# 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.

794 questions
Filter by
Sorted by
Tagged with
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 ...
278 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 ...
324 views

### Ernie and the Chocolate Bomb

Last week when I visited Ernie he complained that his trousers had shrunk in the wash. I suggested an alternate possibility – he was putting on weight. So he decided to go on a diet and to support him,...
423 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 ...
257 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 ...
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 ...
121 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 ...
1k views

### Minimum cells to fill grid without consecutive neighbours

Imagine you have a m x n grid which is initially colored white. you can fill in a cell with black color if and only if there are no immediately neighboring black cells (no black cells to the left/...
389 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 ...
131 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 ...
95 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 ...
286 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 ...
281 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 ...
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,...
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 ...
444 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?
339 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 ...
284 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 ...
199 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 ...
173 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 ...
295 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. ...
154 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 ...
759 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 ...
255 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 $1$
154 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 ...
479 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 ...
445 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 $T$ ...
63k views

### Cutting a cake into 8 pieces

Say, you are given a cake which you must share with 7 others. So, you must cut the cake into 8 pieces. But, you are only allowed to make 3 straight cuts. You cannot move pieces of the cake after the ...
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 ...
206 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, ...
728 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?
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 ...
293 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?
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!
115 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 ...
83 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!
86 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 ...
493 views

### Balance the nails

Someone I know handed me this puzzle, I have seen a couple of solutions for it that follow the instructions and don’t involve bending the nails, etc. Can you figure out how to balance the 6 nails on ...
847 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 ...
762 views

### Pythagorean triplets wheat field

A rectangular field has width $a$ and length $a+1$. We cut it into 3 triangles that all have integer side lengths. If all triangles have a different area, then what’s the minimum value of $a$? Please ...
618 views

### What are the fewest weights you need to balance any weight from a triangular seesaw?

I saw this question: What's the fewest weights you need to balance any weight from 1 to 40 pounds? Suppose you want to create a set of weights so that any object with an integer weight from 1 to ...
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 ...
555 views

### Laser Beams in Helsinki Skies

(The first two chapters are just for flavour, you can safely skip to the TL;DR near the end.) This puzzle in situated Helsinki for a reason: for the city's "200 years as capital" celebration, the ...
2k views

### Ten miles south, east, north and west

I'm standing on the surface of the Earth. I walk ten miles south, ten miles east, ten miles north and ten miles west. I end up exactly where I started. Where on earth am I?
156 views

### Geometric logos 2

Following the idea of Geometric logos. Which famous company/software logos include the following geometric shapes (listed alphabetically by company)? Three congruent parallelograms forming a hexagon ...
2k views

### Size of a square in a square

Given a unit square (blue in the picture), pick a point on one edge and label it A. Label the distance from the nearest corner to A as x. Pick one of the corners opposite A and label it B. Call the ...