# Questions tagged [combinatorics]

A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]

723 questions
Filter by
Sorted by
Tagged with
263 views

### Kings on a chessboard

Let $n$ be a positive integer. You are given $4n^2$ kings and a $4n\times4n$ chessboard. You have to place the kings on the chessboard such that each row and column contains exactly $n$ kings, and no ...
113 views

### minimum liars- truth tellers dwarves pairs [closed]

Each of the 88 dwarfs living on the farm has at least one friend. Of these dwarves, 44 are honest and always truthful, and 44 are liars and always lie. When Keloğlan visited the farm, 44 of the dwarfs ...
806 views

### 4x4 4-color Golomb square

This is a variation of my previous puzzle Can you paint a $4 \times 4$ grid with $4$ colors, such that for every color the Euclidean distance* between any pair of cells of that color is distinct? Good ...
1k views

### How many "mathletic" couples were having dinner at the table?

Once, a number of couples, each one of them happening to be composed of a mathematician and an athlete (hence 'mathletic'), wanted, in order to diversify communication, to sit down at opposite sides ...
3k views

### How many different non congruent polygons can you make on a 3x3 dot grid?

There is a 3×3 dot grid. How many different non-congruent polygons can you make on the grid? Rules: All vertices of the polygon must be on the grid Only non self intersecting polygons Only polygons ...
302 views

### Sum in Magic star puzzle

I have the following problem: Place the first 11 natural numbers in the circles so that the sum of the four numbers at the tops of each of the five sectors-beams of the star equals 25. I came up with ...
1k 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 ...
256 views

### A Christmas Tennis Tournament

Three friends, Charles, Hugh, and Freddy, played a one-month tennis tournament, just one match between two of them every single day during last December. During the tournament, whoever lost the day's ...
679 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 ...
499 views

### Find the most unfortunate compact combination of coins to have in LOLandia

You live in LOLandia. Its currency is called 'lulz' and comes in the form of coins and paper banknotes. The smallest paper banknote has a nominal value of 500 lulz. There are six types of coins, each ...
214 views

### My High School's Reunion

My high school is celebrating 30 years since graduating its first class and is planning to invite for lunch 20 alumni, 600 in all, from each of those classes. Hosts are planning to sit everyone in ...
296 views

### Integers containing all ten digits

It is known that most positive integers contain at least one copy of each of the ten digits. What is the largest n such that at most 50% of the integers in the set [1,2,3,...,n] contain at least one ...
11k views

### How many paths are there through a chess board? [closed]

A pawn is placed on the lower left corner square of a standard 8 by 8 chessboard. A 'move' involves moving the pawn, where possible, either: one square to the right, one square up, or diagonally one ...
314 views

### Can you distribute the balls equally into 2 boxes?

You have 2 boxes and an even number ($2n$) of balls in the first box. Your goal is to distribute the balls equally into the two boxes, so that each box contains $n$ balls. You must obey the following ...
152 views

### How to solve this problem on overlapping?

In cases of problems involving order and ranking where there are two indices (namely left and right) there is a particular chance of overlapping. Let us take an example to justify this: Ranjan is ...
2k views

### Two out of a dozen cartons have Easter eggs. Two people try to find one Easter egg carton, each using a different strategy. Who is expected to win?

I have found a counter intuitive puzzle. I have read the answer given at the source and understand it completely. But, what I am unable to understand is why my intuition turned out to be wrong. ...
637 views

### Two dimensional Mastermind

You have probably played the classic game of Mastermind with 4 pegs and 6 colours. It turns out that the codebreaker can always find the pattern in 5 moves or fewer. Now consider the 2D version of the ...
703 views

### XV Sawtooth Sudoku

Please find below a variant Sudoku puzzle, based on a combinatorics problem I was having a look at. The timing is right, as I recently saw @BeastlyGerbil back in chat, and I know that user is a big ...
3k views

### Can you survive this infinite zombie attack?

You're surrounded by infinitely many zombies. You're at the origin, and zombies occupy the points $(100i,100j)$ for all integer $i, j$ except the origin, as shown below: You and the zombies move ...
2k views

