Questions tagged [combinatorics]
A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]
700
questions
6
votes
1answer
246 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
133 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
90 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
214 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 ...
19
votes
1answer
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 ...
20
votes
2answers
2k 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
659 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
793 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
722 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
105 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
84 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
251 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
201 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
940 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
680 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
78 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
312 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
486 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
99 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
590 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
909 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
132 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
154 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
153 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
339 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
231 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
127 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
193 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
93 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
841 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
459 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
902 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
830 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
200 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
494 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
85 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!
9
votes
3answers
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 ...
5
votes
2answers
192 views
Running Out of Digits, Level 3
The challenge idea is credited to HelloWorld1337.
You initially have x of each digit from 0 to 9. This means you have x * 10 digits in total. This count for each digit is shown in the table below.
...
6
votes
2answers
454 views
Covering a 15x15 grid with rectangles
You are given a 15x15 grid and asked to cover it with rectangles whose dimensions are a power of 2. For example you can use rectangles 8x1 and 4x4, but not 2x3. The rectangles must cover every cell of ...
10
votes
1answer
426 views
Grid infection with diagonal adjacencies
A community consists of 81 houses laid out in a 9 x 9 square grid. Every household is friends with their eight orthogonal and diagonal neighbors (except for the houses on the perimeter which have only ...
15
votes
5answers
655 views
Domino tiling on 8x8 checkerboard with four squares removed
I once posted this problem on the (now deleted) Area 51 Math Puzzles proposal. It was well-received there, but obviously I didn't get an answer. I still don't know the answer, and I'm not even sure if ...
2
votes
4answers
202 views
The Greenhouse Problem version 2
This is an extension of Nilster's great puzzle: The Greenhouse Problem
The task is the same, but this time sprinklers cover only a 3x3 square around them. For completeness, here is the full set of ...
2
votes
1answer
172 views
How long will it take to hand out the shuffled papers?
A teacher has $n$ students sit in a circle in her classroom. She holds in her hands a perfectly shuffled stack of the students' graded homework, with Juan's on top. She is currently standing in front ...
4
votes
2answers
285 views
Mathematics for the English major
An entry in Fortnightly Topic Challenge #48: Unusual tag mix
I was looking at the unusual tag mixes post, and one of the ones listed is combinatorics and english. I thought "who's going to be ...
1
vote
2answers
211 views
Dividing the first 10 primes into groups whose sum is prime [closed]
Take the first 10 primes. Can you divide them into $g$ disjoint groups, such that the sum of numbers in each group is prime. In particular can you make this work for every value of $g$ in the range $[...
