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.

Filter by
Sorted by
Tagged with
16
votes
2answers
860 views

The Square's Center

The warden of the local prison was in a good mood - after a long winter, it was finally spring. The weather was warm and the sun shined over the prison. The warden decided, as is customary in the ...
8
votes
2answers
919 views

Overlapping Gang Territories

There are $7$ gangs in (a greatly simplified version of) Los Angeles. Each of them claims half of Los Angeles as territory. Let's say a place is "hot" if a majority of gangs claim it. How small can ...
2
votes
2answers
227 views

Is there a better solution to parallel segments problem?

There is a puzzle: Find a finite set of points with the following property: 1. Points are placed in space (not on a single plane) 2. If you take any pair of points A and B there are different ...
10
votes
3answers
1k views

Do a barrel roll! (i.e. a Euclidean plane rotation puzzle)

One of my favorite Putnam problems due to a slick solution. $R$ is at $(3, 4)$ on the cartesian plane. To try to confuse $R$, the devious $S$ decides to rotate $R$ about the point $(1, 0)$ by $36^\...
-2
votes
6answers
1k views

Minimum number of steps required to cut this bar

What is the minimum number of steps required to cut this chocolate bar which is a single piece into at least 30 single pieces? 1)Image for chocolate bar 2)Image for single piece ...
2
votes
1answer
281 views

Skylark of Valeron [closed]

In E.E. 'Doc' Smith's classic Skylark of Valeron, Dick Seaton and party are beset in space by disembodied intelligences. These intelligences working as they do on the 6th order of forces are able to ...
7
votes
3answers
2k views

Partition a cube into 6 congruent tetrahedra

A tetrahedron is a solid shape with four triangular faces, not necessarily regular or identical. Show how to partition a solid cube into 6 tetrahedra that are congruent, meaning identical up to ...
11
votes
1answer
561 views

Ants on a Hula Hoop

There are $24$ ants scattered around a hula hoop of circumference $3$ meters . They each randomly, and independently, choose to face clockwise or counterclockwise, and then simultaneously start ...
-3
votes
2answers
136 views

Points following an axiom [closed]

Your aim: Mark any finite number of points on a plane. It should meet this axiom. Axiom: Original way of stating it: If a line (infinite) is drawn passing through exactly n (n>0) points, any line ...
12
votes
4answers
808 views

Pursuit Problem II: Surrounded in Marauders' Circular Cove

(This is the sequel to this puzzle. It has a similar setup, but believe me, the solution is very different. Be careful! The answer is counterintuitive. It shocked me at first.) You are a pirate. ...
18
votes
6answers
2k views

Pursuit Problem: Mutineer trapped on an island

You are a pirate. One of your crew ran off in the night, swam to your secret island, and dug up your life's treasure. You need to capture her. You may deploy $n$ pirate ships to patrol the boundary ...
9
votes
5answers
663 views

Wolves and a Hare on a tiny planet

On a tiny spherical planet there exist $N$ wolves and 1 hare. The planet is so small any of these creatures can circle it in exactly 1 day. No creature needs to sleep or eat. The wolves communicate ...
15
votes
4answers
10k views

Drawing something using one pen stroke

Can you determine if it's possible to draw a geometric figure (made up from shapes like rectangles, triangles, and other regular shapes) with one pen stroke and not drawing the same line twice. I am ...
15
votes
2answers
575 views

What is the Universal Translator censoring?

