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# Questions tagged [combinatorics]

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

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5 votes
3 answers
394 views

### What is the expected time this will take?

There are n white balls and m black balls in a bag. Select 2 balls randomly, if they're the same color, repaint both balls white, else repaint both balls black, then replace both balls in the bag. ...
• 171
11 votes
7 answers
2k views

### The hat-check problem

Six men go to the theatre and leave their hats at the cloakroom. During the play there is a power cut, the six men leave, but it's dark, so the hats are handed back to them at random. Someone says: ...
• 321
2 votes
1 answer
377 views

### How many of the 16 cells of the grid could contain the black dot?

Beginner puzzle This puzzle is intended to be suitable for people who are new to puzzle solving. Clarification: Both experienced solvers and new solvers are welcome to post solutions to this puzzle. ...
• 15.8k
15 votes
1 answer
2k views

### The shortest way from A1 to B1

On the square A1 of a regular chessboard is a regular dice, face 1 up and face 2 in front. The only allowed move is to rotate the dice by 90 degrees to an adjacent square. Find the shortest way from ...
• 581
5 votes
2 answers
893 views

### 8x8 grid with no unmarked L-pentomino

What is the minimum number of cells on a 8x8 chessboard that need to be marked so that the unmarked cells do not contain an L-pentomino? An L-pentomino looks like ...
• 1,498
1 vote
0 answers
100 views

### How many squares in a large square [closed]

How many squares of all sizes can be found on a large square made of 10000 small squares? I don't want to try them out or count them. I tried finding a pattern, as in a square made of 1^2 small square ...
1 vote
0 answers
124 views

### Strategy for determining graph traversal rules yielding exactly one distinct graph traversal under every permutation [closed]

I am looking for advice/guidance/solutions which can help me with the following problem: Imagine a spatial graph in the x,y-plane arranged in an m x n grid e.g. a 4x4 grid Each vertex in the graph ...
• 111
2 votes
1 answer
280 views

### Strategy for solving "Hypatian Enigma" puzzle (all lines add to 38)

The "Hypatian Enigma" puzzle consists of 19 hexagonal blocks numbered 1 through 19. The blocks are arranged in 5 rows. The first and last rows have 3 blocks, the second and fourth row has 4 ...
• 121
5 votes
1 answer
375 views

### Filling a rectangular grid with holes using tetrominoes

There is a rectangular grid of $R$ rows and $C$ columns. $R \times C \bmod 4$ of the cells are painted black, and all other cells are white. In other words, there are at least 0 and at most 3 black ...
• 17k
13 votes
2 answers
612 views

### The HH vs HT question

My puzzle is based on this tweet (image): Flip a fair coin 100 times—it gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, Alice gets a point; for each HT, Bob does, ...
• 32.6k
2 votes
0 answers
160 views

### A quantum physics puzzle [closed]

A new mathematical approach to quantum physics via graphs was found recently, see this paper for more details. It particular, it refers to papers by Anton Zeilinger, a Nobel laureate in physics of ...
• 231
0 votes
1 answer
212 views

### Tile dominoes in a 3x10 space [closed]

How many ways are there to tile 1x2 (unmarked) dominoes in a 3x10 space? This is a harder version of Tile dominoes in a 2x10 space, since that was too easy.
• 989
20 votes
3 answers
1k views

### Keys and Locks Puzzle

Let $a,b,n$ be positive integers in which $a,b\le n$. You are locked in a room, with $n$ distinguishable keys and $n$ distinguishable locks in it. You know that each lock can be unlocked by a unique ...
• 6,314
5 votes
2 answers
534 views

### Rubik's Cube with no two squares of the same color on any horizontal, vertical or diagonal line

Consider the 12 lines depicted below which all run through the centers of the 9 color squares of a single face of Rubik's Cube. Can you find a scramble such that on each line all squares have ...
• 1,861
5 votes
2 answers
533 views

### Number of 3x3 One Up puzzles

Rodolfo Kurchan created a wonderful new grid puzzle called One Up that you can play on his website. There is one main rule: Each horizontal and vertical sequence of N cells between walls, must contain ...
• 36.9k
2 votes
2 answers
396 views

### Tournament Scheduling Puzzle

I have an interesting real life problem that can be turned into an interesting puzzle pertaining to a tournament that can be represented in this way: I have 24 people which are assigned numbers 1 to ...
3 votes
5 answers
1k views

### Professor Rackbrane: Count the triangles

Professor Rackbrane has just given me the following puzzle as an example of those that interested his party at Christmas. Draw a pentagon, and connect each point with every other point by straight ...
• 15.8k
7 votes
1 answer
593 views

### The governor's birthday puzzle

The prison governor surveys the three petty thieves in his panopticon. "Next week is my birthday. I've a mind to free some of you, if you can solve a little puzzle of mine." "I'm going ...
• 912
3 votes
3 answers
278 views

### All poker hands from a single deck

This question suggested itself to me – and I found a solution – after I solved Can you balance this poker deck?. Take out two aces from the standard 52-card deck. Your challenge is to partition the ...
• 7,820
4 votes
1 answer
283 views

### Can you balance this poker deck?

You are given the digits 0 to 9 in 4 poker suits. Distribute them onto the 8 highest poker hands (make one of each) of 5 digits each. They are Royal Flush (9 to 5 of one suit) Straight Flush (...
• 389
14 votes
5 answers
2k views

### General attacking chessboard squares

This is a general version of this beautiful puzzle. Place any number of standard chess pieces on a 8x8 chessboard, such that there is at least 1 empty square attacked by exactly 1 piece, at least 1 ...
• 36.9k
-4 votes
1 answer
125 views

### Binary Grid Challenge and X & O Conundrum

You are given a 5x5 grid with some cells filled with either "X" or "O". Your goal is to fill the remaining cells with "X" or "O" following these rules: Each row ...
• 109
1 vote
1 answer
209 views

• 17k
6 votes
1 answer
284 views

### Permuting officers during a Chess960 game

There are... let me see... ah yes 960 different possible starting positions in Chess960. Suppose the players never move a pawn, or make a capture, but simply move their officers so that eventually ...
• 1,919
11 votes
6 answers
2k views

### The shorter the message, the larger the prize (version II)

This is a successor question to The shorter the message, the larger the prize . For completeness I will include the entire question even though only the numbers have changed. Solutions to this puzzle ...
• 8,057
14 votes
1 answer
1k views

### The shorter the message, the larger the prize

Andrei and Belle have been set a task by their “friend”, Carroll. Carroll has promised them money depending on how well they do. Carroll will give a 99 bit array to Andrei and a different one to ...
• 8,057
5 votes
1 answer
600 views

### What is the number of ways to spell French word « chrysanthème »?

As many people know, theoretically a lot of words have more than one way to be spelled. I just want to provide a single example from English language: the word "fish". As Bernard Shaw noted, ...
• 605
17 votes
5 answers
4k views

### 5 chess pieces dominating a 5x5 grid

This is a puzzle based on work by Rodolfo Kurchan. Can you place a pawn, a knight, a bishop, a rook and a king on a 5x5 chess grid, such that every empty cell is attacked by at least one piece? Note ...
• 36.9k
69 votes
1 answer
3k views

### Alphabet snake, master of camouflage

The alphabet snake is a master of camouflage. It finds a section of text in an old book or newspaper... ...crawls upon it... ...and disappears. Now see if your camouflage skills can match those ...
• 14k
5 votes
1 answer
347 views

### Nuts and screws

Imagine that you are given a box with n nuts and n screws. Each screw have different size (diameter) and on each screw there is ...
• 263