Questions tagged [combinatorics]
A puzzle based on combinatorics, which is the study of counting discrete structures.
161
questions
60
votes
9answers
96k 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 ...
10
votes
4answers
10k views
N balls and a scale
The question of twelve balls and a scale is probably the best-known example of the "find the ball of a different weight" problem. But does it generalize? Is there a general way to find a weighing ...
41
votes
3answers
8k 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 ...
12
votes
4answers
1k views
Nerds, Jocks, and Lockers
Here's an oldie but goodie from The Daily WTF; I paraphrase to avoid copyright issues:
A middle school math teacher, who also happened to be the P.E. coach, made the following deal with the non-...
16
votes
3answers
23k views
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 ...
32
votes
18answers
47k 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....
15
votes
1answer
1k views
One beer too many
You're standing outside your apartment building after a late night out, with perhaps one beer too many, and you realize you have completely forgotten the code to get in. Luckily, you're a mathematical ...
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?
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 ...
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 ...
9
votes
4answers
3k views
How many different non congruent polygons can you make on a 3x3 dot grid?
There is a $3\times3$ dot grid. How many different non-congruent polygons can you make on the grid?
Rules:
All vertices of the polygon must be on the grid
Only non self intersecting polygons
Only ...
34
votes
9answers
4k 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 ...
26
votes
4answers
14k 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 ...
41
votes
3answers
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 ...
16
votes
1answer
2k views
A classical combinatorial puzzle
It is a classical puzzle by Edsger Dijkstra.
Not quoting the original problem but changing it into bag and balls, the puzzle is:
A bag contains some black and white balls. The following process is to ...
12
votes
5answers
2k views
99 Bags of Apples and Oranges
You have $99$ bags, each containing various numbers of apples and oranges. Prove that there exist $50$ bags among these which together contain at least half the apples and at least half the oranges.
...
6
votes
2answers
3k views
Minimum moves to have all coins face Heads up
Given a circular list of coins, that all have Tails facing up.
In each move, if we flip the coin at position $i$, then the coins at positions $i-1$ and $i+1$ get flipped as well. That is, consider: H ...
9
votes
3answers
1k views
Calendar Cubes are Impossible!
An easy one for the middle of the week!
Jane and John were discussing a business idea. John wanted to make a little set to keep track of the date. It was to include two cubes with a single digit on ...
8
votes
2answers
792 views
Termite eating through a large cube composed of 27 smaller cubes while not moving diagonally
The is a large cube formed by gluing together 27 smaller cubes of uniform size (see figure). A termite starts at the center of a face of any of the outside cubes and bores a path that takes him once ...
7
votes
3answers
7k views
Flip one of the 64 coins on chess board [duplicate]
I know the answer of this puzzle. I want to know does this puzzle works perfectly for every n × n chess board? Is there a upper bound to n?
What is the upper bound for n, if m coins are allowed to ...
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 ...
47
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 ...
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 ...
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?
20
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. ...
22
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 ...
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 ...
20
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 ...
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 ...
15
votes
6answers
2k views
15 Distinct Weights' Sorting
There are $15$ balls. Each of them has a different weight.
You want to sort them according to their weights.
You have a friend who will help you with his scale to do this.
At each weighing process, ...
24
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 ...
14
votes
2answers
758 views
How many ways to pick the fruit
Alice and Bob have a farm in the form of a $20 \times 20$ grid, each square in the grid containing the same amount of fruit. They both live in the lower-right square. Luckily for them, Alice is a ...
6
votes
2answers
560 views
Paint 7 cells of a 7x7 grid
Can you paint 7 cells of a 7x7 grid such that the largest unpainted rectangle has area of 6 cells?
Good luck!
36
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 ...
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
5answers
1k views
Board with all 2020s
Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain ...
14
votes
2answers
1k views
Crazy Dice - Unusual Dice with Usual Sum
Can you design two six-sided dice (different from the standard ones), where each face has a nonzero number of dots, so that the probability distribution of their total is the same as for two standard ...
9
votes
1answer
2k views
Three button calculator
A calculator has only 3 buttons. The first multiplies the current value by 3, the second adds 2 and the third subtracts 2. Starting with 0 what is the least number of presses you need to reach 100?
5
votes
2answers
3k views
Finding the number of ways of crossing a river
There are $X$ stones in the river.The stones are placed in such a way that a person can jump from one stone to the next one, or skip one and jump to the one after that. Find the number of ways in ...
4
votes
3answers
1k views
Total no of squares on a Chess Board
Is there any formula than calculates the total number of squares on chessboard?
For example in a $8\times8$ chessboard, there are squares of sizes $1\times1$, $2\times2$, $\ldots$, $8\times8$.
So I ...
12
votes
2answers
2k views
How many squares can you make with equal ranged points?
This question is directly related to How Many Squares on the Peg Solitaire.
Is it possible to formulate with a given dimension of equal ranged points $m\times n$ where $m,n\geqslant 2$?
For example;...
10
votes
3answers
1k views
Painting a 4x6 grid with 2 colours
Can you paint a 4x6 grid with 2 colours such that it doesn't contain any rectangles whose corners are all the same colour? Can you do it without a computer? Rectangles must be 2x2 or greater and ...
8
votes
3answers
2k views
Coloring an n by n grid with four colors
This is a generalization of Place 4x12 detainees on a 7x7 grid of cells.
The goal is to color the squares of an $n\times n$ grid with four colors such that
at most one square is uncolored
no two ...
7
votes
2answers
686 views
Peaceable Bishops on an 8x8 grid
Place an equal number of red, white and black bishops on a 8x8 chess grid, such that no two bishops of different colours attack each other. What is the largest number of bishops you can place? Bonus ...
6
votes
3answers
444 views
Orchard planting problem for circles
The classic Orchard planting problem asks for the maximum number of 3-point straight lines attainable from a configuration of $n$ points drawn on a plane.
Here we are interested in a variant of this ...
5
votes
1answer
635 views
Pythagorean coins
To make payments, the Pythagoreans use coins in no more than three denominations. The three denominations are in whole Oboloi amounts, and the sum of the squares of the two smaller denominations ...
4
votes
2answers
455 views
8x8 grid with no unpainted pentominoes
What is the smallest number of cells you need to paint in an 8x8 grid, such that it contains no unpainted pentominoes? Can you find multiple solutions? Note that a pentomino is a set of 5 adjacent ...
2
votes
1answer
266 views
A robot visiting every edge of a 3x3 grid
A robot is placed on the top-left vertex of a 3x3 grid. At each move the robot can take one step (up, down, left or right) along the edge of the grid to the adjacent vertex, but he cannot go outside ...
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 ...
10
votes
2answers
3k views
How many Strobogrammatic numbers are there from 0 to 99999
0,1,2,5,8,11,69,96 are Strobogrammatic numbers.
We call a Strobogrammatic numbers if:
When it is typed on a calculator, and the calculator is spun 180 degrees, the number visually looks the ...
