Questions tagged [combinatorics]

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

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2 votes
1 answer
194 views

Good and bad numbers of remaining mines

You've been tasked with finishing solving this Minesweeper board: "How many mines remain?", you ask. "I'm just choosing that now, actually. Tell you what: I was going to consider every ...
8 votes
2 answers
475 views

How many Nonconsecutive Sudoku solutions are there?

Consecutive Sudoku is a variant with the additional rule that orthogonally adjacent numbers are consecutive if and only if there is a dot/bar on the line between them. A Nonconsecutive Sudoku is one ...
  • 9,910
7 votes
1 answer
502 views

Two arcs equal three arcs

(Gonna answer my own question, as is encouraged.) To set the stage: an arc (or a Jordan arc) is a non-self-intersecting curve with two distinct endpoints. (For those who are familiar with topology, it'...
6 votes
1 answer
740 views

Infected squares warmup: infect a 7x7 board with 21 squares

You can consider this a "warmup" to my other question about infected squares. On a $7\times7$ square, some cells are infected; if a cell shares an edge with $3$ infected squares, it becomes ...
7 votes
2 answers
407 views

Infected cubes puzzle in 3D with threshold 4

(This question was previously posted on Math SE, but received no answers.) 3D infected cubes puzzle with threshold $4$: On an $n\times n\times n$ cube, some cells are infected; if a cell shares a ...
9 votes
3 answers
1k views

Tiling a chessboard

Say I have an eleven by eleven chessboard, so there's 121 squares total. I remove the centermost piece so there's 120 pieces. I want to tile the chessboard with 1x4 or 4x1 pieces in a way that none of ...
  • 203
6 votes
3 answers
306 views

More Genuine and Fake Coins

I have 36 identical coins of which four, all weighing the same, are known to be fake. Fake coins are either all heavier than genuine coins, or all lighter. At most how many weighings on a balance ...
0 votes
0 answers
38 views

Impossible tiling of board using dominoes [duplicate]

prove that no matter how you tile a 6 x 6 board using 2 x 1 tiles, there would always be a vertical or horizontal line separating the board. Separation here means that no tile would be cutting across ...
11 votes
4 answers
1k views

Prime lights out

You start with a 4x4 grid filled with zeroes. If you press a cell then the cell and all its neighboring (horizontally and vertically) cells will have their numbers increased by 1. What is the most ...
0 votes
2 answers
144 views

How can 8 , 10 or 12 teams rotate through 7 or 8 games without overlaps? [closed]

We are scheduling a big scout event with children, and we have 7 or 8 games organized for them to rotate and play with each other. Set up a game schedule that follows these rules: There are 3 ...
13 votes
1 answer
2k views

The Lufthansa Lottery

In order to pass free time while striking for better pay, some Lufthansa workers organise a lottery where each ticket picks three distinct numbers from $1$ to $11$ inclusive the draw picks five ...
  • 6,350
0 votes
1 answer
157 views

Number of 6-person events so all groups of 3/10 people have dined together [closed]

Assume 10 people numbered 1-10 have to be invited for dinner events. However, the hotel can accommodate only 6 at a time. Therefore, they will be invited in batches until all groups of 3 people have ...
  • 103
1 vote
1 answer
186 views

Generalization of twelve balls and scale problem

This problem is a generalization of Twelve balls and a scale problem. So I can solve and understand how things are going if we have 12 balls or 9 balls but how do I generalize? If say we have $3^n$ ...
  • 613
19 votes
3 answers
1k views

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-...
6 votes
2 answers
348 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 ...
3 votes
1 answer
330 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 ...
  • 1,095
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 ...
  • 141
1 vote
2 answers
359 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 ...
  • 27
4 votes
1 answer
215 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 ...
2 votes
1 answer
592 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 ...
15 votes
1 answer
809 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
134 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 ...
  • 5,881
-2 votes
1 answer
163 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 ...
5 votes
2 answers
478 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. ...
7 votes
2 answers
430 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, ...
  • 333
2 votes
1 answer
149 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 ...
  • 31.7k
4 votes
3 answers
474 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. ...
2 votes
1 answer
195 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 – ...
  • 6,350
2 votes
2 answers
398 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 ...
  • 6,350
10 votes
2 answers
886 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 ...
4 votes
2 answers
406 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 ...
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 ...
5 votes
5 answers
527 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 ...
5 votes
1 answer
230 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 ...
7 votes
2 answers
318 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 ...
7 votes
1 answer
348 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 ...
  • 5,279
-2 votes
2 answers
183 views

How to solve this problem on overlapping? [closed]

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. ...
10 votes
2 answers
773 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 ...
16 votes
2 answers
754 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 ...
  • 25.3k
8 votes
2 answers
573 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 ...
7 votes
2 answers
819 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?
3 votes
2 answers
231 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 ...
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 ...
-6 votes
1 answer
89 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. ...
user avatar
6 votes
1 answer
658 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 ...
5 votes
3 answers
411 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 ...
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, ...
  • 5,279
6 votes
4 answers
568 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 ...
  • 5,042
9 votes
2 answers
197 views

A grid-line of nuclear balls

Imagine a semi-infinite grid-line in which every box can hold any number of balls. ...
  • 301

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