Questions tagged [combinatorics]

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

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5
votes
2answers
553 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
2answers
208 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 ...
9
votes
4answers
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 ...
-7
votes
1answer
68 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. ...
7
votes
1answer
617 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 ...
4
votes
3answers
292 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 ...
9
votes
9answers
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, ...
6
votes
3answers
400 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 ...
8
votes
2answers
159 views

A grid-line of nuclear balls

Imagine a semi-infinite grid-line in which every box can hold any number of balls. ...
6
votes
1answer
365 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 ...
2
votes
1answer
145 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 ...
4
votes
1answer
107 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 ...
5
votes
3answers
261 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 ...
20
votes
2answers
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 ...
21
votes
2answers
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 ...
3
votes
2answers
671 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 ...
14
votes
2answers
815 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,...
9
votes
1answer
732 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.
3
votes
1answer
117 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 ...
2
votes
1answer
92 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 ...
4
votes
1answer
126 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 ...
3
votes
3answers
259 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 ...
6
votes
1answer
209 views

A 3x3 grid with common factors

A $3 \times 3$ grid $G$ is filled with every number from the set $\{2,3,5,6,7,11,14,15,30\}$. Now a new $3 \times 3$ grid $H$ is formed, such that $H_{ij}$ is the number of neighbors of $G_{ij}$ that ...
11
votes
4answers
944 views

Deriving a 3x3 grid from another one

A $3 \times 3$ grid $G$ is filled with every number from $1$ to $9$. Now a new $3 \times 3$ grid $H$ is formed, such that $H_{ij}$ is the number of neighbors of $G_{ij}$ that are greater than $G_{ij}$....
8
votes
3answers
689 views

Perfect magic 4x4 square

Can you fill a 4x4 grid with every number from 1 to 16, such that every row, every column and every 2x2 sub-grid of numbers sum to the same value?
0
votes
1answer
81 views

Unusual 3x3 square

Can you fill a 3x3 grid with every number from 1 to 9, such that the sum of numbers in the first row is equal to the sum of numbers in every 2x2 sub-grid? Can you find multiple solutions?
8
votes
1answer
314 views

Manual tiling with 8 dodecadudes

Here are 8 dodecadudes. (Drawn from the very numerous dodecadrafters, made from 12 half equilateral triangles, dodecadudes are a subset of 770 pieces with sharp points and narrow necks excluded). ...
8
votes
5answers
1k views

Minimize 1×3 tiles on a 5×5 table to block any more 1×3 tiles

What is the minimum number of 1×3 tiles that can be put on a 5×5 table so that no more 1×3 tiles can be put on it? Borders of a tiles are parallel to sides of the table. It is 5 but I can not prove ...
9
votes
1answer
490 views

Plants vs Zombies!

Several plants and zombies (no more than 20 creatures in total) came to the party “Plants VS Zombies”, and it turned out that all the creatures are of different heights. When a plant speaks to a lower ...
11
votes
3answers
1k views

Swapping 3 knights in a 4x4 grid

Can you swap black and white knights in this 4x4 grid? There is one important constraint: at no point can a knight be under attack by an opponent knight. Knights move using standard chess moves (L ...
2
votes
2answers
100 views

Surrounding an L-shaped tromino

You are given an L-shaped tromino, shown below. Can you surround it by 10 more L-shaped trominoes? The new trominoes can be rotated and must touch the original tromino at an edge or a corner. This ...
10
votes
2answers
596 views

Two football teams

Twenty two football players have agreed to split every week into two teams and play a match against each other. Every week, teams will be chosen differently, 11 players in each team, and they will ...
13
votes
2answers
914 views

Placing 9 cars into a 4x4 carpark

A carpark is arranged in a 4x4 grid and has a single entry/exit as shown in the diagram. A car has the size of a single cell of the grid. Cars can move through adjacent empty cells of the carpark ...
1
vote
1answer
170 views

Probability of a successful DT Cannon

In multiplayer Tetris, there are a number of opening setups that let you send powerful attacks very quickly. One popular opening is the DT Cannon, which allows the player to very quickly send a T-Spin ...
3
votes
1answer
161 views

