Questions tagged [combinatorics]
A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]
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The Flippin' Magician's 7-card Grand Finale
This question is a followup to this question by @ais523, which itself was a followup to this question by @Wen1now.
After touring the globe to accolades when performing his 10-card trick and 8-card ...
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votes
2
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Cross the pond, but there's a catch!
There is a square pond, conveniently divided into segments, with coordinate $(0,0)$ in the bottom left and $(10,10)$ is the top right.
You have planks length $2$ and $3$. You start at $(0,0)$ and ...
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Transferring 9 pegs on a 9x9 grid
You are given a 9x9 grid with a set of 9 pegs (red circles) arranged in a 3x3 pattern in the corner, as shown below:
A peg can jump over another adjacent peg in any direction (horizontal, vertical or ...
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votes
1
answer
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Consecutive numbers that are Manhattan distance 5 apart
Can you place numbers from 1 to 36 on a 6x6 grid, such that the distance between any two consecutive numbers ($a$ and $a+1$) is Manhattan distance 5?
Bonus question: can you also make 1 and 36 be ...
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Consecutive numbers that are Manhattan distance 3 apart
Can you place numbers from 1 to 16 on a 4x4 grid, such that the distance between any two consecutive numbers ($a$ and $a+1$) is Manhattan distance 3?
Bonus question: can you also make 1 and 16 be ...
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vote
2
answers
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Painting a 6x6 with 3 colours
Can you paint a 6x6 grid in red, green and blue, such that its every 3x3 sub-grid contains exactly 5 red, 3 green and 1 blue cell?
Good luck!
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4
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3x3 self-descriptive squares
A self-descriptive square is a square grid filled with integers such that:
The sum of the numbers in any row describes the number of times that
row’s rightmost number appears in the square.
The sum ...
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Generating Roman numerals with dice
This puzzle is closely based on this one: Generating numbers with cubes
Now we want to generate Roman numerals by placing up to three 6-sided dice side by side. We are allowed to write multiple ...
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0
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What are my sisters' ages? (With ice cream!) [duplicate]
This is from a book I read as a child.
Steve said to his friend Jessica, "I have 3 sisters. The sum of their ages is the same as my age, and the product of their ages is 36. How old are my ...
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Generating numbers with cubes
I saw an interesting calendar in a shop. It is composed of two cubes with numbers written on their 6 sides. By placing these cubes side by side one can make any day of the month from 1 to 31 (even 32)....
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TripTog's problem with his socks [closed]
Our friendly three-footed alien TripTog has two triplets of socks, which he keeps in a drawer in a room.
Each triplet of socks is labeled 1, 2 or 3, because TripTog is very meticulous about which ...
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Paint 10 cells of a 10x10 grid
Can you paint 10 cells of a 10x10 grid such that the largest unpainted rectangle has area of 10 cells?
Here is a similar question for the 7x7 grid: Paint 7 cells of a 7x7 grid
Good luck!
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votes
2
answers
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Paint 7 cells of a 7x7 grid
Can you paint 7 cells of a 7x7 grid such that the largest unpainted rectangle with grid-aligned sides has an area of 6 cells?
Good luck!
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vote
0
answers
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Triangle of numbers [duplicate]
You can place each number from 1 to 10 into a triangle, such that each number below the first row is the absolute difference of the two numbers above it:
...
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votes
1
answer
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Neighbouring numbers summing to a prime on a 4x4
Can you place every number from 1 to 16 on a 4x4 grid such that every pair of neighbouring (horizontally and vertically) numbers sum to a prime? Note that the generated primes can be reused.
A ...
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votes
3
answers
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Neighbouring numbers summing to a prime on a 3x3
Can you place distinct numbers from 0 to 9 on a 3x3 grid such that every pair of neighbouring (horizontally and vertically) numbers sum to a prime? Can you find multiple solutions? Note that the ...
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votes
4
answers
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Sharing cake among 9 or fewer people
You are expecting guests to your birthday party. You know that there will be at most 8 guests, but you don't know how many will actually come. What is the smallest number of pieces you should divide ...
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1
answer
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Sudokus and combinatorics
Considering your suggestions, I have redrawn the 12 X 12 Sudoku so that the twelve rectangles are now visible. The question remains the same:
Can someone construct a 12 X 12 Sudoku with the following ...
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votes
3
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Painting a 10x10 grid with 3 colours
Can you paint a 10x10 grid with 3 colours such that it doesn't contain any rectangles whose corners are all the same colour? Rectangles must be 2x2 or greater and parallel to the grid's sides. ...
12
votes
3
answers
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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 ...
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votes
1
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Arrange 20 counters and remove 6 form them and not a single square can be indicated?
Arrange 20 counters in the
form of a PLUS(+), as you can see
in the image.
Now, how many
different ways are there in which four
counters will form a perfect
square if considered alone? Thus the
four ...
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votes
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All numbers twice in a 7x7 Minesweeper grid
Can you place mines on a 7x7 Minesweeper grid such that each number from 0 to 8 appears exactly twice?
A similar question about a 5x5 grid:
All numbers in a 5x5 Minesweeper grid
Good luck!
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votes
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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!
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votes
1
answer
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Attacking queens revisited
Can you place 5 queens on a 11x11 chess board such that they can attack every empty square?
It turns out this problem was answered here:
http://golovolomka.hobby.ru/books/gik/04.shtml
Can you find ...
