Questions tagged [combinatorics]

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

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9
votes
2answers
394 views

Maximize my flags

You are given the next list of 48 flags. For each pair of flags that are side by side, you score 1 point per color they share as a frontier. For instance France and Finland score 1 point thanks to ...
8
votes
4answers
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 ...
4
votes
1answer
244 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 ...
5
votes
1answer
472 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™...
10
votes
1answer
688 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™ ...
3
votes
1answer
205 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 ...
-1
votes
2answers
345 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 ...
5
votes
1answer
153 views

Sorting 9 numbers with 9 flips

You want to sort a sequence of numbers into ascending order. You can perform flips: take a sub-sequence of 4 numbers (a, b, c, d) and reverse their order to obtain (d, c, b, a). Can you sort the ...
3
votes
2answers
151 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. ...
3
votes
4answers
167 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
1answer
111 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 ...
1
vote
1answer
130 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 ...
14
votes
5answers
1k 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 ...
2
votes
3answers
469 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
1answer
360 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, ...
19
votes
5answers
983 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 ...
4
votes
2answers
250 views

Lesser derangement on a round table

This is a harder variant of Super-derangement on a round table. There is a round table with 16 seats, each seat labeled with 1 to 16 in clockwise order. Also, there are 16 people, each of whom is ...
6
votes
1answer
249 views

Super-derangement on a round table

There is a round table with 16 seats, each seat labeled with 1 to 16 in clockwise order. Also, there are 16 people, each of whom is assigned a unique integer between 1 and 16 inclusive. Now, the 16 ...
9
votes
2answers
191 views

4x4 grid equations version 2

I decided to make another one of these, because they are fun and this one is rather different. Can you place all numbers from 1 to 16 into cells, such that the following 8 equations hold? Note that ...
10
votes
1answer
611 views

4x4 grid equations

Can you place all numbers from 1 to 16 into cells, such that the following 8 equations hold? Note that the operator "/" only works for non-remainder division, i.e. you can have "8 / 4&...
10
votes
1answer
369 views

The Flippin' Magician's 7-card Grand Finale

This question is a followup to this question by @ais523, which itself was a followup to this question by @Wen1now. After touring the globe to accolades when performing his 10-card trick and 8-card ...
4
votes
3answers
989 views

Most intersections with Olympic rings

The Olympic symbol has 5 rings that intersect at 8 points: What is the most number of intersection points can you achieve by moving the rings?
2
votes
2answers
154 views

3 switches and 4 lights

The are three switches (1, 2 and 3) and four lights (A, B, C and D). Each switch turns on exactly two lights and no two switches turn on identical lights. You know that Lights A, B and C are on when ...
6
votes
4answers
1k views

A robot making increasing steps

A robot starts on a cell in an infinite grid. On the first turn it can move 1 cell horizontally or vertically. On the $n$-th turn ($n>1$) it can move $n$ cells horizontally or vertically, but it ...
5
votes
3answers
211 views

Swapping 6 queens in a 4x4 grid

What is the least number of moves required to swap black and white queens? Queens move using standard chess rules - any number of empty cells vertically, horizontally or diagonally in a straight line. ...
10
votes
1answer
777 views

Swapping 3 rooks in a 3x3 grid

This puzzle was inspired by this one: Swapping rooks in a 4x4 board What is the least number of moves required to swap black and white rooks? Rooks move using standard chess rules - any number of ...
4
votes
1answer
373 views

Die rolling around a 6x6 grid

Each side of a standard 6-sided die is painted with a different color. A 6x6 grid is drawn on paper and the die is placed in one of its corners. At each turn the die can be rolled to an adjacent cell (...
5
votes
2answers
324 views

General orchard planting problem for circles

My previous puzzle asked for the maximum number of 4-point circles attainable from a configuration of $n=10$ points drawn on a plane. I am now interested in generalizations of this puzzle to arbitrary ...
6
votes
3answers
478 views

Orchard planting problem for circles

The classic Orchard planting problem asks for the maximum number of 3-point straight lines attainable from a configuration of $n$ points drawn on a plane. Here we are interested in a variant of this ...
5
votes
2answers
173 views

Highest n where an equal number in all cells is (im)possible

Inspired by Board with all 2020s : Zeroes are written in all cells of a n×n board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. ...
13
votes
1answer
478 views

Is $~\let\r\raise\text{3rd.3}\r.6ex{\small3}\r1.1ex{\scriptsize3}\!\:\r1.6ex{\tiny3}\!\;\r2.2ex.\!\r2.5ex.\!\r2.8ex.\!\!\!\!\!\!(~)~$ too much to ask?

