Questions tagged [combinatorics]
A puzzle based on combinatorial mathematics, which is the study of finite or countable discrete structures.
509
questions
6
votes
2answers
484 views
10x10 grid with no unpainted hexominoes
What is the smallest number of cells you need to paint in an 10x10 grid, such that it contains no unpainted hexominoes? Note that a hexomino is a set of 6 adjacent cells (horizontally or vertically).
...
4
votes
2answers
388 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 ...
4
votes
1answer
157 views
Prime magic star
Can you replace the letters with 10 consecutive primes such that the sum of numbers on each line is equal? I expect this to be solved with a computer.
Good luck!
3
votes
1answer
150 views
Partition a 3x3 square into rectangles [closed]
Yesterday I watched "The man who knew infinity" about the amazing Ramanujan. Inspired by the partitions problem from the movie I came up with a puzzle: In how many ways can you partition a 3x3 grid ...
5
votes
3answers
209 views
Rawrdon Mamsay pays a visit
Now, I should warn you, this is one of my practical problems; meaning I don't know the solution and the answer's probably anticlimactic (like this or that). Still...
My old pal Rawrdon Mamsay is soon ...
12
votes
3answers
2k views
Find number 8 with the least number of tries
You and your friend plays a game. In this game, there are numbers written on the cards from 1 to 12 on one side (so 12 cards in total). The other side is blank. You can write anything on the blank ...
8
votes
1answer
651 views
A curious 5x5 square
Can you fill a 5x5 grid with numbers from 1 to 5, such that every number occurs exactly once in each row, exactly once in each column and exactly once in each broken diagonal (in both directions)? ...
0
votes
2answers
103 views
Painting edges of a 3x3 grid with 4 colours
Can you paint the edges of a 3x3 grid with 4 colours, such that:
The colours of edges of every 1x1 square are different.
The colours of edges adjacent to every vertex are different.
Here is a ...
0
votes
1answer
64 views
Painting edges of a 2x2 grid with 4 colours
Can you paint the edges of a 2x2 grid with 4 colours, such that:
The colours of edges of every 1x1 square are different.
The colours of edges adjacent to every vertex are different.
Good luck!
8
votes
0answers
152 views
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 ...
3
votes
2answers
186 views
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 ...
15
votes
3answers
1k views
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 ...
5
votes
1answer
313 views
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 ...
12
votes
4answers
2k views
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 ...
1
vote
2answers
142 views
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!
8
votes
4answers
1k views
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 ...
17
votes
5answers
1k views
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 ...
1
vote
0answers
80 views
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 sisters?"...
13
votes
5answers
2k views
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)....
0
votes
1answer
73 views
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 ...
7
votes
3answers
793 views
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!
5
votes
2answers
534 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!
1
vote
0answers
35 views
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:
...
1
vote
1answer
85 views
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 ...
4
votes
3answers
489 views
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 ...
10
votes
4answers
705 views
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 ...
4
votes
1answer
111 views
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 ...
5
votes
3answers
295 views
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. ...
9
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 ...
4
votes
1answer
98 views
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 ...
3
votes
2answers
649 views
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!
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!
4
votes
1answer
158 views
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 ...
6
votes
1answer
134 views
Attacking queens
Can you place 3 queens on a 6x6 chess board such that they can attack every square?
Good luck!
3
votes
1answer
68 views
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 ...
3
votes
1answer
82 views
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 ...
4
votes
2answers
163 views
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 ...
7
votes
1answer
151 views
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 ...
4
votes
2answers
125 views
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 ...
-4
votes
4answers
166 views
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 ...
0
votes
3answers
141 views
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 ...
12
votes
5answers
763 views
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, ....
0
votes
2answers
86 views
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.
...
2
votes
2answers
351 views
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 ...
3
votes
2answers
345 views
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 ...
5
votes
1answer
112 views
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 ...
6
votes
2answers
266 views
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) ...
14
votes
7answers
707 views
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 ...
12
votes
3answers
653 views
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 ...
7
votes
4answers
297 views
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 ...
