is a game played between the easter bunny VERSUS a team of 4 kids. I will fully explain the rules of the game below. However, I'd like to start with the preface. I found this problem as a king ...
294 views

### Cut the square cloth!

I have a square cloth with side length $x$ cm, and I am going to cut it into at least $n$ squares with side length $1$ cm for my customer, and also you cannot cut the cloth to thinner pieces (reminded ...
122 views

### Concerning Tetrahedra

As a pyramid with a triangular base, the volume of a tetrahedron, like all pyramids, is $(1/3)*BH$, where $B$ is the base area and $H$ is the height. If one had $3$ square $45$ degree pyramids (square ...
295 views

### Touching triangles at their vertices

What is the minimum number of non overlapping congruent triangles arranged in the plane, such that each vertex of the triangles coincide with exactly three triangles?
97 views

### An angle between diagonals

The diagonals of a square intersect at a right angle. Is that true in three dimension, i.e. have the two diagonals of a cube, each running from one corner of the cube to its opposite corner and ...