**82 graph-theory questions.**

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 ...

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 ...

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 ...

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. ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

I saw the following problem on 4chan and couldn't solve it:
It's very likely to be some kind of troll (no solution).
I'm hoping to see some rigorous proofs that disprove the existence of such a line....

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

Initially, each of 50 Puzzling Stack Exchange users have a single distinct juicy bit of gossip not known to the others.
If $A$ sends an email to $B$, that email can include all the bits of gossip $...

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 ...

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.
...

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 ...

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 ...

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 ...

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?

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: ...

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 ...

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 ...

Can you determine if it's possible to draw a geometric figure (made up from shapes like rectangles, triangles, and other regular shapes) with one pen stroke and not drawing the same line twice.
I am ...

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 ...

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.
...

Is there a way to take a tour through the exhibition that passes through each door exactly once?
Source: chegg.com

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 ...

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 ...

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 ...

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 ...

I'm posting this from my phone again because I've gotten hopelessly lost in Boston, really want to go home, and desperately need the help of some clever people on the internet. Please help me - I'm ...

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 ...

The Four color theorem states that no more than four colors are required for any map. Can it be proved or disproved that 3 colors can be used for United States map?

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 ...

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 ...

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 ...

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 ...

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. ...

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 ...

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?

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 $30$ ...

- mathematics
- combinatorics
- geometry
- number-theory
- logical-deduction
- strategy
- optimization
- story
- chess
- calculation-puzzle
- no-computers
- checkerboard
- arithmetic
- computer-puzzle
- geography
- mazes
- algorithm
- polyomino
- enigmatic-puzzle
- tiling
- real
- matches
- topology
- network
- cipher

- Is a pig-mounted cavalry possible?
- Is this news article from 1912, essentially explaining climate change, real?
- The core of Voldemort's wand came from Fawkes, so how could he successfully use a Killing Curse against Fawkes?
- What do MAD and SAD mean?
- What is "roots and hacks"?
- Distance between two stations
- Should I tilt my DIY self-standing bookshelf backwards to prevent it falling down?
- Number Spiral Problem
- Why does there seem to be a lack of conservative comedy and comedy-news compared to liberal?
- Does there exist a right triangle with area 7 and perimeter 12?
- Why didnâ€™t the Spacecraft used for the Apollo 11 mission melt in the Earthâ€™s Atmosphere?
- I can be "educational" (Depending on your opinion)
- What do you call the act of drinking a whole bottle of, say, water in one go?
- How do I kill a 20th level raging Zealot Barbarian?
- Can I forbid a patented invention from use by the government?
- Make 37+32=55 correct by moving one match
- Are there examples of conjectures supported by heuristic arguments that have been finally disproved?
- How can prostitution be made acceptable in a religiously conservative society?
- When someone's ability to pay is suspect, why would it make sense to give them more expensive credit?
- Does a point charge inside a conducting shell cause redistribution of charge in the shell?
- How to Write an Eldritch Abomination?
- Get users to use the download button instead of right-click save
- What are the mechanical and role-playing advantages of playing a human?
- Does this mathematical technique have a formal name? And why does it work?