Questions tagged [combinatorics]

A puzzle based on combinatorial mathematics, which is the study of finite or countable discrete structures.

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80
votes
18answers
15k views

Nine gangsters and a gold bar

One night nine gangsters stole a gold bar. When the time came for dividing the bar, they faced a problem: two of the criminals put guns to each other's faces. Now it's up to fate whether one of them ...
55
votes
3answers
5k views

All numbers in a 5x5 Minesweeper grid

Can you place mines on a 5x5 Minesweeper grid such that each number from 0 to 8 appears exactly once? Good luck!
52
votes
9answers
75k views

Twelve balls and a scale

You are given twelve identical-looking balls and a two-sided scale. One of the balls is of a different weight, although you don't know whether it's lighter or heavier. How can you use just three ...
52
votes
4answers
5k views

Is this chromatic puzzle always solvable?

I've created a new puzzle from an Alexey Nigin's idea. It consists of a 8x8 board where each square is randomly assigned one of three colors. A movement is defined by picking any two orthogonal ...
46
votes
5answers
6k views

A man possesses a large quantity of stamps

James Joseph Sylvester was one the greatest British mathematicians of the 19th century, who made many fundamental contributions to number theory, combinatorics, and invariant theory. In 1884, he ...
43
votes
3answers
3k views

Relabeling two 20-sided dice without changing their total

Usually, 20-sided dice are labeled with the numbers 1 to 20. When you roll two of these, their sum is a random number between 2 and 40. The total 21 is most likely to occur, while 2 and 40 are the ...
37
votes
2answers
4k views

Pirate democracy at its finest

With our pirate crew becoming too big, the captain grew very concerned about splitting all the treasure - we continued to split it equally, but, of course, each crew member got less and less with the ...
37
votes
3answers
5k views

The coolest checkerboard magic trick

In the small town of Terni (Italy), there's a couple of young friends named Marco and Leonardo, who like to perform magic tricks to a restricted audience of common friends and relatives. They like to ...
36
votes
5answers
5k views

My grandfather's socks

My grandfather has a big drawer where he keeps his socks. The drawer contains more than 900 but less than 1000 individual socks. Each of his socks is black or blue, and there are more blue socks than ...
36
votes
9answers
4k views

Filling an 11-by-11 square

Is it possible to fill the $121$ entries in an $11\times11$ square with the values $0,+1,-1$, so that the row sums and column sums are $22$ distinct numbers?
34
votes
9answers
5k views

Can political debates really work?

In the far-off country of Politica, there are three main parties: the Left, the Right, and the Centre. In the last election, there were 19 million Left voters, 21 million Right voters, and 23 million ...
33
votes
9answers
4k views

Winning Strategy for the Magician and his Apprentice

There are $13$ upside-down opaque cups and $2$ balls, a magician and his apprentice and yourself. You decide under which cups to put the balls, and the objective of the magician is to find the two ...
32
votes
11answers
3k views

Coin weighing with a single weighing device

You have 12 coins which each weigh either 20 grams or 10 grams. Each is labelled from 1 to 12 so you can tell the coins apart. You have one weighing device as well. At each turn you can put as many ...
30
votes
18answers
30k views

Draw a line through all doors

I saw the following problem on 4chan and couldn't solve it: It's very likely to be some kind of troll (no solution). I'm hoping to see some rigorous proofs that disprove the existence of such a line....
30
votes
6answers
2k views

A triangle formed of three letters

Consider a downward-pointing triangle of letters X, Y, and Z formed by the following algorithm. Start with a sequence of 10 letters each of which is X, Y, or Z. Under each row, construct the next row ...
30
votes
8answers
2k views

Wolves and sheep

All the sheep were living peacefully in the Land of Shewo. But suddenly they were struck by a danger. A few wolves dressed up as sheep entered the territory of Shewo and started killing the sheep one ...
30
votes
5answers
6k views

Two spies throwing stones into a river

There is a puzzle about two spies: Two spies must pass each other two secret numbers (one number per spy), unnoticed by their enemies. They have agreed on a method for doing this using only 26 ...
29
votes
13answers
4k views

