Questions tagged [graph-theory]

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

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4 votes
1 answer
164 views

Curious statements about black cells on a grid

Consider a finite rectangular grid consisting of unit squares (cells). Some cells are colored black, and the rest are white. Some definitions: Two black cells are neighbors if they share an edge. Two ...
14 votes
7 answers
3k views

Join six cities with roads

Warmup question: Each of five cities is connected to the others by four roads. Show that it is possible for the roads to intersect only once with exactly two roads crossing over at that single ...
18 votes
5 answers
1k views

Maximize the number of paths

You have exactly 990 edges. Assemble them into a simple undirected graph with two distinguished vertices A and B, such that the number of different simple paths from A to B is as large as you can make ...
6 votes
2 answers
337 views

While 2024 arrives

There are about $9.266 \times 10^{45}$ partitions of 2024, a handful! To each of these partitions corresponds a graph in which the vertices are each of the parts, two of which are joined by an edge if ...
12 votes
3 answers
1k views

Dissecting a square

You are asked to dissect an $N \times N$ square into polyomino pieces such that each piece shares a portion of its boundary with exactly $D$ other pieces, and no piece has area exceeding $N$. This can ...
5 votes
1 answer
204 views

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 ...
17 votes
1 answer
774 views

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?
14 votes
5 answers
3k views

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 ...
32 votes
2 answers
8k views

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 ...
12 votes
3 answers
2k views

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 ...
7 votes
5 answers
927 views

Every tournament has a dominant player

A tournament was played round-robin: each pair of players played a match where one defeated the other. Prove that there was a player for which every other player either lost to them or lost to someone ...
11 votes
1 answer
607 views

Generalisation of tours on chessboards

It's well-known that a knight placed on one square of a chessboard can get to any other square, but a bishop can only reach half the squares from a fixed starting point. Another question on this site ...
1 vote
1 answer
207 views

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 ...
7 votes
3 answers
1k views

Magic-preserving Permutations on a 4x4 Magic Square

Messing around with some magic-square puzzles, I faced the problem of deciding whether some two magical squares are, in fact, the one and same square wearing a different hat. It seemed to me, that for ...
35 votes
5 answers
3k views

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 ...
19 votes
1 answer
672 views

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 ...
-1 votes
1 answer
218 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 ...
29 votes
4 answers
3k views

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 ...
1 vote
4 answers
319 views

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: ...
2 votes
3 answers
237 views

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 "...
10 votes
2 answers
2k views

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 ...
4 votes
1 answer
240 views

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 ...
3 votes
2 answers
253 views

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 ...
-2 votes
1 answer
206 views

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 (...
8 votes
1 answer
348 views

Snowy, snowy night

Notice that each snowflake is composed of seven hexagons, and each hexagon has a word written clockwise around its perimeter.
5 votes
1 answer
262 views

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....
11 votes
1 answer
817 views

It comprises that which composes it

Text version: ...
14 votes
4 answers
2k views

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 ...
8 votes
1 answer
330 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
1 answer
263 views

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 ...
3 votes
1 answer
414 views

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 ...
27 votes
1 answer
2k views

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, ...
4 votes
2 answers
879 views

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: ...
18 votes
7 answers
1k views

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, ...
5 votes
2 answers
246 views

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....
3 votes
2 answers
2k views

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. ...
6 votes
4 answers
427 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-...
6 votes
2 answers
764 views

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 ...
3 votes
2 answers
240 views

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 ...
3 votes
4 answers
282 views

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 ...
12 votes
3 answers
2k views

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 votes
2 answers
333 views

What's the graph relation? #2

What's the relation that joins the nodes? Open the image in a new tab if you'd like to see the diagram with better resolution. Previous What's the graph relation? #1 Hint 1
7 votes
1 answer
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 ...
8 votes
1 answer
220 views

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 ...
9 votes
1 answer
313 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 ...
2 votes
1 answer
258 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
1 answer
432 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
1 answer
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
2 answers
969 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 ...
8 votes
2 answers
1k 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 ...

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