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Questions tagged [graph-theory]

A puzzle built around graphs: sets of nodes joined together by paths.

9
votes
1answer
1k views

How many colors does it take?

This question is from a popular monthly science magazine in my country: You have an 8x8 square where any 3 squares forming a tromino (including reflections and rotations) must consist of three ...
4
votes
2answers
105 views

Identify this type of graph puzzle

There are $V-1$ pieces, each with an identifying symbol. The board is a graph with $V$ vertices and some number of edges $E$. The idea is to move around the pieces so that each piece's symbol matches ...
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votes
1answer
109 views

A Colorful Honeycomb

What strategy can you use to color using only 6 colors the lines in an infinite hexagonal tiling such that no two sides of the same hexagon have the same color?
11
votes
6answers
693 views

I Have To Be With Them!

It's almost time for another year of school! But before school starts, Principal Little needs to form classes. Because there are so many people in a class, the parents are always complaining, asking ...
3
votes
1answer
201 views

Create a map of a game's portals

Given a set of rooms, each with a N, a S, an E, and a W exit/entrance to another of the rooms, create as simple a map as possible that graphically represents their connections. The rooms in question ...
-4
votes
1answer
343 views

A party of jealous guys

I was really happy for the fact that I won the inter-galactic best magician award. So I decided to throw a party of $n$ people (excluding me). The people who came to that party was jealous, really ...
22
votes
1answer
715 views

A Tour Around a Triangle

Place the 18 even integers between 2 and 36 in the empty nodes of this triangular graph in such a way that if a path is drawn by coloring in red all the edges joining any two nodes whose numbers add ...
17
votes
1answer
471 views

A partition of 1000 into nine parts

The sum of nine whole numbers is 1000. If those numbers are placed on the vertices of this graph, two of them will be joined by an edge if and only if they have a common divisor greater than 1 (i.e. ...
14
votes
3answers
364 views

Fearful asymmetry

An asymmetric graph (or identity graph) has every vertex unique: no different relabeling of the vertices leaves the edges unchanged. The trivial graph on one vertex is (trivially) asymmetric. All ...
11
votes
1answer
445 views

Any hope for Humpty Dumpty?

It was inevitable, really... Each fragment of shell has exactly three sharp points, joined by smooth curves. While the King's horses can count reasonably well, his men have been known to confuse ...
4
votes
2answers
238 views

How many nodes in the network?

I don't actually have a solution in mind for these, but it seemed puzzly enough to bring to the table. Seems as though someone must have come up with this before, but if so, I couldn't find it when I ...
5
votes
3answers
306 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 ...
7
votes
2answers
195 views

A partition of 1000 into six parts with least and greatest product possible

Find six positive natural numbers, not necessarily distinct, whose sum is 1000 and which, if placed appropriately on the vertices of the following graph, two of them will be joined by an edge if and ...
16
votes
3answers
2k views

A partition of 100 into nine parts

The sum of $9$ positive natural numbers, not necessarily distinct, is $100$. If placed appropriately on the vertices of the following graph, two of them will be joined by an edge if and only if they ...
22
votes
7answers
5k views

Hacking an electronic keypad

You are a spy trying to break into an enemy facility. The back door is protected by an electronic keypad lock. You know that this particular lock is opened by a four digit code. Any stream of button ...
1
vote
1answer
266 views

Scheduling Meetings

I came across this problem in real life and thought it could be made into an interesting puzzle. I will enjoy seeing how my eventual solution could be improved! Here's the situation. There ...
5
votes
2answers
175 views

Triangle of Safety

Saitama: "The Hero Association called me for a low-level mission, can you meet them as my representative?" Genos: "No." Saitama: "Aww, man.. That's not fun." Then Saitama decided to meet Hero ...
7
votes
1answer
185 views

Trip Routes that Visit 9 of 10 Cities

There are 10 cities on this island. For each pair of cities, they may have a bidirectional path. A trip route is defined as a route which start on a city e.g. $A$, goes to 8 of 9 other cities exactly ...
4
votes
1answer
307 views

A certain partition of 130

Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than ...
6
votes
2answers
216 views

Teacup geometry

Inspired by the three utilities puzzle from prog_SAHIL I'm now posting a similar puzzle that makes use of the topology of a cup with a handle: The question is: How many distinct points can you ...
5
votes
1answer
160 views

A minor rearrangement of the one sided hexominoes in 12 simultaneous shapes

Here are the one sided hexominoes arranged into 12 congruent shapes. But there are one or two flaws: The dark blue hexominoes, which are the symmetric ones, may not occur more than once each in a ...
14
votes
2answers
334 views

Hexominoes into 7 simultaneous congruent shapes

I came up with this puzzle 16 years ago, it was on Ed Pegg's Mathpuzzle site but nobody solved it AFAIK. The 35 hexominoes (which look like this): are to be arranged, in groups of five, into seven ...
12
votes
5answers
1k views

Soccer balls in the stadium

The coach asks you take as many soccer balls as possible and put those balls onto the field with the condition that For any arbitrary set of three balls, at least two of those balls are exactly 10 ...
12
votes
1answer
285 views

Multibranched tree

The Furca Fractalis tree grows in a very special way. Starting with the trunk there are three possibilities to continue growing: It can split in two branches. It can grow one branch and one leaf. ...
12
votes
1answer
293 views

Jigsaw Logic: ?s galore

I am working on a 256 piece jigsaw puzzle, but I am having a lot of trouble. Instead of the picture being a landscape or painting, the final image is just a sixteen by sixteen grid of identical ...
3
votes
1answer
398 views

