# Questions tagged [graph-theory]

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

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### The longest path of edges on a 3x3 grid

A robot is placed on some vertex of a 3x3 grid. At each move the robot can take one step (up, down, left or right) along the edge of the grid to the adjacent vertex, but it cannot go outside the grid. ...
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277 views

### Cover an n times n grid with non-diagonal non-intersecting n-1 shortest paths

This puzzle was given to me by PhD student colleagues. Suppose that you have a $n\times n$ grid. Is it possible, for a given $n$ to cover all its $n^2$ nodes with $n-1$ non-diagonal and non-...
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### Splitting the integers 1 to 36

Split the integers 1 to 36 into two sets, A and B, such that any number in set A has a common divisor greater than 1 with no more than two other numbers in A, but for every number in B there are at ...
138 views

### How to arrange the colored cells in game grid?

Puzzle: In a game grid some cells are missing. Each line has only one colored cell with a label (a number greater than zero). This is an example grid and the number of columns/rows can be less than ...
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### Planar Investigator

Use logical deduction to place a different digit from 1 to 9 in each circle below so that 8 of the arrows form the primes 23, 31, 41, 53, 59, 79, 89, and 97. (We view an arrow starting at digit A and ...
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### Travel in the USA

If you decide to travel from state to state in US in alphabetical order how many states can you cover if: The state you are in must share a border with the previous state. The last state in your list ...
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### Longest infinite loop of 5 states

This is based on a question I posed in The Nineteenth Byte: What group of 5 states have the longest total name, under the constraint that you must be able to travel from one state to another in the ...
1k views

### Can you stop the falling piano with 23 nets?

MIT's Baker House has a tradition of dropping an irrepairable piano six floors every Drop Day, the last day one can drop a class without penalty (the 2022 date is 19 April). This year, in order to ...
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### A Knight's Tour

A lonely chess knight stands on a cell somewhere in the first row of a 3x13 board, and elsewhere there is a castle. The knight takes a tour of all the remaining 37 cells of the board, missing just the ...
286 views

### Be Paired or Be Square

8 white and 8 black dots are drawn on a piece of paper. Parcly and Tori take turns drawing edges, always between white and black dots not already adjacent (so the graph is always bipartite); the first ...
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244 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 ...
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403 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 ...
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### 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 ...
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### 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 ...
540 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 ...
758 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 ...
• 309
188 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 ...
416 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 ...
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 ...
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### 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 ...
• 1,465
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### 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) ...
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### 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 ...
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 ...
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### 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 ...
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### 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 ...
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76 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 ...
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198 views

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 ...
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362 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 ...
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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 ...
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### 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 ...
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126 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 ...
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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 ...
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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 ...
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119 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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### 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 ...
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73 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 ...
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1 vote
255 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 ...
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148 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 ...
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884 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 ...
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### 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 ...
167 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 ...
215 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
98 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)
177 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?
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### 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?...
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### 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 ...
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### 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:
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### 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 ...
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527 views

### Dragon summoning spell

The parchment shown below got stained with something. See if you can determine the obscured parts.
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