15 Balls Sorting

This is a variant of 15 Balls Weighing. You have 15 balls of 15 different weights, but the weights are so similar you can't tell them apart by feel. The balls are also identical by any other sense ...
26
votes
3answers
2k views

One hundred tiles

One hundred tiles are arranged in a $10 \times 10$ square. Each tile is black on one side and white on the other side. Two types of move are allowed: Flip over all four tiles in any $2 \times 2$ ...
26
votes
1answer
3k views

The tough one from “A Brilliant Young Mind” (2014)

Great movie by the way. I'm quoting from memory, so I may get the wording wrong. The positive integers are each colored Red, Yellow or Green. Prove that for any such coloring, there must exist ...
25
votes
2answers
4k views

Two chessmasters at work

Viswanathan Anand plays a chess game against Magnus Carlsen. Anand plays white and Magnus plays black. They use a non-standard digital double chess clock that is counting up from zero (instead of the ...
25
votes
2answers
2k views

Rooks on a 15x15 chessboard

On a 15x15 chessboard there are 15 rooks that do not attack each other (via ordinary rook moves). Then each of the rooks makes one move like that of a knight. Is it possible that after all this is ...
24
votes
4answers
11k views

Plant 9 trees in 10 rows of 3

"Tree-planting" puzzles are also known as "points and lines" puzzles. The English puzzle author and mathematician Henry Ernest Dudeney was very fond of them. In 1917, Dudeney published a collection ...
24
votes
4answers
2k views

Savage Road Signs

There is a highway that starts in the city of Savage. You must must place distance marker signs on this highway for the outgoing traffic. According to highway code, there must be a distance marker ...
23
votes
7answers
5k views

Hacking an electronic keypad

You are a spy trying to break into an enemy facility. The back door is protected by an electronic keypad lock. You know that this particular lock is opened by a four digit code. Any stream of button ...
23
votes
8answers
4k views

How to choose at least half of everything

Some number of gold, silver, and copper coins are scattered in $N$ chests. You may look into each chest and count each type of coin in them, and then select $M$ of the chests. Your goal is to have at ...
22
votes
3answers
4k views

bored of eating soup

A man orders spicy noodle and leek soup from a restaurant, but gets bored while eating. When he gets bored, there are exactly 100 noodles in the soup. Because he is bored, he decides to play a game ...
21
votes
4answers
4k views

A magic trick of David Copperfield

David Copperfield puts $52$ cards numbered $1$ to $52$ into three top hats. One of these top hats is red, one is blue and one is yellow, and each of them receives at least one card. Then David ...
21
votes
6answers
2k views

A game with 52 cards

Alice and Bob play the following game with two (identical) standard decks of $52$ cards. First Alice secretly arranges one deck of $52$ cards in a long row on the table. All cards are face-down, and ...
21
votes
4answers
2k views

Even and Odd game

You are playing a game with your friend on a $7$x$7$ grid board. In every turn, you begin by putting a $0$ (zero) on any empty square on the board, and then your friend puts a $1$ (one) on a ...
20
votes
5answers
980 views

Three Dice minimum value

You shall form three dice, placing 18 distinct integers on the faces of three cubes. Your goal is to be able to obtain all the integers between 1 and 216, inclusive, as the sum of the integers on the ...
20
votes
2answers
2k views

One Hundred Lockboxes of Wood and Steel

A bit beyond perceptions reach, I sometimes believe I see that life is two locked boxes, each containing the other’s key. ― Piet Hein You have one hundred lockboxes, fifty made of steel, ...
19
votes
6answers
4k views

How many digits can you create with only one seven segment display?