Swapping eggs puzzle

You can try this with pencil and paper, or make it a physical puzzle you can try out with your kids if you have one of those large 3x6 egg cartons laying around. Imagine you have 3 rows of 6 spots. ...
3
votes
1answer
155 views

Non-increasing arrangement

We are arranging the numbers from 1 to 8 in an order so that three consecutive terms cannot be increasing. For example, 12345678 isn’t allowed but 81436572 is. How many ways are there to do it? Please ...
2
votes
0answers
68 views

Fair and square island hopping [duplicate]

If amateur fiction is not your thing skip to the bottom. As IP (Implausible Physics) expert for DREAM, the Department for Reckless Engineering and Advanced Megalomania you have been tasked by sheikh ...
4
votes
1answer
344 views

Two knight tours on a 4x4 grid

Two knights are placed on opposite corners of a 4x4 grid. Can you move* each knight 7 times, such that each cell of the grid is visited exactly once by exactly one of the knights? *Note that a knight ...
1
vote
2answers
233 views

Minimize the longest King chain on a 6x6 ternary grid

This puzzle is an extension of this one: Minimize the longest King chain on a 5x5 binary board Given a grid filled with numbers, we define a King chain to be a path on the grid such that: The path ...
0
votes
1answer
128 views

Minimize the longest King chain on a 7x7 binary grid

This puzzle is an extension of this one: Minimize the longest King chain on a 5x5 binary board Given a grid filled with numbers, we define a King chain to be a path on the grid such that: The path ...
5
votes
0answers
194 views

Choosing squares on a square board [closed]

I have an $8 \times 8$ board. On the board, I want to choose 2 unit squares in each column and row such that none of the chosen squares are touching. This means they cannot share a side or a corner. ...
3
votes
2answers
94 views

Dividing the first 10 numbers into two groups with similar product

Can you paint all numbers from 2 to 10 with red and blue colour, such that the product of all red numbers is as close as possible to the product of all blue numbers?
13
votes
7answers
846 views

Minimize the longest King chain on a 5x5 binary board

Given a grid filled with numbers, let's define a King chain to be a path on the grid such that the path can be traversed with chess King's moves (moving to one of 8 adjacent cells at a time), the ...
10
votes
2answers
463 views

Sixteen chess pieces on a square board

It is well known that the eight main chess pieces cannot cover a chess board. Suppose I have two sets of the eight main pieces. What is the size of the largest chess-like square board all of whose ...
14
votes
2answers
2k views

Three queens and two rooks covering the chess board… again!

Three queens and two rooks can be placed on a chess board so that all empty squares are under attack, as has been shown here: 3 queens and 2 rooks covering a 8x8 chess board. What if we require that ...
7
votes
2answers
905 views

21 knights covering a 11x11 chess board

Can you place 21 knights on a 11x11 chess board, such that every empty cell is under attack? Good luck! Here is a similar question for 10x10: Knights covering a 10x10 chess board
5
votes
1answer
832 views

3 queens and 2 rooks covering a 8x8 chess board

Can you place 3 queens and 2 rooks on a 8x8 chess board, such that every empty cell is under attack? Good luck!
3
votes
2answers
202 views

Determine minimal number of moves to find cells on a square table 10×10 in which a treasure is hidden

In a 10x10 square table, two neigbouring 1x1 cells contain a hidden treasure. John needs to guess these cells. In one move he can choose some cell of the table and can get information whether there is ...
2
votes
2answers
497 views

Lightbulbs in a 3×3 square

Suppose we have a $3\times 3$ arrangement of lightbulbs and we switch them on/off randomly (probability $½$). What is the probability the no adjacent bulbs are on? My attempt was: Let $1= $ on and $0 =...
1
vote
2answers
86 views

No four cells forming a rectangle

You are given a 5x5 square grid with 25 cells. Can you paint 12 cells, such that no 4 painted cells form the corners of a rectangle with sides parallel to the edges of the grid? Good luck!

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