Questions tagged [combinatorics]

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

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8
votes
1answer
301 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 ...
-2
votes
2answers
140 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 ...
32
votes
7answers
2k 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. ...
11
votes
2answers
610 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 ...
17
votes
2answers
682 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 ...
21
votes
3answers
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 ...
2
votes
3answers
2k views

How Many Times will 1 Appear on the Broken Clock?

You just got a new clock, but it's broken. Your friend tells you he can fix it, but he needs a little bit of data from it. One thing he needs is how many times 1 appears in the run of 2 days. The ...
8
votes
2answers
506 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
2answers
603 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?
36
votes
5answers
4k views

Magnets on a whiteboard

Alice enjoys placing magnets on a magnetized whiteboard. This day, she placed all 16 magnets in her possession on the board in a rectangular fashion. o o o o o o o o o o o o o o o o "...
3
votes
2answers
216 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 ...
6
votes
3answers
425 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 ...
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 ...
-6
votes
1answer
76 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. ...
10
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 ...
7
votes
1answer
631 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
301 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, ...
9
votes
2answers
174 views

A grid-line of nuclear balls

Imagine a semi-infinite grid-line in which every box can hold any number of balls. ...
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 ...
6
votes
1answer
376 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 ...
5
votes
3answers
283 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 ...
4
votes
1answer
120 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 ...
2
votes
1answer
197 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 ...
1
vote
2answers
233 views

Painting a 6x6 with 3 colours

Can you paint a 6x6 grid in red, green and blue, such that its every 3x3 sub-grid contains exactly 5 red, 3 green and 1 blue cell? Good luck!
14
votes
2answers
832 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,...
3
votes
2answers
681 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 ...
4
votes
1answer
135 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 ...
9
votes
1answer
747 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
128 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
108 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 ...
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 ...
33
votes
10answers
5k views

Winning Strategy for the Magician and his Apprentice

There are $13$ upside-down opaque cups and $2$ balls, a magician and his apprentice and yourself. You decide under which cups to put the balls, and the objective of the magician is to find the two ...
3
votes
3answers
270 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 ...
7
votes
1answer
220 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
949 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 $G_{ij}$....
3
votes
1answer
209 views

How to think about permutation puzzles?

I know this is a very general question, but these kinds of puzzles are the only ones I can't figure out on my own. Rubik's cube, Twiddle, 16... (the last two are in Simon Tatham's Puzzle collection.) ...
8
votes
3answers
713 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
82 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
339 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). ...
9
votes
1answer
497 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 ...
17
votes
5answers
886 views

Maximize the number of paths

You have exactly 990 edges. Assemble them into a simple undirected graph with two distinguished vertices A and B, such that the number of different simple paths from A to B is as large as you can make ...
8
votes
3answers
670 views

How Many Squares on the Peg Solitaire

We have a well known peg solitaire which is not played yet as seen below: At most how many squares can you make by joining the points as exemplified below? Note: No ball (point) in the middle! Don't ...
12
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
103 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 ...
1
vote
2answers
358 views

Find the least number of objects from a jar when those have two colors

I've been going in circles with this question which belongs to certainty about something. The original source of this problem is unknown. I found it in a textbook who doesn't have an author but rather ...
10
votes
5answers
1k views

2 fake coins from a pile of 30 coins

You need to find two fake coins from a pile of 30 coins. You know that a fake coin has a different weight to a real coin, but you don't know whether it is lighter or heavier. You also know that all ...
10
votes
2answers
601 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 ...
45
votes
4answers
8k views

The coolest checkerboard magic trick

In the small town of Terni (Italy), there's a couple of young friends named Marco and Leonardo, who like to perform magic tricks to a restricted audience of common friends and relatives. They like to ...
13
votes
2answers
918 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 ...

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