# Questions tagged [combinatorics]

A puzzle based on combinatorics, which is the study of counting discrete structures.

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### How many friends does Tiffany have?

Tiffany has 14 classmates; all of her classmates have a different number of friends in the class. How many of them are friends with Tiffany? (If A is a friend of B, then B is a friend of A.)
330 views

### 3 Colors of Chess Pieces Attacking Each Other Once Each

Yes, it's another "Chess Pieces Attacking Each Other" puzzle. This time we have 3 colors. Your goal is to place as many of the same type of chess piece (excluding pawns since you can't define the "...
316 views

### Savage Road Signs (Part 2)

Please read part 1 or this might be confusing Since part 1, you have replaced the stolen stickers and your daughter has forgiven you. The highway ended up being a full 700km long, so you are happy ...
695 views

### Knights covering a 10x10 chess board

What is the minimum number of knights you need to place on a 10x10 chess board, such that every empty cell is attacked by at least one knight? Good luck!
3k views

### Coin weighing with a single weighing device

You have 12 coins which each weigh either 20 grams or 10 grams. Each is labelled from 1 to 12 so you can tell the coins apart. You have one weighing device as well. At each turn you can put as many ...
6k views

### Is this chromatic puzzle always solvable?

I've created a new puzzle from an Alexey Nigin's idea. It consists of a 8x8 board where each square is randomly assigned one of three colors. A movement is defined by picking any two orthogonal ...
2k views

### Four buttons on a table

I was asked lately (in an interview) to solve this puzzle, which is similar to the 4 cups on table puzzle. In a certain room there is a rotating round table, with 4 symmetrically located ...
2k views

### Numerical Boggle

You are probably familiar with the word game Boggle, where you need to construct words by concatenating letters from a grid. Here we will play a numerical version of the game. The rules are as follows:...
864 views

### Rotationpuzzle in hex - The journey beyond the tomb

This is the 3rd themed puzzle of the tomb. (See puzzle 1 and puzzle 2) It is fully independent of the other two and just linked by the common story. After you set the colour-puzzle dials to the right ...
1k views

### N-dimensional Tic-Tac-Toe variant

Consider the game surface to be an infinite N dimensional Cartesian lattice. The rules are X moves first, but O gets to move ...
827 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 ...
1k views

### Automatically a Knight, Knave, and Joker

Let M be a finite positive integer. It's exact value is not known. Suppose we have three classes of automaton, all of which accept a bit stream as input, produce a bit stream as output (one bit per ...
924 views

### The impossible digital sum

There are 10 digit numbers you are supposed to use shown as below; And there is a very special addition where every digit is used only once. As you see, most of the digital signals (blue squares) are ...
603 views

### How Many Squares on the Peg Solitaire

We have a well known peg solitaire which is not played yet as seen below: At most how many squares can you make by joining the points as exemplified below? Note: No ball (point) in the middle! so ...
5k views

### All numbers in a 5x5 Minesweeper grid

Can you place mines on a 5x5 Minesweeper grid such that each number from 0 to 8 appears exactly once? Good luck!
1k views

### A row of 2015 red and white chips

There is a row of 2015 chips, of which 2014 are white and one is red. You are allowed to make moves of the following type: "Choose one red chip, and flip the colors of its two neighboring chips (from ...
2k views

### A Guide to the Number Rotation Puzzle

This is an extension of What is the strategy to solve Simon Tatham's Twiddle? in that it explicitly goes beyond the default gamemodes of Twiddle The Number Rotation Puzzle (NRP) is a combination ...
4k views

### Two chessmasters at work

Viswanathan Anand plays a chess game against Magnus Carlsen. Anand plays white and Magnus plays black. They use a non-standard digital double chess clock that is counting up from zero (instead of the ...
1k views

### Finding Doctor No

James Bond is invited to a party with altogether $128$ participants (including Bond himself, the host, and the hostess). At the beginning of the party the host takes James Bond aside and asks him to ...
799 views