8 Train Stations

You are going to build $8$ train stations and the railroads with it in an area. But you are asked to build these stations and their railroads in a very efficient way where there has to be the least ...
13
votes
1answer
1k views

Hunter chasing a fox on a graph

This is a variant of the sleeping princess puzzle. There are fifteen foxholes, connected by underground tunnels as shown below. A fox is sleeping in one of them. Every day, a hunter checks one of ...
8
votes
2answers
2k views

Color this map using only 4 colors (easy)

According to graph theory, in order to color any map so that 2 touching regions don't have the same color, 4 distinct colors are enough. Can somebody color the following map?
9
votes
2answers
361 views

Looking for another partition of 100

The sum of ten, not necessarily different, positive integers is 100. If placed adequately on the vertices of this graph, two of them will be joined by a line if, and only if, they have a common ...
6
votes
2answers
1k views

Is there a simple algorithm for solving Kami 2 puzzles?

I'm finding my life has been consumed by Kami 2, partly because I seem to have achieved some "insight" and am able to solve the puzzles reliably, almost always on the first try. The rules are simple: ...
8
votes
2answers
1k views

A mystery partition of 100

Given a multiset (a set with repeated elements allowed) of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if ...
10
votes
1answer
473 views

Labelling a graph with a partition of 100

Label the vertices of this graph with positive integers (repetitions allowed) whose sum is 100 in such a way that any pair of vertices are joined by an edge if (and only if) they have labels with a ...
2
votes
3answers
359 views

Find the least number of Dragons required

The following graph represents the positions at Castle Dragonstone. Each edge indicates that the positions are within sight of each other. This is not transitive; i.e., you can't see all the way along ...
2
votes
2answers
164 views

Find a tour through the exhibition [duplicate]

Is there a way to take a tour through the exhibition that passes through each door exactly once? Source: chegg.com
18
votes
1answer
395 views

Interconnectivity

As I was walking through my university's mathematics department, I came across a poster pinned to the notice board. Underneath, it simply said 'In memory'. However, I could not connect the poster to ...
8
votes
1answer
387 views

A perfect metro map

You are working for a company and asked to create a perfect metro map where there will be as many stops as possible. But there are two constraints which limits the number of tracks (railroads) and ...
11
votes
5answers
728 views

Can the idiot's route be less expensive than the genius' route?

In a certain country, there are $n$ cities. Between every pair of cities, there is a fixed travel cost to go from one city to the other. An idiot and a genius both decide to tour this country by ...
20
votes
2answers
722 views

Honeydripping around the clock

      What path could a honeybee follow, beginning and ending at top center, visiting every empty cell exactly once and dripping 2 drops of honey into the last cell? Start ...
6
votes
2answers
140 views

A problem about oriented face in Square grid

Consider a $n \times n$ square grid (finite) (a square is divided into smaller squares by lines parallel to its sides). The boundary of the square is oriented, (clockwise or anticlockwise) that is, a ...
4
votes
2answers
181 views

Four mutually adjacent US states (not the Four Corners) [closed]

The question Is there a proof that a map of the United States requires 4 colors? was answered by showing that Nevada has five neighbors, each adjacent to the next (an "odd wheel"). "Adjacent" here ...
21
votes
2answers
3k views

Help the prisoners

Given a 5×5×5 cube of identical cubical cells (total 125 cells). In each cell there is a prisoner. There are doors from each cell to the adjacent cells (not diagonally). Their task is to ...
16
votes
5answers
748 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 ...
9
votes
4answers
432 views

Mystery on the trail to Dutch Flat

On a recent amble, I happened upon a sign $\small \raise2mu ( {\normalsize\sf\color{#d90} A} \raise2mu )$ that seemed to indicate a mistake — or— a mystery. ...
2
votes
4answers
224 views

Create a special Playing Schedule - Logical/Mathematical Solution

Last Week in Training (I'm a Cycleball player) a logial problem/puzzle tricked us. And I'm wondering if there exists a logical solution for the next time. Cycleball is played in pairs (2 Players vs. ...
8
votes
2answers
758 views

Eight queens on the chessboard with mirrors

A 8-by-8 chessboard has two mirrors added on its left and right margin. The mirrors reflect the queen moves so that a queen threatens additional squares on the board. A queen threatens all squares in ...
5
votes
3answers
406 views

Crossing Frog Lake

Our Red Frog wants to get to the Orange Frog, but he can only jump right and down, but over multiple lilies if he chooses, although he can't stay still. How many ways can he do it?
14
votes
5answers
515 views

To each his own

You are a graduate student in theoretical mathematics, dabbling every so often in some interesting but equally useless computer science theory. From the beginning of the year, Professor Carl Hayden ...
6
votes
3answers
269 views

Professor Halfbrain and the tennis club

The other day, I met with professor Halfbrain and professor Erasmus in the coffee house. Professor Erasmus told us that he had been working on a schedule for a tennis club with $30$ senior and $30$ ...
41
votes
1answer
3k views

Crippled King Crossing a Canyon

A chess king has been injured in battle against an evil wizard, and can no longer move northeast or southwest. This king is on the North rim of a canyon, and must flee to safety on the South rim. ...
16
votes
2answers
2k views

Professor Halfbrain and the fantasy knight

Professor Halfbrain owns a $99\times99$ board for fantasy chess, whose rows are numbered consecutively from $1$ to $99$ and whose columns are also numbered consecutively from $1$ to $99$. A fantasy ...