Questions tagged [combinatorics]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
21 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
4answers
253 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 ...
4
votes
0answers
90 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 ...
7
votes
3answers
469 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
2answers
157 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 ...
6
votes
2answers
132 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.
25
votes
4answers
3k 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 ...
16
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
864 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). ...
7
votes
1answer
287 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
100 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
96 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
252 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 ...
11
votes
1answer
605 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
176 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
171 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
376 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
1answer
115 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 ...
7
votes
1answer
217 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
475 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
212 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
828 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
325 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
320 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
374 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
99 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
348 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
933 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 ...
21
votes
8answers
5k views

Queens attacking exactly one queen

What is the most number of black and white queens that you can place on a standard 8x8 chess board, such that each queen attacks exactly one opponent queen?
5
votes
2answers
230 views

Almost a sudoku puzzle

Given an $N \times N$ square grid, with $N$ is a positive integer and $N \geq 5$. You need to write on the square cells numbers from $1$ to $4$(some square cells can be left blank) so every row and ...
10
votes
1answer
205 views

A Wife to Honor - The waltz of the electrons

Here's a noodle-scratcher (hopefully) for you math puzzlers. The Stage Let $n$ be an integer $\geq 2$. A grid is laid out as in Fig. 1, with $n$ source (S) nodes, $n$ drain (D) nodes, and a network ...
6
votes
2answers
249 views

A very tiny probability puzzle [closed]

Imagine there are six football teams (A,B,C,D,E,F). They play each other in a league once only (on neutral ground). To determine the score, two $d4-1$, i.e. the outcome is $\{0,1,2,3\}$ are rolled, ...
17
votes
4answers
2k views

Paint 21 Squares of a 7×7 Board Without Forming a Rectangle

Got a nice puzzle from my friend, when he was competing in IWYMIC/IMC 2011. Paint $21$ of the $49$ squares of a $7 \times 7$ board so that no four painted squares form the four corners of a rectangle....
4
votes
4answers
362 views

Rotating numbers in a 3x3 grid

We have a 3x3 grid numbers like so 9 8 7 6 5 4 3 2 1 Four numbers in any 2x2 sub-grid can be rotated clockwise or anti-clockwise. For example a b c d rotated clockwise becomes c a d b Is it ...
2
votes
1answer
71 views

Rotating numbers in a 2x3 grid

We have a 2x3 grid numbers like so 6 5 4 3 2 1 Four numbers in any 2x2 sub-grid can be rotated clockwise or anti-clockwise. For example a b c d rotated clockwise becomes c a d b Can you obtain ...
8
votes
4answers
516 views

Running Out of Digits, level 2

The challenge idea, and images are credited to Andrew. You initially have 100 of each digit from 0 to 9. This means you have 1000 digits in total. This count for each digit is shown in the table ...
10
votes
3answers
1k views

Math puzzle - Running Out of Digits

You initially have 100 of each digit from 0 to 9. This means you have 1000 digits in total. This count for each digit is shown in the table below. Now start counting by ones, from 1. Each time you ...
-6
votes
1answer
208 views

Puzzling set of numbers [closed]

Let's have the following numbers $23, 40, 42, 44, \sqrt{43},\ 128i, 130, \sqrt{172}\ $. What is the relationship between these numbers, taken four at a time? There are only two combinations when you ...
-3
votes
1answer
261 views

Traps on Sudoku grids [closed]

Let's have a 9x9 Sudoku grid. Where the dots are shown above, you are allowed to put the numbers 2, 4, 6, 8. Each of these appears seven, seven, seven, and six times. It does not matter which number ...
14
votes
2answers
1k views

Make a hexiamond star by hand

Using some or all of the hexiamonds (pictured), make a star. You may flip pieces. The usual tiling rules apply, no overlaps, no gaps. Use only one or none of each piece. Answer is unique. Target shape ...
-1
votes
1answer
69 views

Painting a grid with 3 colours such that there are no right-angled triangles of one colour

What is the largest rectangular NxM grid (by area) that can be painted with 3 colours, such that no three cells of the same colour form a right-angled triangle. N and M must be 4 or greater. We only ...
3
votes
3answers
316 views

A 3x3 grid of numbers with unique row and column medians

Can you place every number from 1 to 9 into a 3x3 grid such that the median of every row and column is a unique value? The median of a row is the number that is greater than one number and smaller ...
7
votes
2answers
287 views

A 4x4 grid of numbers with unique row, column and diagonal ranges

Can you place every number from 1 to 16 into a 4x4 grid such that the range of every row, column and two main diagonals is a unique value? The range of a row is the difference between its maximum and ...
5
votes
2answers
151 views

A 3x3 grid of numbers with unique row and column ranges

Can you place every number from 1 to 9 into a 3x3 grid such that the range of every row and column is a unique value? The range of a row is the difference between its maximum and minimum values (...
3
votes
1answer
169 views

Two dice with same probability for each sum mk2

Inspired by Two dice with the same probability for each sum? To cheat in a game of sums, you get yourself a pair of magic dice. That pair behaves in a wonderful way where each individual die is fair (...
5
votes
6answers
1k views

Two dice with the same probability for each sum? [duplicate]

A friend invites you to play a game. The game is using two standard six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 each. As usual, the dice are considered distinguishable, i.e. throwing a 1 with ...

1
2 3 4 5
12