You will not be asked to devise a function to find the third-least of $ 3 \raise .6ex {\small 3} \raise 1.1ex { \scriptsize 3}\!\: \raise 1.6ex { \tiny 3} \!\; ...
6
votes
3answers
919 views

Flipping coins in a circle

We have a set of N coins that are all placed in a circle. They all have "Tails" as their face up side. The coins are all distinct and have numbers (1,2,3...N) written on them. In each move, ...
5
votes
4answers
1k views

The coolest checkerboard magic trick. Version 2

Version 1: The coolest checkerboard magic trick You and your friend are imprisoned. Your jailer offers a challenge. If you complete the challenge you are both free to go. The rules are The jailer ...
4
votes
1answer
141 views

Orchard planting problem for squares

The classic Orchard planting problem asks for the maximum number of 3-point straight lines attainable from a configuration of $n$ points drawn on a plane. Here we are interested in a variant of this ...
7
votes
3answers
625 views

10x10 grid with no unpainted hexominoes

What is the smallest number of cells you need to paint in an 10x10 grid, such that it contains no unpainted hexominoes? Note that a hexomino is a set of 6 adjacent cells (horizontally or vertically). ...
5
votes
2answers
443 views

Different numbers in all cells of a 3x3 board

This puzzle is inspired by this one: Board with all 2020s Zeroes are written in all cells of a 3×3 board. Pressing a cell increases by 1 the number in this cell and all cells having a common side with ...
5
votes
3answers
333 views

Different numbers in all cells of a 4x4 board

This is a harder version of this puzzle: Different numbers in all cells of a 3x3 board Zeroes are written in all cells of a 4×4 board. Pressing a cell increases by 1 the number in this cell and all ...
18
votes
5answers
1k views

Board with all 2020s

Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain ...
2
votes
1answer
276 views

A robot moving on a grid

In the spirit the question I propose the puzzle: A robot is placed on a vertex of a grid. At each move the robot must take three steps along the edge of the grid. After each step the robot must turn ...
3
votes
1answer
1k views

A robot visiting every edge of a 4x4 grid

This is a harder version of this puzzle: A robot visiting every edge of a 3x3 grid A robot is placed on the top-left vertex of a 4x4 grid. At each move the robot can take one step (up, down, left or ...
2
votes
1answer
3k views

A robot visiting every edge of a 3x3 grid

A robot is placed on the top-left vertex of a 3x3 grid. At each move the robot can take one step (up, down, left or right) along the edge of the grid to the adjacent vertex, but he cannot go outside ...
2
votes
3answers
347 views

Can you minimise the arithmetic average?

Let $n$ be a positive integer. There are $2n$ $1$s written on the whiteboard. John repeats the following procedure $3n$ times, as follows: Choose two numbers $x,y$ on the board, then replace each of ...
5
votes
3answers
835 views

Will you be the first to get free?

It is your first day in prison and you are approached by a guard having a hunch for puzzles. He tells you that he gives every new prisoner the chance to be freed if they can present him with a version ...
8
votes
1answer
166 views

L-tromino pair!

Amy is playing with different polyominoes. She suddenly thinks of a problem as follows. Choose two positive integers $m,n$. If we can use only L-trominos to tessellate a $m\times n$ rectangle with no ...
9
votes
1answer
341 views

Multiplying to reverse digits

Today I noticed that $294$ is a multiple of $49$, which is the last two digits of $294$ reversed. How many other numbers have this property? That is, how many three-digit numbers have a factor which ...
0
votes
7answers
2k views

Finding the number of poisoned bottles

This is a well-known problem (discussed here and here), but I am adding a twist to it. A king has 100 bottles of wine and poisons $K$ of them, where $0 \leq K \leq 100$. You have a supply of rats and ...
2
votes
4answers
506 views

Reconstructing points based on the sum of their coordinates version 2

10 points are drawn on a piece of paper with the following rules: Each point has integer coordinates (𝑥,𝑦) that are between 1 and 10 inclusive. For each point there is exactly one other point with ...
1
vote
2answers
326 views

Reconstructing points based on the sum of their coordinates

9 points are drawn on a piece of paper with the following rules: Each point has integer coordinates $(x,y)$ that are between 1 and 10 inclusive. For each point there is exactly one other point so ...
7
votes
2answers
370 views

Unlock the safe!

There is a (very insecure) safe, which has three digits in the lock. Each digit can only be $0,1,2$. The user choose a password made up with three $0,1,2$ digits, and the safe can be unlocked if at ...
3
votes
2answers
208 views

Count the squares [closed]

My professor at college loves geometry and discrete mathematics. He gave us a question let see if you can solve it. He asked us ...

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