Questions tagged [combinatorics]

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

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3 votes
4 answers
238 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
1 answer
185 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 ...
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4 votes
2 answers
308 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 ...
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1 vote
2 answers
218 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 $[...
1 vote
3 answers
241 views

5x5 grid with no tetrominoes containing repeating colors

Paint the cells of a 5x5 grid with 𝑛 colors, such that every possible tetromino found in the grid uses 4 different colors. What is the smallest value of 𝑛 possible in such a coloring? Here is a ...
3 votes
3 answers
506 views

4x4 grid with no trominoes containing repeating colors

Paint the cells of a 4x4 grid with 𝑛 colors, such that every possible tromino found in the grid uses 3 different colors. What is the smallest value of 𝑛 possible in such a coloring?
6 votes
4 answers
626 views

8x8 square with no adjacent numbers summing to a prime

Can you fill a 8x8 grid with numbers from 1 to 8 such that: Every number occurs exactly once in each row and in each column (Latin square). No two adjacent (horizontally or vertically) numbers sum to ...
3 votes
1 answer
99 views

4x4 square with no increasing triples

Can you fill a 4x4 grid with numbers from 1 to 4 such that: Every number occurs exactly once in each row and in each column (Latin square). No row or column contains 3 adjacent numbers that are all ...
8 votes
2 answers
1k views

How many descendants can this spaceship crew produce?

A spaceship is on a very long voyage. It starts with a crew of 4 women and 4 men, none of whom are related by blood. How many descendants at most can this 8-person crew produce without inbreeding? ...
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3 votes
1 answer
210 views

Moving coins in a grid

Here is a great puzzle from Ed Pegg Jr. Place two coins in the center cell of the following grid. Now you can choose a coin X and move the second coin Y one cell in the direction of the arrow under ...
1 vote
1 answer
165 views

Cut the string!

There are five pieces of blue string on the table with different lengths, the total length of which is 30 cm. There are also five pieces of red string with different lengths, the total length of which ...
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4 votes
2 answers
249 views

Place 4 players to make 6 distances between pairs

Is it possible to place 4 players on a football field in such a way that the 6 distances between every pair of them are 1, 2, 3, 4, 5, 6 meters? Source: Moscow Math Olympiad 2001 (Look Inside to Page ...
  • 71
8 votes
2 answers
418 views

Rack 'Em Up! 🎱

In a game of English eight-ball pool, a set of 15 balls are arranged or 'racked' in the shape of an equilateral triangle. In order for the balls to be racked fairly, they must be arranged like so: <...
  • 1,221
16 votes
3 answers
1k views

The maximum period of dancing program

Sixteen people named A, B, ..., P are standing in line in the order ABC...P. They "dance", or swap places, according to some predefined instructions. ...
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13 votes
3 answers
959 views

8 soldiers lining up for the morning assembly

There are 8 soldiers, gathering and lining up every morning for their military service. The commander at the head of these soldiers demands that the morning lineup of these soldiers be arranged ...
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3 votes
2 answers
192 views

Generalized rectangular tilings with no "fault lines"

I recently came across this question: One rectangle, indivisible The goal is, by tiling 2x1 rectangles, to create a larger rectangle that cannot be split into 2 smaller rectangles. But my question is ...
4 votes
2 answers
378 views

The 7 face up/down card

Note: This puzzle is a very old puzzle I got from the Internet, however I changed it a bit to be more interesting. INSTRUCTIONS You have got 7 blank cards. You are playing with a friend of yours. Your ...
  • 1,147
-3 votes
1 answer
161 views

a 17X17 grid filled with trominoes of three different colors

Let's have an 17x17 grid. We can fill this grid with 96 trominoes of three different colors, 32 trominoes of each color. On this particular grid the empty single square is the position A1. By visual ...
6 votes
1 answer
360 views

Alice and Bob playing Neighboring Sums Game

Alice and Bob are playing the neighbouring game which is originally single game to get the highest point at the end. You start with an empty 4x4 grid. At each turn you can choose an empty cell and ...
  • 29.5k
7 votes
2 answers
412 views

