Questions tagged [combinatorics]

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

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15 votes
2 answers
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The universal ticket

I am submitting a very interesting problem from a French mathematical recreation site: http://www.diophante.fr/problemes-par-themes/g-probabilites/g2-combinatoire-denombrements/1434-g248-le-billet-...
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6 votes
2 answers
324 views

Taking turns adding a number 1,2,3 to a 3x3 matrix without repeating numbers in the rows or columns: does the first player always win?

Alice and Bob are playing a game on an initially empty 3x3 matrix. They take turns, and each turn: They add a number in {1,2,3} to an empty cell. They are not allowed to repeat a number in a row or ...
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3 votes
1 answer
310 views

How many distinct pentominos can be placed on a 8×9 board?

Upon proving optimality of an 8-pentomino solution for an 8×8 board, I was curious to see whether there is a 9-pentomino solution for an 8×9 board, namely a way to arrange 9 distinct pentominos within ...
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  • 1,073
14 votes
3 answers
2k views

How many distinct pentominoes are possible to place on an 8 x 8 board?

Rules Place some pentominoes into an 8 x 8 grid. They do not touch each other. They can touch only diagonally (with corner). Pentominoes cannot repeat in the grid. Rotations and reflections of a ...
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  • 141
1 vote
2 answers
352 views

How many different tiles are there when each corner may have 0-6 dots, each of which may have 0-6 dots?

There are four corners to each tile. Each corner can be empty, or contain an arrangement of dots (1-6) like the sides of a dice. Within each of these dots can be a further arrangement of 1-6 dots, or ...
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  • 27
4 votes
1 answer
203 views

More stepping stones

Start by placing prime numbers 2, 3 and 5 anywhere on an infinite square grid. Now you can place a prime number $p$ subject to the following rules: It must be greater than all the previous numbers ...
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2 votes
1 answer
236 views

Probability that there will be no mutual best friendships? [closed]

Here is a problem: There are two groups of n users, 'A' and 'B'. Each user in A is friends with those in B, and vice versa. Each user in A will randomly choose a user in B as their best friend and ...
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15 votes
1 answer
792 views

Exterminating blobs on a grid

On an infinite square grid, some of the squares are occupied by little creatures called blobs. Cute as they are, it is your mission to exterminate all of them! You only have two methods at your ...
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2 votes
0 answers
116 views

Maximize my flags - 2x2 version

Because Maximize my flags was not solved to optimality by the community, perhaps because the coding required was too harsh, I present you Maximize my four flags. The rules are exactly the same as in ...
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  • 5,557
-2 votes
1 answer
160 views

What is the right triplet of characters? [closed]

Below you will find three questions. Answer each question in order with no spaces between them. So if the answer to the first question is a, the answer to the second question is b, and the answer to ...
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  • 105
5 votes
2 answers
450 views

How many Wordle images are there?

Wordle (https://www.powerlanguage.co.uk/wordle/) has recently become a well-known word guessing game. The rules are simple: … a five-letter word is chosen that players aim to guess within six tries. ...
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7 votes
2 answers
419 views

How many squares can a limp queen move to?

Consider a large chessboard. A limp rook is a chess piece that moves one step orthogonally, but it turns $90$ degrees after every move. The limp rook makes some moves, not crossing over its own path, ...
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  • 333
2 votes
1 answer
109 views

Harary's generalized Tic-Tac-Toe; Winning strategy for Skinny on a 7 x 7 board?

Disclaimer: The purpose of this post is a ask question, not to offer a puzzle. Still, there are some puzzles here for the reader's pleasure. Disclaimer 2: This question was also asked on Math Stack ...
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  • 31.4k
4 votes
3 answers
463 views

Number merging game

You are given a grid filled with numbers. If a number $n$ is orthogonally adjacent (horizontally or vertically) to another number $n$ then you can pick it up and place it on top of the second number. ...
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2 votes
1 answer
185 views

How useful is Marijn's Bluff?

