Questions tagged [graph-theory]

A puzzle built around graphs: sets of nodes joined together by paths. Use with [mathematics]

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2
votes
1answer
217 views

Prove the existence of a triangle such that all of its sides are of the same color [closed]

Seventeen points have been picked in a plane, and each pair of points has been connected by a line segment of one of three colors: red, yellow, or green. Prove that there are three points which are ...
3
votes
1answer
380 views

Genies' chess on a 10×10 board

The work of Hearth Taxel revealed some other results related to genies' chess. For example, there is an arrangement $A$ of pawns on a 10×10 board such that no 3×3 submatrix is empty and $A$ is ...
13
votes
1answer
1k views

Cracking the Cryptic Logo

The well-known Cracking The Cryptic YouTube channel has a logo consisting of 12 circles joined by 16 straight lines running horizontally, vertically or diagonally. What is the significance of this ...
4
votes
2answers
292 views

The Divisibility Graph... Again!

The divisibility graph of a set of positive integers is the graph whose vertices are the integers, two of which are joined by an edge if one divides the other. What is the smallest positive integer ...
7
votes
2answers
530 views

Divisibility Graph

The divisibility graph of a set of integers is the graph whose vertices are the integers, two of which are joined by an edge if one divides the other. What is the largest integer N such that the ...
3
votes
1answer
732 views

Most efficient way for people along the edges of a grid to move to the center

I'm considering a $2k\times 2k$ square grid ($k\in\mathbb Z^+$) with $8k$ highly rational people standing along the vertices forming the perimeter. All of these people want to go to the centre of the ...
2
votes
2answers
183 views

Powerful Octagon

Place different integers on the vertices of an octagon so that the sum of the integers in any two vertices joined by one of its edges is a power of 2. Do so in such a way that the largest integer used ...
7
votes
2answers
407 views

Another Rook's Tour of the Chessboard

Place numbers 1 to 64 in the cells of this 8 x 8 board in such a way that consecutive numbers occupy neighboring cells (either vertically or horizontally). Shaded cells must be occupied by prime ...
22
votes
3answers
1k views

Knight tour on a racetrack

Help the chess knight complete four clockwise laps on this racetrack, so that he lands on every square and never lands on the same square twice! The final square the knight lands on will be the same ...
-4
votes
1answer
140 views

Show that every finite directed acyclic graph has at least one source vertex [closed]

Easy puzzle courtesy of a paper I'm reading rn: Show that every finite directed acyclic graph has at least one source vertex. That is, a vertex such that all the directed edges incident to it are ...
7
votes
1answer
275 views

Existence of index-uniform Hashi puzzles

On the left, we have a starting configuration for a game of Hashi, and on the right, its solution: That is to say, the goal is to make connections (planar, and traveling only in cardinal directions) ...
2
votes
1answer
269 views

Trees from integers [closed]

A set of distinct positive integers is said to be a prime tree of integers if the graph obtained by letting the integers be its vertices, two of which are joined by an edge if (and only if) their sum ...
-2
votes
1answer
148 views

Trails on a grid filled with skinny tetrominoes

Let's have a 10x10 grid with 12 empty bases. The rest of the grid is filled with skinny tetrominoes. The 5 regular tetrominoes are marked with a red color and the 2 reflections are marked with a green ...
7
votes
2answers
1k views

Coloring positive integers 'black or white'

Each of the positive integers from 1 to n is colored either black or white. You can repeatedly choose a number m and recolor m together with those numbers, which are not coprime to m. At the beginning ...
2
votes
1answer
117 views

Fetching Alchemist, Excavation I

This is a puzzle in the Fetching Alchemist series. It has been generated especially for Puzzling Stack Exchange. Please note that, in my opinion, imperfect solutions should be up-voted so long as they ...
0
votes
1answer
74 views

Fetching Alchemist, Grand Potion I

This is a puzzle in the Fetching Alchemist series. There's no selling in this puzzle, just one potion to brew, but with a lot of ingredients. Please note that, in my opinion, imperfect solutions ...
2
votes
3answers
197 views

Advanced Fetching Alchemist II

This is a puzzle in the Fetching Alchemist series. From now on, you complete quests at the place you start at as well. Please note that, in my opinion, imperfect solutions should be up-voted so long ...
8
votes
3answers
340 views

Toroidal Pipes Puzzle: T's and Bulbs Only

A continuation in the Pipes puzzle series. Problem statement for math nerds: Let $G(N)$ denote the graph consisting of cardinally adjacently linked lattice points on an $N \times N$ toroidal grid. For ...
2
votes
1answer
103 views

Advanced Fetching Alchemist I

This follows the same rules as previous Fetching Alchemist puzzles, except you choose where you start, and you may now return to your starting place after leaving it. How to Play You are looking for ...
2
votes
1answer
87 views

Fetching Alchemist IV

This is the fourth puzzle in the Fetching Alchemist series, and is another puzzle that is exclusive to Puzzling SE until solved. This one might be a little too easy for those of you who have already ...
4
votes
1answer
125 views

Fetching Alchemist III

This is the third puzzle in the Fetching Alchemist series, and I am experimenting with a new format here. This time, I won't tell you in advance what the perfect score is. The first guess may be ...
3
votes
1answer
125 views

Fetching Alchemist II

This is a puzzle from the Expert section of my game Fetching Alchemist, visually modified for presentation here. It is a variant of the Travelling Salesman problem where you are trying to complete a ...
2
votes
1answer
164 views

