# Questions tagged [combinatorics]

A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]

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### Nimber mnemonic combinatorial puzzle

Please see my previous question for more background. The following represents an unfolded version of PG(3,2) with 1 as the center point: Given that each number must be an end point of a line which ...
204 views

### Nimber Mnemonics

Note I originally tried to ask a variation of this question on math.stack; however 1 commenter pointed out that math.stack is not a puzzle site, which made me think maybe the fine folks of puzzling ...
231 views

### Longest subsequences and shortest longest ones

This challenge is about permutations of the integers 1 to 30 with longest increasing subsequence length 3. An important part of the definition is that a subsequence is not necessarily contiguous or ...
161 views

### If there are 6 men and 6 women around a table, what's the probability that both groups are joined in a single cluster each? [closed]

Suppose we have twelve people: six men and six women. They randomly sat around a circular table. What's the probability that both male and female groups accidentally formed a single conjoined cluster ...
142 views

### Counting puzzle #1: Function combinations

Not in conjunction with my function optimization puzzles, also sorry for the extremely difficult discrete mathematics puzzle So as you may or may not know, I have recently uploaded 2 function ...
344 views

### 5x5 grid with a special colouring

Can you paint the cells of a 5x5 grid in 5 colours such that for each cell its colour and the colour of its orthogonal (horizontal and vertical) neighbours are all different?
2k views

### Playing Mastermind against an angel and the devil

This puzzle is based on a card game. There are 7 suspects and 3 of them committed a crime. The game contains 35 cards that contain the 35 possible choices of 3 out of the 7 suspects. One card is drawn ...
7k 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!
1 vote
230 views

### How many 4x4 Latin Squares are there?

I thought of this problem when playing Sudoku. Let A = {1,2,3,4}. I have to make a 4x4 box (i.e. the size of A in both dimensions) and fill it with data such that ...
327 views

### Permutations with given longest increasing subsequence

How many permutations of 1 to 20 are there with 2,5,6,9,13 as a longest increasing subsequence? (It may be tied with others.)
1 vote
1k views

### 16 queens puzzle

Can you place 8 white queens and 8 black queens on an 8x8 grid, such that no two queens of the same colour occupy the same row, column or diagonal?
393 views

### Closed path on a dodecahedron

Your task is to draw lines between edges on a regular pentagon such that if you tile a dodecahedron with 12 identical copies of that pentagon you get a single closed line which does not intersect ...
2k views

### 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 ...
152 views

### Tile 1x2 dominoes in a 2x10 space

How many ways are there to tile unmarked 1x2 dominoes in a 2x10 space? Bonus: What if the dominoes were identical and had pips on their front (face-up), so they could be distinguished by 180 degree ...
647 views

### Pay each amount with at most two coins

Euro cent coins come in the denominations 1, 2, 5, 10, 20 and 50 cents. You are inconvenienced by the fact that you need a lot of coins to pay each amount up to 100 cents. To pay 99 cents, you need 6 ...
270 views

### Permuting officers during a Chess960 game

There are... let me see... ah yes 960 different possible starting positions in Chess960. Suppose the players never move a pawn, or make a capture, but simply move their officers so that eventually ...
422 views

### Tiling a dodecahedron

The surface of a dodecahedron is tiled with 6 of the shown tiles, each tile covering two faces of the dodecahedron. In how many essentially different ways this can be done? Two tiled dodecahedrons are ...
1 vote
188 views

### Visiting all strings by swapping

Consider the following strings ...
399 views

### Counting Tic-Tac-Toe draws on larger grids

Alice and Bob play a game of Tic-Tac-Toe on a grid of size $N \times M$. The rules of this game are the same as the original Tic-Tac-Toe: Alice plays first (white); Bob plays second (black). On each ...
2k views

### Prime lights out

You start with a 4x4 grid filled with zeroes. If you press a cell then the cell and all its neighboring (horizontally and vertically) cells will have their numbers increased by 1. What is the most ...
2k views

### The shorter the message, the larger the prize (version II)

This is a successor question to The shorter the message, the larger the prize . For completeness I will include the entire question even though only the numbers have changed. Solutions to this puzzle ...
1k views

### The shorter the message, the larger the prize

Andrei and Belle have been set a task by their “friend”, Carroll. Carroll has promised them money depending on how well they do. Carroll will give a 99 bit array to Andrei and a different one to ...
591 views

