Questions tagged [graph-theory]
A puzzle built around graphs: sets of nodes joined together by paths. Use with [mathematics]
225
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Puzzling Pelican Pebbles
Story Setup
It's Percy the Pelican's first day running the front desk of his master's magical pebble shop. His job is to fetch pebbles from the stock room to fulfill customer orders.
The stock room ...
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Poetry of the stars
Prof. Levenshtein is meant to be teaching us astronomy, but they fancy themselves as a poet. Can you figure out the name they've given to each star?
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Are there always 2 teams such that they have together defeated every other team
In a tournament without draws, every two of the nine teams play against each other exactly once. Must there always be two teams such that every other team has lost to either or both of them?
From the ...
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An immortal ant on a gridded, beveled cube divided into 3458 regions
This puzzle takes place on the surface of the following gridded, beveled cube:
The surface of this cube is divided into 3458 small regions separated by black lines. Of these regions, 3450 of them are ...
- 5,580
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votes
2
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Is this Hashi puzzle unsolvable?
I found this puzzle in my dad's bathroom. The book is "Things To Do While You Poo On The Loo". I think the puzzle is malformed. Here is a Penpa link to an online solver. Is there actually an ...
- 504
1
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How many ways are there to mark a way to walk around every edge of the triforce?
A triforce for the purposes of this question is a plane figure with an equilateral triangle at its center, with one additional vertex connected to each pair of original vertices (forming an additional ...
- 179
19
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World Tour of Planet Rhombicosidodecahedria
This is the planet Rhombicosidodecahedria:
This lovely planet has 62 countries, each with its own distinct history and culture. By an amazing coincidence, the countries all happen to coincide ...
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votes
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Rolling cube on an infinite chessboard
Imagine a six-sided die, D6, the right size to exactly occupy a square on a chessboard.
The die can move to any adjacent square, but does so by rolling rather than sliding, so the topmost side of the ...
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Coordinating trains? [closed]
In the picture below, each node represents a train station. On each node there is a train.
Two trains can change the location / node they are in, if they are connected by an arrow.
The puzzle is this:
...
35
votes
5
answers
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A Queen and her Pawns
Place a queen and as many pawns as possible on a chessboard so that the queen has just one way of capturing all the pawns in precisely as many moves as there are pawns. Pawns do not move and do not ...
- 16.9k
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3
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Detecting Connected Components on an Infinite Graph after Modification
This puzzle was inspired by thinking about how to implement a system like Factorio's power grid.
Start with an infinite connected undirected acyclic graph.
Graph = A set of nodes (called "...
- 2,434
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Can you eat a 4-dimensional Rubik's Cube?
If you follow the traditional Rubik's cube eating conventions,
Start by eating any piece except the central one
Next, eat a piece orthogonally adjacent to the previously eaten piece
(repeat)
The last ...
- 76k
4
votes
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Non-separable sudoku numbers
The standard rules for sudoku say that you have a 9×9 grid and need to put in every digit from 1 to 9 in a way that each digit occurs exactly once in each row, column and 3×3 box.
So the grid can be ...
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Directed Graphs from numbers
Bob and Alice play a game. Bob sends a sequence of positive numbers to Alice and using that information she forms a directed graph.
For each number in the sequence, she splits it into two non-empty ...
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What is the solution to this room maze? I still don't know!
There are 3 rooms (two on top, one on bottow- like an upside down pyramid) with 12 doors, each room with 5 doors (three doors are shared between multiple rooms). You have to make one solid line (...
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Snowy, snowy night
Notice that each snowflake is composed of seven hexagons, and each hexagon has a word written clockwise around its perimeter.
- 13.5k
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Who wins this game of graphs?
Albert and Bob are playing a game. This time it works like this: there are n points and Albert can ask whether or not $2$ points are connected. Bob then decides whether or not the points are connected....
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Labyrinth of Teleporters
You find yourself in an empty room, with a few distinctly numbered elevated platforms on the floor; your only possession is a pebble that can easily be picked up and placed down. You step on one of ...
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Tetromino tilings of a 4x5 rectangle with minimal diversity
This is a relatively easy manual tiling puzzle. In fact the tiling is all done for you, you just have to specify how many of each of the 26 given tilings to use. The puzzle is:
Using N complete sets ...
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3
votes
1
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Largest word tree
I was inspired by this awesome puzzle. Here is an image of a word tree borrowed from there:
In a word tree every path from the root to the leaves must form a distinct word. The size of the tree is ...
- 33.2k
27
votes
1
answer
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Two mystical trees
Behold, the mystical Tree of B:
Notice that every path from trunk to canopy forms a word. You should see 16 words (from left to right: BLOOM, BLOOD, BLOWN, BLOWS, BLAND, BLANK, BLASÉ, BLAST, BROOK, ...
- 13.5k
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D&D dice for literary people
Put a letter on each face of an icosahedron such that a five-letter word can be read clockwise around each vertex. Specifically, these words:
...
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Efficient Mowing at PSE
Your task:
Find the most efficient mowing path around the dark green bushes that mows (passes over) all of the grass (light green).
For those who cannot view the image above, there are 9 rows of 16, ...
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Robot painting a $K_5$
A robot starts at a node of a fully connected graph of 5 nodes (shown below). Each turn the robot can move across an edge and paint it in one of two colours - blue for odd turns and red for even turns....
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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. ...
- 33.2k
6
votes
4
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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-...
- 5,963
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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 ...
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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 ...
- 1,699
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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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12
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3
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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 ...
7
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1
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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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8
votes
1
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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 ...
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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 ...
- 6,996
2
votes
1
answer
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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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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 ...
- 6,996
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1
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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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4
votes
2
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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 ...
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votes
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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 ...
- 16.9k
3
votes
1
answer
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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
2
votes
2
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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 ...
- 16.9k
7
votes
2
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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 ...
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22
votes
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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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votes
1
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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 ...
- 2,818
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votes
1
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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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2
votes
1
answer
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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 ...
- 16.9k
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votes
1
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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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2
votes
1
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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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