Questions tagged [combinatorics]

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

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4
votes
2answers
168 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). ...
3
votes
2answers
164 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
107 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
129 views

Partition a 3x3 square into rectangles [on hold]

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
177 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 ...
11
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
628 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
81 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
51 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
107 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
162 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 ...
14
votes
2answers
848 views
+50

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
293 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
136 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
75 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
1k 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
67 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
761 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
500 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
32 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
59 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
451 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 ...
9
votes
2answers
496 views
+50

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
98 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
260 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
90 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
615 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
150 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
122 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
65 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
76 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
152 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
144 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
112 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
161 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
138 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
758 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
346 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
328 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
91 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
263 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
655 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
608 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
292 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 ...