Pentomino tiling on wrap-around 5x5 grids

It is known that P pentominoes cannot tile a 5x5 square board. Q1: If the east and west edges of the 5x5 square board are "wrapping around" (if you move a piece through one of the edges, the ...
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-2 votes
1 answer
268 views

How many shapes can you form with squares? [closed]

There is a 6 by 6 dot-grid. You will draw two squares by joining the dots. The squares cannot have common dots/points or areas. Rotations or reflections of a drawing are considered distinct. In How ...
4 votes
1 answer
289 views

Grids with trominoes

Let's have two 8x8 grids. By visual inspection we see they are filled with trominoes of three different colors. There are 7 trominoes of each color. On the grids the trominoes are not allowed to touch ...
4 votes
1 answer
239 views

n rows and 18 columns

I haven't posted for a long long time, so here is an interesting combinatorics problem! There is a table with 𝑛 rows and 18 columns. Each of its cells contains a 0 or a 1. The table satisfies the ...
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8 votes
4 answers
889 views

Neighboring sums 5x5 game

You start with an empty 5x5 grid. At each turn you choose an empty cell and place a value in it. The placed value is given by the following rules: If the chosen cell has no neighboring (horizontal or ...
13 votes
4 answers
1k views

Neighboring sums 4x4 game

Here is an interesting game. You start with an empty 4x4 grid. At each turn you can choose an empty cell and place a value in it. The placed value is given by the following rules: If the chosen cell ...
2 votes
1 answer
319 views

Forming pairs of trominoes on an 8X8 grid

On an 8x8 grid I put 21 trominoes of thee different colors. Each group of 7 trominoes has one color. By visual inspection we see the trominoes cover the whole surface except the single empty square ...
5 votes
3 answers
414 views

Optimal Strategy for Matching Pairs

I found a reality TV show recently that I thought would make a fun puzzle. On the show are 10 men and 10 women that have been "matched by experts" (ie. randomly paired). Their goal is to ...
  • 1,061
-3 votes
1 answer
162 views

Self-intersecting polygonal chains in a hexagon [closed]

This is continuation of this Q&A. Given a regular hexagon with center at point O: Question: How many self-intersecting polygonal chains are there that connect 7 points? The self-intersecting ...
  • 1,701
6 votes
1 answer
205 views

Plus-sized amoeba escapes

As an extension to @WhatsUp 's question here, the rules of which are included below, with the following differences: In one of the squares, there lives an amoeba (marked as a circle in the following ...
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21 votes
2 answers
851 views

Amoebas escaping the prison

There is an infinite grid of squares. In one of the squares, there lives an amoeba (marked as a circle in the following pictures). Amoebas cannot move, but they can perform their unique action: an ...
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3 votes
2 answers
221 views

Prime stepping stones

Start by placing number $1$ anywhere on an infinite square grid. Now place numbers $2, 3, 4, \ldots, K$ in order. A number $k$ can be placed if the following rules hold: It must be adjacent (...
0 votes
2 answers
106 views

Car registration similarity [closed]

In my city, car registration plates contain 3 numbers (0 to 9) and 3 letters (A to Z). Today I've noticed that my neighbour's car has the same registration as my car except for one character. Should I ...
10 votes
2 answers
518 views

What is the minimum number of problems in the pool? [closed]

Using a pool of problems, 16 tests will be formed, following certain conditions: Every test should have the same number of problems. Any problem should be included in at most 8 tests. Every group of ...
13 votes
1 answer
607 views

The tip of a colorful triangle

Original source: Problem 1 of British Informatics Olympiad 2017, Round 1 You're given a bunch of red (R), green (G), and blue (B) balls. I arrange some balls on a line. Then I ask you to complete the ...
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7 votes
1 answer
200 views

My two button microwave

Long ago, I encountered a microwave with a display in the "HH:MM:SS" format. But instead of a number pad, you entered the desired time through two buttons: An "up" button, which ...
9 votes
1 answer
260 views

