Questions tagged [combinatorics]
A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]
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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 ...
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2
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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 ...
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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'...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2
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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 ...
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1
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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 ...
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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 ...
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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 $3^n$ ...
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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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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
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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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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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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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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
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1
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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
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1
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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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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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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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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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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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1
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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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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
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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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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
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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
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2
answers
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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
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2
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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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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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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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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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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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2
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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 ...
user75346
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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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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
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2
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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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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
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2
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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
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2
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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
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4
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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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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. ...
user40368
6
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1
answer
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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
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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
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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
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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
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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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