Questions tagged [combinatorics]
A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]
792
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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 ...
- 1,317
70
votes
7
answers
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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 ...
user88
68
votes
3
answers
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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!
- 33.2k
68
votes
1
answer
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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 ...
- 13.5k
53
votes
4
answers
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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 ...
- 1,003
51
votes
5
answers
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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 ...
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47
votes
4
answers
10k
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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 ...
- 2,555
45
votes
4
answers
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A colorful dodecahedron
Divide a "base" edge of a regular pentagon into three equal parts. Then draw two lines from the base to the center of the other edges such that the lines do not intersect. This splits the ...
- 1,599
44
votes
3
answers
4k
views
Controlling a robot blindfolded on a 9x9 grid
A robot is located somewhere inside a 9x9 grid shown below, but you don't know where it is. You can send commands to the robot to make it move one cell left, right, up or down. Shaded areas and edges ...
- 33.2k
43
votes
3
answers
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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 ...
- 32.1k
41
votes
2
answers
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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 ...
- 6,802
38
votes
9
answers
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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 ...
- 116k
38
votes
5
answers
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Magnets on a whiteboard
Alice enjoys placing magnets on a magnetized whiteboard.
This day, she placed all 16 magnets in her possession on the board in a rectangular fashion.
o o o o
o o o o
o o o o
o o o o
"...
- 2,237
37
votes
9
answers
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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?
- 45.4k
36
votes
5
answers
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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 ...
- 8,304
34
votes
11
answers
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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 ...
- 7,060
34
votes
10
answers
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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 ...
- 341
34
votes
9
answers
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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 ...
- 1,198
33
votes
18
answers
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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....
- 1,079
33
votes
8
answers
3k
views
Two out of a dozen cartons have Easter eggs. Two people try to find one Easter egg carton, each using a different strategy. Who is expected to win?
I have found a counter intuitive puzzle. I have read the answer given at the source and understand it completely. But, what I am unable to understand is why my intuition turned out to be wrong. ...
- 2,818
32
votes
14
answers
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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 ...
- 7,438
31
votes
6
answers
2k
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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 ...
- 116k
31
votes
5
answers
6k
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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 ...
- 16.1k
30
votes
6
answers
5k
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A robot surviving on top of a 3x3 platform
A robot sits in the central square on top of a 3x3 platform. The robot can move up, down, left or right, but if it steps off the platform it will crash and die. You can preprogram the robot to make a ...
- 33.2k
30
votes
3
answers
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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 $2 \times 2$ ...
- 2,591
29
votes
4
answers
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 ...
- 33.2k
29
votes
5
answers
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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 ...
- 11.2k
28
votes
1
answer
4k
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 ...
- 441
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votes
4
answers
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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 ...
- 3,808
25
votes
12
answers
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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 ...
- 32.1k
25
votes
2
answers
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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 ...
- 2,346
25
votes
8
answers
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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 ...
- 16.1k
24
votes
7
answers
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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 ...
- 32.1k
24
votes
2
answers
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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 ...
- 45.4k
24
votes
10
answers
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7 mathematicians around the clock in prison
In a very strange kingdom, 7 mathematicians (let's call them Ann, Ben, Cid, Dan, Eve, Flo and Guy) were sent to prison because they did a calculation which was correct, but was not in favor of the ...
- 1,589
24
votes
1
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Selectively neglected collection
These mannequins are complete and ready for display.
These parts were found in a storage closet. Create four additional mannequins by assembling the parts appropriately and designing a suitable ...
- 13.5k
23
votes
4
answers
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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 ...
- 8,304
23
votes
5
answers
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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:...
- 33.2k
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votes
5
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What's the fewest weights you need to balance any weight from 1 to 40 pounds?
Suppose you want to create a set of weights so that any object with an integer weight from 1 to 40 pounds can be balanced on a two-sided scale by placing a certain combination of these weights onto ...
user88
23
votes
3
answers
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 ...
- 2,586
23
votes
4
answers
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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 ...
- 1,313
23
votes
4
answers
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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 ...
- 247
22
votes
8
answers
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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?
- 33.2k
22
votes
6
answers
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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 ...
- 45.4k
21
votes
4
answers
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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?
- 2,346
21
votes
5
answers
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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 ...
- 4,992
21
votes
5
answers
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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 ...
- 211
21
votes
2
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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, ...
- 32.1k
21
votes
4
answers
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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 ...
- 45.4k
21
votes
1
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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 ...
- 2,346