Questions tagged [combinatorics]

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

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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 ...
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 ...
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 ...
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™...
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™ ...
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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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&...
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 ...
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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?
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 ...
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 ...
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. ...
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 ...
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 (...
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 ...
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 ...
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. ...