Lieutenant, launch the polar probes! Launching probes, sir. Failure, sir. Probes did not land at the poles. Well!? Where are they? Unknown, sir. Somewhere off-axis. They did land on ...
6
votes
4answers
2k views

Cutting from a cube (visualization test)

I had asked this question on Facebook. Imagine a cube. Put it on a flat surface, so that four vertices are at the bottom and four vertices are on top. Select any vertex on top, and connect it to ...
3
votes
2answers
433 views

Gold and Silver Cubes

An aged King, after losing his only son in battle, determines to divide his kingdom amongst his wisest advisors. He presents two solid cubes, one made of gold, the other made of silver, and a wooden ...
4
votes
3answers
693 views

Changing perspective

Where am I? If I'm stuck in the middle, but can look around as much as I'd like; I can see 120, 24 times. These 24 are actually arranged into groups of 3 across 8 complete imaginary ●'s of view. ...
7
votes
4answers
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/...
18
votes
9answers
3k views

Tiling a Hexagon with Diamonds

A regular hexagon is divided into a triangular grid, and completely tiled with diamonds (two triangles glued together). Diamonds can be placed in one of three orientations. Prove that, no matter how ...
53
votes
8answers
5k views

Find a straight tunnel

There is a circular area with radius 1 km. And there is a tunnel, which is just under the surface, but invisible - unless you dig. It is known that the tunnel goes under the area (at least touches it ...
5
votes
1answer
346 views

Deceptive Dissection

Divide the below figure into $5$ equal pieces (same shape, same size, possibly reflected). $\qquad\quad\qquad$ I believe this is one of Martin Gardner's, but I could not find the source.
1
vote
1answer
171 views

What are the weights of the bangles?

John the jeweler is planning to make bangles from solid 14 karat gold. As shown below, his design will: be available in 5 different diameters (D) have a height (h) of 1 inch for all bangles have an ...
3
votes
3answers
1k views

The duck and the wolf [duplicate]

This is quite a famous one, I think, but hopefully not too famous. The phrasing is loosely taken from here. A duck finds itself alone in the centre of a round pond. On the edge of the pond stands a ...
8
votes
3answers
551 views

Periodic sequences confuse all my worms

You live on an infinite plane with a house at each integer lattice point. The plane is inhabited by horrible worms who can only walk in straight lines. You want to paint the houses so that all the ...
14
votes
3answers
1k views

Ants playing tag!

$5$ ants are initially arranged on the corners of a regular pentagonal table (edges are long $L$), as showed in this picture: At some point they all start moving with a constant speed $S$ and, at any ...
8
votes
4answers
278 views

Rectangles with a whole-number measurement

A rectangle is proper if its width is a whole number, or its height is, or both. Show that if a rectangle can be cut into a finite number of proper rectangles, then that rectangle is itself proper. ...
15
votes
4answers
544 views

Darkness in Euclidea

Darkness has descended on the infinite plane of Euclidea. The only sources of light are $4$ lighthouses; they were built long ago, and cannot be moved. The bulbs of these lighthouses can be rotated ...
2
votes
8answers
191 views

Pass the Baton competition 6 members teams

The following diagram depicts a Pass the Baton tournament (Note: due to a merge, some of the answers below use this image instead.) Each team consists of 6 members, 2 at the central point $E$, and ...
7
votes
2answers
1k views

Clash of arrows

During the last country fair, you have seen a weird arrow-shooter machine. Its shape is a regular polygon with $N$ sides of length $1$ and it fires one bolt from each vertex at the same time. The ...
7
votes
1answer
786 views

A purposefully obtuse Euclidean geometry riddle from an old interactive fiction game

I've been playing Praser 5, an old interactive fiction game (like Zork) made a few decades ago, and available here. In it, you wander from area to area solving riddles posed by mythical animals. Many ...
15
votes
1answer
1k views

Which Way did the bicycle go?

I found this puzzle which simply asks which way was the bicycle going, from its wheel marks on the mud shown in the image Now there is a simple way to find which mark(line) was made by the front ...
9
votes
4answers
2k views

How many different non congruent polygons can you make on a 3x3 dot grid?

