Questions tagged [combinatorics]

A puzzle based on combinatorics, which is the study of counting discrete structures.

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2
votes
3answers
395 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
141 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, ...
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 ...
17
votes
5answers
910 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
236 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
229 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
167 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
529 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&...
3
votes
3answers
939 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?
-8
votes
0answers
131 views

A puzzle with seventy-two countries

Let's have two groups of countries. Each group has 36 countries. The first letter of the countries in the first group start with A,B,C,E,G,S. The second group of countries has as first letter T,I,K,L,...
2
votes
0answers
397 views

A robot moving on a grid. Part 2

This is an extension of the discussion A robot is placed on a grid point. At each move the robot must take three steps along the edge of the grid. After each step the robot must turn right. Lengths of ...
2
votes
2answers
142 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
182 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
717 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
349 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
1answer
116 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 ...
13
votes
1answer
434 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} \!\; ...
5
votes
2answers
306 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
845 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, ...
4
votes
1answer
135 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 ...
6
votes
3answers
443 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
164 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. ...
5
votes
3answers
326 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 ...
5
votes
2answers
437 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 ...
2
votes
1answer
224 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 ...
17
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 ...
3
votes
1answer
207 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
229 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
335 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
803 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
137 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
270 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
1k 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
500 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
317 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
349 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
195 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 ...
21
votes
5answers
2k views

Numerical Boggle

You are probably familiar with the word game Boggle, where you need to construct words by concatenating letters from a grid. Here we will play a numerical version of the game. The rules are as follows:...
7
votes
3answers
212 views

Perfect power nim

Let $m,n$ be positive integers. Ann and Ben has $m$ stones, and each of them takes exactly the perfect power of $n$ stones ($n^k$, where $k$ is a nonnegative integer) in order, starting from Ann. Who ...
13
votes
3answers
676 views

A grid where every combination of two colours appears exactly once

Is it possible to paint the cells of a rectangular grid with $K$ different colours such that: No two adjacent (horizontally or vertically) cells have the same colour, and Every combination of two ...
13
votes
2answers
797 views

A tournament, and a tight personal schedule

A 64-player binary tournament bracket is about to start. You plan to free up your schedule in advance to watch some of the matchups (meaning, you can plan to watch the second semifinal, for example, ...
6
votes
1answer
233 views

Crosswords: Maximum number of words in an n×n grid

What is the maximum number of "words spaces" that can be in an n×n crossword, based on the placement of the shaded squares. Some limitations No word can be less than 3 spaces in length ...
3
votes
3answers
243 views

Most number of equilateral triangles formed by 13 points

What is the most number of equilateral triangles you can form by drawing 13 points on a piece of paper? Each triangle must have 3 equal sides and pass through 3 points. Only equilateral triangles can ...
10
votes
2answers
1k views

Escape from your friend!

I saw this interesting problem in a Mathematics book in Chinese(I translated it): You and your friend is playing a game. There is a square swimming pool, and you are in the middle of it. Your friend ...
20
votes
4answers
2k views

Boys and girls in a circle

There are $28$ students in a class, and each of them are either boy or girl. They sit in a circle, and claim that “The two people next to me are of different gender than each other.” It's known that ...
2
votes
2answers
249 views

Super Blox - level 1.13

Here is a hard puzzle from my game. The aim is to change the color of all blue blocks (squares) to green using the following rules: You can move any block or the red ball to an adjacent empty ...
8
votes
3answers
922 views

Fillomino Tiling…how many 1's?

Suppose a 'Fillomino tiling', much like a completed Fillomino puzzle, consists of a set of polyominoes covering a region without gaps nor overlaps, with no two n-ominoes of the same size touching ...
12
votes
5answers
2k views

Maximise your gold!

You met a genie. He gets $150$ magic lamps out, which are numbered from $1$ to $150$. You have to colour each lamp red or blue. After colouring, the genie will count the number of triples $T$ of magic ...
8
votes
2answers
4k views

Alice and Bob play a game

The rain was still falling and Alice and Bob were terribly bored of having to stay inside the caravan, so they decided to play a game. The game is that Alice chooses a number $x$ in the interval [1,n] ...

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