### How many possible starting positions are uniquely solvable for a nonogram puzzle?

This type of puzzle goes by many names: Nonogram, Picross, and Griddlers are all mentioned on the Wikipedia page, Simon Tatham calls it Pattern, I was introduced to it as Descartes Rainbow, ... The ...
1k views

### A closed path on the Rubik's cube

Is it possible to draw a closed path on the surface of a standard $3\times3\times3$ Rubik's cube such that the path traverses each of the $54$ little squares exactly once, and such that the path ...
548 views

### Breaking Balance (Part C)

For a starting number of otherwise identical coins there are among them TWO IDENTICAL counterfeit coins which are either heavier or lighter than the rest. Using a three-pan balance (described in ...
698 views

### Four Magic Ellipses

These four ellipses represent four sets and all the possible ways they can intersect (a Venn diagram, in other words). There are 8 regions inside each ellipse, and 15 regions altogether. Is it ...
1k views

### Possible pawn combinations

This may seem simple, but I have a problem calculating it. It may be because it's Monday morning. How may possible valid combinations of one color pawn (white or black, your choice) positions are ...
926 views

### The coolest checkerboard magic trick. Version 2

Version 1: The coolest checkerboard magic trick You and your friend are imprisoned. Your jailer offers a challenge. If you complete the challenge you are both free to go. The rules are The jailer ...
6k views

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 ...
7k views

### Flip a Fair Coin

I found this question and became curious, can anyone tell me the answer and prove it, i know it seems fairly simple but just thought an explanation of this would make an interesting case. Flip a fair ...
4k views

### Guess five binary digits!

Person A thinks of a 5 digit binary number. Person B tries to guess the number. B can guess a 5 digit binary number and A will respond with the number of correct digits (digits in the right place). ...
609 views

### Sort 9 train cars on 3 paths

On the three paths of a station are A, B, and C types of train cars as shown in the figure. A locomotive driver (L) can attach from 1 to 9 train cars to a locomotive at any time, move them to the ...
1k views

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### How many hexagonal paths?

Here is a hexagonal tiling, borrowed from Wikipedia. I start in any hexagon on the left hand side. I end at any hexagon on the right hand side. I can only travel to the right, not up, down or ...
4k views

### Alice and Bob play a game

The rain was still falling and Alice and Bob were terribly bored of having to stay inside the caravan, so they decided to play a game. The game is that Alice chooses a number $x$ in the interval [1,n] ...
943 views

### Colored balls in a 4x4 grid

Colored balls are placed in a 4x4 grid. A move consists of swapping two adjacent (horizontally or vertically) balls. What is the least number of moves required to form 4 connected components*, one for ...
202 views

### Generalized color balls in a 4x4 grid

This is a generalization of the Colored balls in a 4x4 grid puzzle that was proposed by Darrel Hoffman. Colored balls from 4 different colors are placed in a 4x4 grid. There is at least one ball from ...
283 views

### Dissect a square into 3:2 non-congruent integer-sided rectangles

(Similar to the recent 3:1 rectangle question) Tile a square completely with rectangles which have aspect ratio 3:2, integral side lengths and all different sizes. In other words selected from 2x3, ...
591 views

### Labeling wires in a *damaged* bundle

Variant of: Labeling wires in a bundle At a remote location, you just finished trenching a data cable across a large plot of land. The cable has 64 individual wires that are not color-coded or ...
1k views

### A party puzzler

At a party, everybody is friend with exactly $22$ of the other persons present. Whenever two persons are friends, they do not have any friends in common. Whenever two persons are not friends, they ...
300 views

### Holo-puzzle (1)

A boundary of red and blue squares is given. Can you fill in the interior such that each 5-square pattern consisting of an interior cell plus its four nearest neighbours always contains an even number ...
308 views

### General orchard planting problem for circles

My previous puzzle asked for the maximum number of 4-point circles attainable from a configuration of $n=10$ points drawn on a plane. I am now interested in generalizations of this puzzle to arbitrary ...
1k views

### Dominos on a checkerboard

What's the maximal number of dominos (2x1 tiles) that can be placed on a checkerboard (8x8 square) so that every domino covers exactly 2 squares of the checkerboard and no two dominos form a 2x2 ...