Questions tagged [combinatorics]

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

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12
votes
1answer
419 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
245 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
769 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
130 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
408 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
157 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
305 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
387 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
164 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
177 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
120 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
328 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
797 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
132 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
262 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
495 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
311 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
337 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
191 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
1k 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
209 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
673 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
794 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
232 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
237 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 ...
9
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
920 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
3k 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] ...
2
votes
1answer
117 views

Super Blox - level 1.7

I wrote a free puzzle game for Android phones, called Super Blox. The aim of each level is to change the colour of all blue blocks (squares) to green using the following rules: You can move any block ...
5
votes
2answers
395 views

Super Blox - level 1.8

I wrote a free puzzle game for Android phones, called Super Blox. The aim of each level is to change the colour of all blue blocks (squares) to green using the following rules: You can move any block ...
4
votes
1answer
128 views

Kings on a chessboard

Let $n$ be a positive integer. You are given $4n^2$ kings and a $4n\times4n$ chessboard. You have to place the kings on the chessboard such that each row and column contains exactly $n$ kings, and no ...
14
votes
2answers
603 views

Sort 9 train cars on 3 paths

On the three paths of a station are A, B, and C types of train cars as shown in the figure. A locomotive driver (L) can attach from 1 to 9 train cars to a locomotive at any time, move them to the ...
6
votes
2answers
189 views

Phone pattern security

My phone is unlocked using a security pattern. This is a path drawn through a 3x3 grid of dots with the following rules: The path can start at any dot The path visits neighbouring dots: horizontally,...
15
votes
4answers
2k views

Creating the hardest 10x10 maze

You are given an empty 10x10 grid. You are allowed to paint some of its cells as walls (black), while the remaining cells stay empty (white). A robot is programmed to start in the top-left corner of ...
9
votes
3answers
644 views

Knight and Knaves Castle

I was pretty bored in the lockdown so I thought up a mathematics puzzle, which I haven's solved yet, so the community can solve together. Let $n>1$ be a positive integer. There is a square castle ...
8
votes
2answers
207 views

A knight chased by four knights

This is a follow up to A knight chased by three knights Two players are playing a variant of chess on a 11x11 grid. The first player controls a white knight that starts in the centre square. The ...
10
votes
3answers
701 views

A knight chased by three knights

Two players are playing a variant of chess on a 8x8 grid. The first player controls a white knight that starts in the top-left corner. The second player controls three black knights that start in the ...
4
votes
3answers
370 views

Creating the hardest 7x7 maze

This puzzle is based on Creating the hardest 6x6 maze You are given an empty 7x7 grid. You are allowed to paint some of its cells as walls (black), while the remaining cells stay empty (white). A ...
8
votes
2answers
167 views

Intersecting shapes on a flat surface

What is the maximum number of enclosed regions that you can create by drawing two circles and two triangles on a flat surface? Try answering with mathematical arguments.
26
votes
4answers
4k views

Creating the hardest 6x6 maze

You are given an empty 6x6 grid. You are allowed to paint some of its cells as walls (black), while the remaining cells stay empty (white). A robot is programmed to start in the top-left corner of the ...
17
votes
3answers
3k views

How many ways can you find the word DIAMOND in this diamond?

In how many different ways may the word DIAMOND be read in the arrangement shown? You may start wherever you like at a D and go up or down, backwards or forwards, in and out, in any direction you like ...
11
votes
5answers
900 views

Wizard creating a jewelry

I have a puzzle game which I am not sure how to prove that I have the right answer. The Puzzle is the following: We have a wizard which makes very special jewelry (a straight line with beads). ...
8
votes
1answer
310 views

Frog in the Well [duplicate]

A frog is trapped in a well, just 1 meter below the lip. On sunny days, the well is dry, and the frog is able to climb up 1 meter. On rainy days, the well is wet and the frog slides down 1 meter. If ...
4
votes
1answer
111 views

Finding the Missing Word in a Crossword

On each of the 25 cells of this board, place one of the letters A, C, M or S so that, in alphabetical order, nine of the ten words that can be read down or across are the following: AACAC AMCAS ...
6
votes
1answer
109 views

Mathematics Puzzle - Number Circle

The numbers 1, 6, 8, 13, 15, and 20 can be placed in the circle below, each exactly once, so that the sum of each pair of numbers adjacent in the circle is a multiple of seven. In fact, there is more ...

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