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votes
1
answer
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Attacking queens
Can you place 3 queens on a 6x6 chess board such that they can attack every square?
Good luck!
3
votes
1
answer
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Arranging people to be in a group with every other person at least once
There are 12 people. These people start in 4 groups - each with 3 people. They swap groups 4 times, so they are in a total of 5 groups. Is it possible for each person to be in a group with every other ...
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votes
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rearrange the numbers as specified rules - Combinatorial puzzle
1 8 9
4 7 0 3 5 2 6
Arranged them so that the numbers in a way
that the 3-sides added up alike-that
is, to 16.
Can you ...
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3
votes
2
answers
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Rearrange these 9 digits - combinatorics puzzle
4 3 2 7 1 9 6 5 8
Can you rearrange these 9 digits so that in all of the 8 directions the difference between one of the digits and the sum of the remaining two shall always be the same?
In the ...
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7
votes
1
answer
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Manually mixing words
Inspired by my struggles with this puzzle.
In that puzzle, I needed to mix two words, meaning blend them together without changing the order. For example, "ab" mixed with "cd" would give us 6 new ...
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votes
2
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12 balls 1 scale with 4 groups
So, we all know the famous 12 balls 1 scale riddle. It has been here repeated many times, however, the provided solutions always start with splitting the twelve balls into three groups of four. This ...
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Go for the Gold
You are given a bag containing 1 and 2 ounce gold rounds.
You need to draw one coin at a time till they Sum up to ten rounds.
How many different ways you can achieve that?
What is the quickest path ...
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Approximate this big number using a binomial [closed]
Mr. Magico is a greater believer in this number:
$$2^{50}=1,125,899,906,842,624$$
He also like to play cards, although he isn't fussy about the size of his deck, and nor does he care how many ...
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Cup and Trade: The Perfect Nutmeg Soup
Your package from Orinoco has finally arrived!
It's the Master Chef's Environmentally-Friendly Measuring Cup Set. It comes with 64 measuring cups having a volume of 1 cup, 1/2 cup, 1/3 cup, 1/4 cup, ....
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It is not as Simple as it Looks to get the right Alignment
Of these US Coins (Quarter and Dime Combo).
Initial arrangement of the coins as shown in the picture is as follows... Q D Q D Q D Q
Objective is to attain the final configuration... Q Q Q Q D D D.
...
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votes
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How many codes are possible?
The line - codes we are looking at consist of black and red lines. These lines can have width 1 or 2. Black and red lines are taking turns, black line, red line, black line, ... The code ends and ...
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votes
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5 cars in a roundabout traffic
Five cars are driving in a roundabout traffic at the same moment. Each comes from an other direction, and drives less than one full round. Also each car leave the roundabout traffic in an other ...
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votes
1
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A Prime Length Rope into Prime length ropes
We have a rope with a prime unit length, and we need to divide this rope into $9$ ropes by cutting it with prime and/or 1 unit lengths. After cutting the rope, you are supposed to find any kind of ...
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votes
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Savage Road Signs (Part 3)
You only need to have read Part 1 to understand this question, reading Part 2 will only help understanding the epic storyline.
Your daughter refuses to talk to you even though you have (once more) ...
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10 coins, 3 of them are fake
Inspired by some great weighing puzzles here (This being one of my favorites), I just made another weighing puzzle - I'm not quite sure how difficult or easy this one is.
You are given 10 coins, 7 of ...
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How many possible starting positions are uniquely solvable for a nonogram puzzle?
This type of puzzle goes by many names: Nonogram, Picross, and Griddlers are all mentioned on the Wikipedia page, Simon Tatham calls it Pattern, I was introduced to it as Descartes Rainbow, ...
The ...
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Savage Road Signs (Part 2)
Please read part 1 or this might be confusing
Since part 1, you have replaced the stolen stickers and your daughter has forgiven you. The highway ended up being a full 700km long, so you are happy ...
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16 people, make 3 consecutive round of 4 players
I am struggling with this issue for a tournament I am planning.
Take 16 people.
Consider a 4-player game.
To be clear, in each round all 16 players must play the game once.
I want to make at least 3 ...
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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 ...
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Finding numbers having exactly two distinct digits
We have $10^K$ road signs (numbered 0 through $10^K−1$).
For each valid $i$, the sign with number $i$ has the integer $i$ written on one side and $10^K−i−1$ written on the other side.
We need to ...
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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 ...
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Restoring order in a deck of playing cards (II)
Michelle has a deck of 52 playing cards in a pile with their backs facing up. She takes the first 2 cards in the pile, turns them over, and places them at the bottom of the pile. She now takes the ...
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3
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Restoring order in a deck of playing cards (I)
Michelle has a deck of 52 playing cards, stacked in a pile with their backs facing up. She takes the first 2 cards in the pile, turns them over, and places them at the bottom of the pile. She now ...
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A Guide to the Number Rotation Puzzle
This is an extension of What is the strategy to solve Simon Tatham's Twiddle? in that it explicitly goes beyond the default gamemodes of Twiddle
The Number Rotation Puzzle (NRP) is a combination ...
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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 ...
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How to Modernize Student Council
You are the math teacher at a high school and you are in charge of organizing Student Council for the whole school next year. Your boss, the Principal, read a research paper on Student Councils and ...
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