Let say you have a digital indicator which consists of seven lights as you see below. You are trying to make as many distinct digits as possible but there are two requirements for that: At least one ...
19
votes
4answers
3k views

Mutilated chessboard

Remove the square in the top-left corner of a $2015 \times2015$ chessboard. Can the remaining mutilated chessboard be tiled with $1\times4$ and $4\times1$ rectangles?
19
votes
5answers
7k views

Flip a Fair Coin

I found this question and became curious, can anyone tell me the answer and prove it, i know it seems fairly simple but just thought an explanation of this would make an interesting case. Flip a fair ...
19
votes
4answers
1k views

The frog concerto

A pond contains $24$ waterlilies that are arranged in a rectangular $2\times12$ grid (that is, two rows with twelve waterlilies). One evening $24$ frogs give a croaking concerto for the residents of ...
19
votes
1answer
1k views

Dominos on a checkerboard

What's the maximal number of dominos (2x1 tiles) that can be placed on a checkerboard (8x8 square) so that every domino covers exactly 2 squares of the checkerboard and no two dominos form a 2x2 ...
19
votes
6answers
2k views

My Sixteen Friendly Students

I have sixteen students in my class who sit in four rows of four. Each week they sit in a different order. After a number of weeks every student has sat next to every other student, next meaning ...
19
votes
2answers
2k views

“Legally” filling a 10x10 table with 10 of each digit

Professor Halfbrain has spent his entire weekend by filling $10\times10$ tables with the digits $0,1,2,3,4,5,6,7,8,9$ so that each digit occurs exactly $10$ times. According to the professor, such ...
19
votes
5answers
1k views

Island of Knights, Knaves and Spies

There is an island with $N$ inhabitants (for example $A_1, A_2, \dots, A_N$), each of them is either a knight, a knave, or a spy. As usual: knights will always tell the truth upon answering a ...
19
votes
4answers
1k views

Magician's hide and seek with 8 cards

This question is a followup to this question by @Wen1now. After demonstrating the previous trick, the magician decided to make things a bit harder, discarding some cards so that only eight were left. ...
19
votes
1answer
428 views

Filling a 13-by-13 square

Is it possible to arrange the integers $1,2,3,\ldots,169$ in a $13\times13$ square, so that in every $2\times2$ square the sum of the four numbers is divisible by $170$?
18
votes
6answers
3k views

Guess five binary digits!

Person A thinks of a 5 digit binary number. Person B tries to guess the number. B can guess a 5 digit binary number and A will respond with the number of correct digits (digits in the right place). ...
18
votes
3answers
2k views

Tiling a rectangle with nine squares

A rectangle is tiled by nine squares with side lengths $2,5,7,9,16,25,28,33$ and $36$ (without overlapping and without gaps). What are the side lengths of the rectangle? What does the tiling look ...
18
votes
6answers
1k views

The Ball Factory Worker

A woman works at a ball factory. She's instructed to open up a ball creation kit composed of one red rubber ball and two strips of digit stickers from $0$ through $9$. She's instructed to do the ...
18
votes
4answers
2k views

The 8-dimensional vegetable kebab

You are given two of each from the array of 8 vegetables numbered 1 to 8 as shown above. So in total you have 16 veggies(8 pairs). Your task is to make the longest kebab (sequence of vegetables ...
18
votes
4answers
2k views

Consecutive Towers of Hanoi

Consider the following variant of the Towers of Hanoi puzzle. There are six pegs. One of the pegs has a stack of $n$ differently sized disks, sorted by size so the smallest disk is at the top. All ...
18
votes
3answers
2k views

Cube made of 2x2x1 blocks which blocks all light

An Egyptian pharaoh wants to build a monument to the Sun God, Ra. He wants this to be a solid stone cube, which is $20$ hectocubits tall, long and wide. His engineers get right to work planning this....
18
votes
9answers
3k views

Tiling a Hexagon with Diamonds

A regular hexagon is divided into a triangular grid, and completely tiled with diamonds (two triangles glued together). Diamonds can be placed in one of three orientations. Prove that, no matter how ...
18
votes
3answers
2k views

Puzzling 101: Sums of seventeen numbers

After reading a bit in Meta and the help section, I'm still not sure whether this qualifies as a "good" puzzle in the spirit of the site, but the votes will show - I liked it when I encountered it, ...