### What is the number of ways to spell French word « chrysanthème »?

As many people know, theoretically a lot of words have more than one way to be spelled. I just want to provide a single example from English language: the word "fish". As Bernard Shaw noted, ...
1k views

### Five professors and nine dishes

Here's yet another puzzle adapted from a puzzle book (in my case, with edits to make some of the specifications of the puzzle more clear because I didn't really understand them the first time I read ...
4k views

### 5 chess pieces dominating a 5x5 grid

This is a puzzle based on work by Rodolfo Kurchan. Can you place a pawn, a knight, a bishop, a rook and a king on a 5x5 chess grid, such that every empty cell is attacked by at least one piece? Note ...
3k views

### 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 ...
331 views

### Nuts and screws

Imagine that you are given a box with n nuts and n screws. Each screw have different size (diameter) and on each screw there is ...
900 views

### Every tournament has a dominant player

A tournament was played round-robin: each pair of players played a match where one defeated the other. Prove that there was a player for which every other player either lost to them or lost to someone ...
1k views

### Save now! All the digits at half the price

... or double the price depending on where you're coming from Consider the set $PD10$ of pan-digital ten-digit numbers, i.e. positive whole numbers whose decimal representation has each of the digits ...
231 views

### Counting combinations with two dice

You are given two identical standard dice as shown below. You can stack them one on top of the other, or place them touching side by side. In all cases the face of one die must fully touch the face of ...
658 views

### Cable with mixed wires

Let's say you have a cable that has n wires. Each wire on the left side corresponds to one wire on the right side. However you cannot distinguish between the wires ...
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. ...
1 vote
207 views

### How many ways are there to mark a way to walk around every edge of the triforce?

A triforce for the purposes of this question is a plane figure with an equilateral triangle at its center, with one additional vertex connected to each pair of original vertices (forming an additional ...
401 views

### Clock hands get it Right

I was asked this question in an entrance exam. In one day, how many times the hour hand and the minute hand of a clock are at right angles to each other? My answer was 48. My reasoning was that during ...
310 views

### n*n*n Rubik's cube algorithm

Is there a universally working (I mean, regardless of n) algorithm for Rubik's cube n×n×n ? It is acceptable to divide ...
1k views

### 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 ...
1k views

### Magic-preserving Permutations on a 4x4 Magic Square

Messing around with some magic-square puzzles, I faced the problem of deciding whether some two magical squares are, in fact, the one and same square wearing a different hat. It seemed to me, that for ...
136 views

### Rotating teams through stations without repeating a topic?

I am putting together a gallery walk activity and want to rotate 6 teams through 4 unique “topics.” This activity will take place in a rectangular room. There will be 6 “stations” set up. Each station ...
357 views

### Micropoker: small hands on deck

Raise your hand if you are ready for micropoker, which minimalistically resembles 5-card poker. The deck has just 8 cards with 2 suits of 4 cards each. A hand is dealt as 3 cards that are final, with ...
469 views

### How many five-digit numbers have digit product 6! [closed]

The product of the digits of a five-digit number is $6!=720$. How many such numbers are there?
708 views

### Attacking Hyenas

$N$ Hyenas are standing on a plane region in a forest. At $t=-1$, they see dead meat nearby. Being selfish, at $t=0$, each Hyena attacks the Hyena which is closest to it. All pairwise distances ...
357 views

### Do non-trivial Skolem squares exist?

Define a Skolem sequence to be a permutation of the sequence of 2n numbers 0, 0, 1, 1, 2, 2, ..., n-1, n-1 in which there are no numbers between the two 0s (the 0s are in adjacent positions), there is ...
2k views

### Do Langford squares exist?

A Langford sequence is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which there is one number between the two 1s, there are two numbers between the two 2s, and more generally ...
178 views

### Creating a clever hemisphere

Given five points on a sphere, can you always draw an equator such that four or more points lie on one hemisphere? How? Points on the equator count as being on either side.
2k views

### Wizard of subsets

Can you change this into this in three moves? You are the wizard of subsets. With only your mind, you can grab any subset of the 16 stone blocks and move them one unit in any direction (north, ...
603 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 ...
646 views

### Running Out of Digits, level 2

The challenge idea, and images are credited to Andrew. You initially have 100 of each digit from 0 to 9. This means you have 1000 digits in total. This count for each digit is shown in the table ...
477 views

### Different numbers in all cells of a 3x3 board

This puzzle is inspired by this one: Board with all 2020s Zeroes are written in all cells of a 3×3 board. Pressing a cell increases by 1 the number in this cell and all cells having a common side with ...