Generalized color balls in a 4x4 grid

This is a generalization of the Colored balls in a 4x4 grid puzzle that was proposed by Darrel Hoffman. Colored balls from 4 different colors are placed in a 4x4 grid. There is at least one ball from ...
8 votes
4 answers
1k views

Colored balls in a 4x4 grid

Colored balls are placed in a 4x4 grid. A move consists of swapping two adjacent (horizontally or vertically) balls. What is the least number of moves required to form 4 connected components*, one for ...
6 votes
1 answer
524 views

All values in a 6x6 Super Minesweeper grid

In a Super™ Minesweeper grid each cell is either a mine or a value. A value in row $𝑟$ and column $𝑐$ represents the total number of mines located in row $𝑟$ or column $𝑐$ Can you fill a 6x6 Super™...
4 votes
1 answer
279 views

Most 5s in a 5x5 Super Minesweeper grid

In a Super™ Minesweeper grid each cell is either a mine or a value. A value in row $𝑟$ and column $𝑐$ represents the total number of mines located in row $𝑟$ or column $𝑐$. What is the most number ...
10 votes
1 answer
787 views

All values in a 5x5 Super Minesweeper grid

In a Super™ Minesweeper grid each cell is either a mine or a value. A value in row $r$ and column $c$ represents the total number of mines located in row $r$ or column $c$. Can you fill a 5x5 Super™ ...
-1 votes
2 answers
391 views

How many triangles can you obtain using the 6 vertices and center of a regular hexagon?

Let's say there is a regular hexagon with center at point O. Question 1. How many triangles can you obtain using the 6 vertices and its center? Question 2. What is the largest number of different ...
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3 votes
1 answer
230 views

Variation of 100 Prisoners' names in boxes

100 Prisoners' Names in Boxes The following puzzle is a variation of the above puzzle. Names in Boxes The names of 4 prisoners are placed in 4 wooden boxes , one name to a box, and the boxes are ...
3 votes
2 answers
161 views

Presidential Election

This puzzle was inspired by the current 2020 US presidential election. You are running for president in a country with 10 states. To win a state you must conduct more rallies than your opponent. ...
1 vote
1 answer
121 views

Special arrangement of 16 cards

This puzzle is from Martin Gardner. You are given 16 cards containing all aces, kings, queens and jacks from a standard deck of cards. Can you arrange them in a 4x4 grid such that each row and each ...
3 votes
4 answers
228 views

Given pairs of weights find individual values

The problem is as follows: A kid has five marbles. These marbles have different weights and the child weighs them in pairs in all possible ways. He records the weights in his notebook. These are the ...
1 vote
1 answer
133 views

Visiting primes on a line

Recently I have been playing a great mobile game called Dicast: Rules of Chaos and it has inspired me to make this puzzle. This puzzle proceeds on an infinite number line, where each integer is ...
3 votes
3 answers
535 views

Trapping fairy chess pieces

This puzzle is based on this wonderful puzzle. A fairy chess piece is placed on an infinitely large chess board with no edges. It can only visit each square once. What is the smallest number of moves ...
5 votes
1 answer
394 views

Sudoku Logic From Another Planet

This is the brutally hard Tatooine Sunset Sudoku puzzle by Philip Newman ... except the Noble Happy Star has goofed! Two of the digits have been swapped and there are multiple solutions. Fortunately, ...
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14 votes
5 answers
2k views

Stepping Stones 1, 2, 3

I came across this beautiful puzzle and decided to create my own version. Start by placing numbers 1, 2 and 3 anywhere on an infinite square grid. Now place numbers 4, 5, 6 ... $m$ in order, subject ...
19 votes
5 answers
1k views

Basic Numerical Boggle

In this post, we were introduced to the game of Numerical Boggle on a $6 \times 6$ board, the rules of which are as follows Each cell must contain a single digit from $0$ to $9$. Starting in one cell ...
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