Parcly and Tori Taxel, after having wished genies' chess into existence and played around with it – noticing the link to Zarankiewicz's problem and getting an OEIS entry published in the process – ...
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2 votes
2 answers
381 views

Two genies and their kind of chess

While playing chess Parcly and Tori Taxel, best friends and genies, got bored and transformed all the pieces into pawns to make pretty patterns. They found this 22-pawn arrangement where every 3×3 ...
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  • 3,681
10 votes
2 answers
854 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 ...
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4 votes
2 answers
390 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 ...
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12 votes
2 answers
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 ...
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5 votes
5 answers
515 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 ...
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4 votes
1 answer
227 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 ...
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7 votes
2 answers
310 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 ...
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7 votes
1 answer
333 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 ...
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  • 4,883
-2 votes
2 answers
165 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 ...
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32 votes
7 answers
3k 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. ...
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10 votes
2 answers
741 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 ...
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16 votes
2 answers
725 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 ...
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  • 23.9k
8 votes
2 answers
567 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 ...
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7 votes
2 answers
628 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?
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3 votes
2 answers
225 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 ...
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10 votes
4 answers
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 ...
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-6 votes
1 answer
84 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. ...
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6 votes
1 answer
651 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 ...
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5 votes
3 answers
406 views

Special team in a soccer tournament

$N$ teams play in a single round-robin soccer tournament. A game has 3 possible outcomes: team 1 wins, team 2 wins or a draw. Is it possible that one team achieves more wins than any other team and ...
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9 votes
9 answers
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, ...
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  • 4,883
6 votes
4 answers
533 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 ...
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  • 4,723
9 votes
2 answers
186 views

A grid-line of nuclear balls

Imagine a semi-infinite grid-line in which every box can hold any number of balls. ...
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  • 301
6 votes
1 answer
390 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 ...
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2 votes
1 answer
217 views

Balls in baskets

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 ...
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4 votes
1 answer
140 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 ...
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  • 135
5 votes
3 answers
300 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 ...
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  • 19.9k
20 votes
2 answers
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 ...
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  • 6,977
21 votes
3 answers
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 ...
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  • 4,883
4 votes
2 answers
690 views

Too many school assignments

This year we have to make our school assignments in pairs. With each classmate must be made exactly one of those assignments. Exactly 30% of the assignments will be made by a pair of girls. How many ...
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  • 4,767
16 votes
2 answers
967 views

7x7 Golomb square

Can you paint $7$ cells of a $7 \times 7$ grid such that the Euclidean distance* between any pair of painted cells is distinct? Good luck! *The Euclidean distance between cells $(r_1,c_1)$ and $(r_2,...
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10 votes
1 answer
769 views

Who will win in a game of writing 3 consecutive Xs on a 2022 × 1 board?

Ana and Bob alternately write Xs on a 2022 × 1 board. The winner is the one who makes 3 consecutive Xs. Who has the winning strategy if Ana plays the first move? Describe such a strategy.
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  • 860
3 votes
1 answer
180 views

Snake game on a 9×9 grid

You are playing a snake game. The snake starts in the top-left corner of a grid. Each cell of the grid is either empty or a wall. Each turn you can press a key to move the snake in one of four ...
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2 votes
1 answer
157 views

Snake game on a 6×6 grid

You are playing a snake game. The snake starts in the top-left corner of a grid. Each cell of the grid is either empty or a wall. Each turn you can press a key to move the snake in one of four ...
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4 votes
1 answer
167 views

Is it possible to calculate group 3's factor of 3 in Thistlethwaite algorithm?

https://www.jaapsch.net/puzzles/thistle.htm I'm trying to generate 29400 ($8C4^2 * 6$) indices for each one of the cube states in G2. $8C4^2$ = 4900 is for solving the corner and edge pieces (forming ...
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4 votes
3 answers
289 views

Not selling 100 pencils

A shop sells pencils only in boxes of fixed size. It cannot sell 100 pencils. It can sell any larger amount of pencils. It has one of each size box on display. question 1: What is the minimum amount ...
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