Questions tagged [combinatorics]

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

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3
votes
2answers
180 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
204 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
668 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
783 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
223 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
231 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 ...
21
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
248 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
916 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
116 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
392 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
123 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
598 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
187 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
636 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
203 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
695 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
366 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
164 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
305 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
106 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
108 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 ...
10
votes
2answers
272 views

Add a divisor! A game

Let $k$ be a positive integer. Amy and Ben are playing a game, with the number $1$ written on the whiteboard initially. Amy and Ben do the following in order, starting with Amy: Suppose the number on ...
11
votes
1answer
635 views

Perfect Golomb Circles

A Golomb ruler of order $n$ is a straight line with $n$ marks (at integer locations) such that no two pairs of marks are the same distance apart. We can extend the concept to circles. Place $n$ marks ...
0
votes
0answers
37 views

Brute force a keypad with minimal keystrokes [duplicate]

Senario Say you have a keypad whose password is some two digit code which you do not know, say 34. Entering digits in succession on this keypad eg. ...
3
votes
1answer
207 views

Puzzle - Turn all the lights on

A machine has 2020 lights and 1 button. Each button press changes the state of exactly 3 of the lights. That means if the light is currently on, it turns off, and if the light is currently off, it ...
4
votes
3answers
184 views

The death prism

One day, you are caught by a evil wizard. He presents you with a prism, and says, "You can ask me to turn this prism to any $n$-angled right prism. Then you shall fill in $1$ to $3n$ with no ...
7
votes
3answers
395 views

Can you fill in the missing numbers in this unfriendly magic square?

An unfriendly magic square is a magic square whereby the difference between neighbouring numbers always is subject to a minimum, i.e. Each number in the matrix is unique. Each row, column and the two ...
1
vote
2answers
149 views

Cheapest strategy on average to find specific items in a set

The problem You have a set on N (around 15-30) seemingly-identical objects, which in fact comprises a small number D (unknown, between 1-3) of defective objects and the rest (N-D) are good. There is a ...
7
votes
1answer
221 views

Largest set of independent hexominoes

What is the largest set of hexominoes that can be found in which no two of them are such that one can be converted into the other by cutting out one of its component squares (thus obtaining a ...
8
votes
1answer
482 views

Recipe for diced shark fin

How may the following shark-fin-shaped Goal distribution be cooked up with 44 ingredients in a roll recipe R based on the roll sum of two 6-sided dice with non-standard allotments of dots? Goal: ...
4
votes
3answers
220 views

4-coloured queens attacking every opponent queen once

Can you place 6 queens from four different colours (24 in total), such that each queen attacks exactly one queen of each colour? They may attack as many queens of their own colour as these are ignored....
8
votes
2answers
838 views

Making longest line in 10 by 10 grid

How many blocks can you pass through at most in a 10 × 10 grid. The rules are: You cannot go over a line You cannot lift the pencil You cannot allow the blocks you have passed through ...
7
votes
2answers
337 views

Finding the most flexible of all 35 hexominoes

Of all 35 hexominoes, which is (or are) the most flexible of all, that is, the one (or ones) that can be converted into the most other hexominoes just by cutting out one of its component squares (thus ...
7
votes
4answers
324 views

3 Colors of Chess Pieces Attacking Each Other Once Each

Yes, it's another "Chess Pieces Attacking Each Other" puzzle. This time we have 3 colors. Your goal is to place as many of the same type of chess piece (excluding pawns since you can't define the "...
11
votes
1answer
378 views

Generating all hexominoes by cutting and pasting

Place the 35 hexominoes around a circle in such a way that if two of the hexominoes find themselves next to each other, it is because one of the two can be obtained from the other by cutting out one ...
2
votes
1answer
108 views

Happy Birthday, Don!

My friend Johan de Ruiter has made a nice puzzle for Donald Knuth's Birthday. The numbers indicate how many connections that candle has. Can you solve it? Original link: https://www-cs-faculty....
8
votes
2answers
1k views

Stacking pancakes for your wife

You are cooking pancakes for your lovely wife. You want to sort the pancakes such that they increase in diameter as you move from the top to the bottom of the stack. The only operation you can perform ...
7
votes
4answers
353 views

Chess pieces attacking exactly N chess pieces

In the spirit of completion and the style of: Discrete Peaceful Encampments: 9 queens on a chessboard Queens attacking exactly one queen Queens attacking exactly four queens Knights attacking exactly ...
16
votes
5answers
3k views

Knights attacking exactly three knights

Can you place 14 black and 14 white knights on a standard 8x8 chess board, such that each knight attacks exactly 3 opponent knights? Bonus question: can you do it with 15 black and 15 white knights? ...
9
votes
2answers
936 views

Queens attacking exactly four queens

Can you place 14 black and 14 white queens on a standard 8x8 chess board, such that each queen attacks exactly 4 opponent queens? Good luck! Here is a related question: Queens attacking exactly one ...
13
votes
5answers
2k views

Chess pieces attacking exactly once

Inspired by this question. Actually the same but in a more generic manner. What is the maximum number of chess pieces of the same type (e.g. kings, bishops, rooks, knights) which can be placed on a ...

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