Fetching Alchemist I

This is a puzzle from the Expert section of my game Fetching Alchemist, visually modified for presentation here. It is a variant of the Travelling Salesman problem where you are trying to complete a ...
0
votes
1answer
114 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 ...
3
votes
2answers
217 views

Pirates dividing booty around a circular table

A group of pirates have plundered one of his majesty's cargo ships and they all carried as much gold coins as each one could find. When they get back to their ship, they sit at a round table and pass ...
2
votes
0answers
72 views

Fair and square island hopping [duplicate]

If amateur fiction is not your thing skip to the bottom. As IP (Implausible Physics) expert for DREAM, the Department for Reckless Engineering and Advanced Megalomania you have been tasked by sheikh ...
1
vote
2answers
247 views

Minimize the longest King chain on a 6x6 ternary grid

This puzzle is an extension of this one: Minimize the longest King chain on a 5x5 binary board Given a grid filled with numbers, we define a King chain to be a path on the grid such that: The path ...
0
votes
1answer
139 views

Minimize the longest King chain on a 7x7 binary grid

This puzzle is an extension of this one: Minimize the longest King chain on a 5x5 binary board Given a grid filled with numbers, we define a King chain to be a path on the grid such that: The path ...
13
votes
7answers
870 views

Minimize the longest King chain on a 5x5 binary board

Given a grid filled with numbers, let's define a King chain to be a path on the grid such that the path can be traversed with chess King's moves (moving to one of 8 adjacent cells at a time), the ...
0
votes
1answer
105 views

Gaby's 21 students sitting around a circle [closed]

Gaby numbers her 21 students with the primes between 11 and 97. She now asks them to sit around a circle making sure that any two of them sitting next to each other have either their tens or units ...
3
votes
1answer
164 views

My sixteen graph theory students

I will have sixteen students in my graph theory course this semester. In our first session I asked each of them with which of the other 15 students in the class they were already acquainted before the ...
3
votes
0answers
210 views

Infected Cylinder and Torus

A variant of the well known Infected Checkerboard problem. If we've a 𝑛x𝑛 square, then we fold it along top and bottom row to form a cylinder. A cell in this cylinder becomes infected if at least ...
1
vote
1answer
95 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)
4
votes
1answer
172 views

Visiting streets, not houses

The section points are houses and lines are streets, all with one unit length. What is the fewest number of units you must travel to visit every street at least once?
22
votes
1answer
614 views

New Year Graph Puzzle

In the graph below, each node is coloured either red or yellow, except for the white node in the bottom left, which I've marked with an X. Can you tell me what the white node marked with X represents?...
-1
votes
2answers
86 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 ...
14
votes
1answer
432 views

Currency connections

Two Forex traders are trying to communicate about their trades. They send each other images with a hidden meaning like this one: Can you work out which currencies they are trading? Hint:
8
votes
2answers
868 views

Frog game on a dandelion graph

There is some noise in the local pond. A group of frogs wants to host a birthday party! There is a total of 22 lily pads in the pond, each housing a single frog. They are labelled as numbers from 0 to ...
9
votes
2answers
504 views

Dragon summoning spell

The parchment shown below got stained with something. See if you can determine the obscured parts.
7
votes
1answer
194 views

Four color a map - but go light on the fourth color!

Here's a map, which I found here: Your challenge is to four-color this map while minimizing your use of the fourth color. More specifically, color the map with four colors so each region is a ...
3
votes
1answer
91 views

How many different ways can you color the diagram?

This question is a continuation of this one. Here are nine squares, connected by lines. Each square must be colored, and two squares connected by a line must be colored differently. Question 1. What ...
12
votes
1answer
374 views

Infection (Information Dissemination) Puzzle

There are 2020 people in a room. One person has COVID. After each minute, each person $\mathrm{P}$ is paired with some other person $\mathrm{Q}$ who was never paired with $\mathrm{P}$ before, and they ...
2
votes
0answers
480 views

A robot moving on a grid. Part 2

This is an extension of the discussion A robot is placed on a grid point. At each move the robot must take three steps along the edge of the grid. After each step the robot must turn right. Lengths of ...
2
votes
1answer
276 views

A robot moving on a grid

In the spirit the question I propose the puzzle: A robot is placed on a vertex of a grid. At each move the robot must take three steps along the edge of the grid. After each step the robot must turn ...
7
votes
2answers
381 views

Seventeen positive integers

Find 17 positive integers such that no four of them have, pairwise, a common divisor greater than 1, but, likewise, no four of them are, pairwise, relatively prime. Do so in such a way that the ...
5
votes
1answer
102 views

The sandpiles on the beach

We call a convex polyhedron a sandpile if one of the faces, called base, has a common edge with all the other faces. Furthermore, each vertex is linked with three edges. We consider the sandpiles by ...
1
vote
2answers
175 views

LGBT+ SpeedDating

Congratulations! You are now the lucky franchise owner of 'SpeedDating Inc'. We are the number one organiser of speed-dating events in WhereverYouAre. We would love for you to start organizing new ...
7
votes
2answers
226 views

Phone pattern security

My phone is unlocked using a security pattern. This is a path drawn through a 3x3 grid of dots with the following rules: The path can start at any dot The path visits neighbouring dots: horizontally,...
8
votes
2answers
190 views

Intersecting shapes on a flat surface

What is the maximum number of enclosed regions that you can create by drawing two circles and two triangles on a flat surface? Try answering with mathematical arguments.
11
votes
4answers
829 views

Draw this planar graph

The playing board for this puzzle is a short list of digits; there is one rule governing possible moves: The 1 may swap with a digit one place to the right; the <...