### How Many Times will 1 Appear on the Broken Clock?

You just got a new clock, but it's broken. Your friend tells you he can fix it, but he needs a little bit of data from it. One thing he needs is how many times 1 appears in the run of 2 days. The ...
514 views

### Largest rectangle from 20 Lego bricks

You have twenty 2x4 Lego bricks, like the one shown below What is the area of the largest rectangle you can make satisfying the following conditions: All bricks must be connected in a single ...
610 views

### 5x5 binary grid with every 2x2 sub-grid occurring once

Can you paint a $5 \times 5$ grid in two colors, such that each of the $2 \times 2$ possible sub-grids ($2^4 = 16$ combinations) occurs exactly once in the grid?
4k views

### Magnets on a whiteboard

Alice enjoys placing magnets on a magnetized whiteboard. This day, she placed all 16 magnets in her possession on the board in a rectangular fashion. o o o o o o o o o o o o o o o o "...
223 views

### 2x4 grid with distinct differences

Can you place numbers from the range $[0,16]$ into a $2 \times 4$ grid such that all orthogonal pairwise differences are distinct? In other words, we want every pair of numbers that lie in the same ...
435 views

### Generalization of the two-surgeons-two-patients-and-two-gloves puzzle

This is the original puzzle with $n=2$. I recommend solving it before this one to get acquainted with the mechanisms. There are $n$ patients in an hospital (let's call them $p_1 \dots p_n$), each of ...
3k views

### 6x6 Minesweeper grid with all threes

Can you place 16 mines on a 6x6 Minesweeper grid such that each number produced is a 3? Bonus: can you find multiple solutions that are not rotations or reflections of each other? Good luck! Related ...
79 views

### Overlapping in Order and ranking

For order and ranking questions there are a couple of the questions which require to find the total number of persons along with maximum and minimum condition which is difficult for me to comprehend. ...
1k views

### No three points in a line

You are given a 4x4 square grid. It has 16 cells and 25 grid intersections. Can you place 10 points at grid intersections, such that no three points lie on the same straight line? Lines can be ...
642 views

### Paint Eleven Squares

I was inspired by this great question: Paint Eight Squares Given a $5 \times 5$ grid of white squares, can you paint 11 of the squares black so that each white square is orthogonally adjacent to ...
306 views

### Special team in a soccer tournament

$N$ teams play in a soccer tournament where each team plays every other team exactly once. A game has 3 possible outcomes: team 1 wins, team 2 wins or a draw. Is it possible that one team achieves ...
3k views

### How many gold coins can you extract from the billionaire?

An eccentric billionaire plays a game with you. She has an urn with 100 gold coins. Each time, you can take any number of coins from the urn. If you take n coins, she will flip a fair coin. If head, ...
181 views

### A grid-line of nuclear balls

Imagine a semi-infinite grid-line in which every box can hold any number of balls. ...
1k views

### Beans under the chessboard

Under every grid cell of a chessboard, I put either one bean or nothing. Now if you choose a (grid) rectangular area on the chessboard, then I will tell you the parity of the number of beans under ...
381 views

### Discounts in a shop

I came across this sign in a shop and thought it could make a nice puzzle. So you can buy items and get discounts depending on how many items you got. You can combine discounts and use as many as you ...
289 views

### The vaccine distribution conundrum

The context is that of a pandemic that is spreading wildly and requiring global vaccination of the population. You are working in a distribution center for the vaccines. One day you have been ...
126 views

### combinations of a strange magic square

The following question has been described to me by my math teacher: The diagram below is to be filled in so that each white square contains a different whole number from 1 to 12 (inclusive) and the ...
203 views

There are 16 baskets: 4 red, 4 blue, 4 green and 4 black. Each basket contains a ball from one of the 4 colours (see image). You can pick up a ball from one basket and swap it with a ball from another ...
236 views

### Painting a 6x6 with 3 colours

Can you paint a 6x6 grid in red, green and blue, such that its every 3x3 sub-grid contains exactly 5 red, 3 green and 1 blue cell? Good luck!