There is a $3\times3$ dot grid. How many different non-congruent polygons can you make on the grid? Rules: All vertices of the polygon must be on the grid Only non self intersecting polygons Only ...
2
votes
1answer
224 views

Klotski Puzzle 3

Another Klotski Puzzle, "The Great D-vide": Rules Here Klotski Puzzle 1 Klotski Puzzle 2
6
votes
1answer
193 views

Klotski Puzzle 2

Another Klotski puzzle! (rules here) Klotski Puzzle 1
12
votes
1answer
914 views

Klotski Puzzle 1

Klotski is a sliding-blocks puzzle game very similar in nature to rush-hour/unblock me. In a given puzzle, a certain number of blocks labeled Z, A, B, C, .... are given. The goal is to move the Z ...
6
votes
2answers
499 views

Block the snake from reaching points

Solution for Version 2 pending.... There is a $100\times 100$ grid. The upper left corner has coordinates $(1,1)$ and bottom right corner has $(100,100)$. A 'snake' starts by occupying a single cell ...
10
votes
3answers
2k views

Precision Tag - can the lion win?

This is a spin-off motivated by Lopsy's interesting variant of Gamow's lion and zebras puzzle. It arose from a line of enquiry that tried to extend the vertical run past 5000km (or characterise the ...
8
votes
3answers
816 views

The Erasmus dissection of a square

Professor Erasmus claims that he is able to cut a square into 100 rectangles by making nine horizontal cuts and nine vertical cuts (parallel to the sides of the square), so that exactly 9 of the ...
6
votes
1answer
273 views

Two potatoes and a loop of wire

You are given two potatoes. You want to make a finite loop of wire so that you can put it on either of them at some location at your choice, so that there are no gaps between the potato and the wire. ...
35
votes
3answers
3k views

Which 3D shape can you make out of this?

The above shape can be folded into a closed 3D shape using no more than 14 distinct folds, with no parts overlapping. What is special about the shape that results? Rules and clarifications: Every ...
6
votes
2answers
828 views

Geocaching Geometry Puzzle

I am looking for help with a math puzzle. The answer is the final coordinates to a geocache. I have put in quite a bit of time and lots of graph paper trying to solve this one. I have tried programs ...
4
votes
1answer
702 views

Points on a cube

Professor Halfbrain has spent his entire weekend by placing colored dots on the surface of a huge wooden cube. His objective was to find large groups of dots that form the vertices of a regular ...
7
votes
5answers
1k views

Polyomino Z pentomino and rectangle packing into rectangle

See my similar question about T hexomino (Polyomino T hexomino and rectangle packing into rectangle) This is exactly same but with other polyomino - Z pentomino. Let's pack some (one or more) Z ...
4
votes
1answer
2k views

Find the angle (hardest easy geometry) [closed]

This is a question which is related to the hardest easy questions. Note that the general solution belongs on math.se and is not solved in simple way. This question is a puzzle and you need to prove ...
1
vote
1answer
803 views

Four similar triangles

The challenge as described hereafter is to create a total of 4 similar triangles by drawing 4 triangle in a scalene, acute triangle - out of the 5 resulting triangles (4 that make the original one) ...
32
votes
10answers
3k views

Variant of lion and 100 zebras

Note: This problem remains unsolved, as of 4 Feb 2018, so do try it out This a variation of this question by @Gamow Suppose there are $100$ lions and $100$ zebras. The lions function together as a ...
8
votes
2answers
980 views

Triangle area equals quadrilateral area

Here is a diagram and challenge description that should be clear and simple to understand.
-4
votes
1answer
146 views

Polygons circumscribed by lines [closed]

You have a plenty of segments of the same length at your disposal. Put a segment and put another to meet at their ends. Set the counterclockwise angle between two segments at 180/5 degrees. If you ...
17
votes
3answers
1k views

Pythagorean walk

It's a hot day. You are in the middle of a flat sandy plane that stretches as far as you can see. Your hands are shackled and you are surrounded by soldiers of the mighty Pythagorean Brotherhood. Ten ...
4
votes
2answers
206 views

A stroll in the park

Professor Erasmus has returned from his saturday walk in the park. He has counted the number of trees in the park and also the number of lines formed by these trees. Professor Erasmus claims that ...