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Questions tagged [combinatorics]

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

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Tile dominoes in a 3x10 space [closed]

How many ways are there to tile 1x2 (unmarked) dominoes in a 3x10 space? This is a harder version of Tile dominoes in a 2x10 space, since that was too easy.
qwr's user avatar
  • 773
19 votes
3 answers
1k views

Keys and Locks Puzzle

Let $a,b,n$ be positive integers in which $a,b\le n$. You are locked in a room, with $n$ distinguishable keys and $n$ distinguishable locks in it. You know that each lock can be unlocked by a unique ...
Culver Kwan's user avatar
  • 5,792
5 votes
0 answers
158 views

Rubik's Cube with no two squares of the same color on any horizontal, vertical or diagonal line

Consider the 12 lines depicted below which all run through the centers of the 9 color squares of a single face of Rubik's Cube. Can you find a scramble such that on each line all squares have ...
Herbert Kociemba's user avatar
5 votes
2 answers
296 views

Number of 3x3 One Up puzzles

Rodolfo Kurchan created a wonderful new grid puzzle called One Up that you can play on his website. There is one main rule: Each horizontal and vertical sequence of N cells between walls, must contain ...
Dmitry Kamenetsky's user avatar
2 votes
2 answers
381 views

Tournament Scheduling Puzzle

I have an interesting real life problem that can be turned into an interesting puzzle pertaining to a tournament that can be represented in this way: I have 24 people which are assigned numbers 1 to ...
Alex DeCarlo's user avatar
3 votes
5 answers
1k views

Professor Rackbrane: Count the triangles

Professor Rackbrane has just given me the following puzzle as an example of those that interested his party at Christmas. Draw a pentagon, and connect each point with every other point by straight ...
Will Octagon Gibson's user avatar
7 votes
1 answer
554 views

The governor's birthday puzzle

The prison governor surveys the three petty thieves in his panopticon. "Next week is my birthday. I've a mind to free some of you, if you can solve a little puzzle of mine." "I'm going ...
user23087's user avatar
  • 770
3 votes
3 answers
261 views

All poker hands from a single deck

This question suggested itself to me – and I found a solution – after I solved Can you balance this poker deck?. Take out two aces from the standard 52-card deck. Your challenge is to partition the ...
Parcly Taxel's user avatar
  • 7,720
4 votes
1 answer
272 views

Can you balance this poker deck?

You are given the digits 0 to 9 in 4 poker suits. Distribute them onto the 8 highest poker hands (make one of each) of 5 digits each. They are Royal Flush (9 to 5 of one suit) Straight Flush (...
Nurator's user avatar
  • 379
13 votes
5 answers
2k views

General attacking chessboard squares

This is a general version of this beautiful puzzle. Place any number of standard chess pieces on a 8x8 chessboard, such that there is at least 1 empty square attacked by exactly 1 piece, at least 1 ...
Dmitry Kamenetsky's user avatar
-4 votes
1 answer
112 views

Binary Grid Challenge and X & O Conundrum

You are given a 5x5 grid with some cells filled with either "X" or "O". Your goal is to fill the remaining cells with "X" or "O" following these rules: Each row ...
Martin.s's user avatar
  • 109
1 vote
1 answer
201 views

Maximum filled days

I have two types of items, $i_1$ and $i_2$. $i_1$ items can be used at most $50$ times and $i_2$ items can be used at most $120$ times. I have $7000$ items $i_1$ and $800$ items $i_2$. Each item $i\in ...
JKHA's user avatar
  • 6,033
5 votes
1 answer
960 views

Anna and Boris play the Red Blue game

Anna and Boris play a game on a 9x9 chessboard. Anna goes first and turns alternate thereafter. In each move, Anna puts a red counter on a vacant square while Boris puts a blue counter on a vacant ...
Will Octagon Gibson's user avatar
2 votes
0 answers
89 views

An array of light-bulbs [duplicate]

There is a grid of 2024 by 2024 light-bulbs. Initially, all bulbs are off. You can switch the state of all 2024 bulbs in a row simultaneously, or all 2024 bulbs in a column simultaneously, and do this ...
Erel Segal-Halevi's user avatar
18 votes
1 answer
670 views

Alphabetic string theory

According to alphabetic string theory, the words we see are merely the visible portions of highly convoluted loops of multi-dimensional alphabetic string. For example, here is what we see in our space-...
SlowMagic's user avatar
  • 13.8k
2 votes
1 answer
277 views

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 ...
stargirl's user avatar
2 votes
1 answer
240 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 ...
stargirl's user avatar
3 votes
1 answer
261 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 ...
Simd's user avatar
  • 7,845
-2 votes
1 answer
174 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 ...
Alexander's user avatar
  • 595
0 votes
3 answers
174 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 ...
CrSb0001's user avatar
  • 2,401
3 votes
3 answers
400 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?
Dmitry Kamenetsky's user avatar
19 votes
3 answers
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 ...
quarague's user avatar
  • 1,863
1 vote
1 answer
291 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 ...
HelptimeCode's user avatar
5 votes
1 answer
348 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.)
Simd's user avatar
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1 vote
2 answers
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?
Dmitry Kamenetsky's user avatar
8 votes
1 answer
429 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 ...
Herbert Kociemba's user avatar
48 votes
4 answers
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 ...
Herbert Kociemba's user avatar
1 vote
1 answer
544 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 ...
qwr's user avatar
  • 773
10 votes
2 answers
675 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 ...
Tilman's user avatar
  • 103
1 vote
1 answer
205 views

Visiting all strings by swapping

Consider the following strings ...
Simd's user avatar
  • 7,845
11 votes
2 answers
465 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 ...
Herbert Kociemba's user avatar
5 votes
1 answer
433 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 ...
Bubbler's user avatar
  • 15.1k
6 votes
1 answer
279 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 ...
Laska's user avatar
  • 1,919
11 votes
6 answers
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 ...
Simd's user avatar
  • 7,845
14 votes
1 answer
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 ...
Simd's user avatar
  • 7,845
5 votes
1 answer
595 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, ...
Alexander's user avatar
  • 595
17 votes
5 answers
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 ...
Dmitry Kamenetsky's user avatar
68 votes
1 answer
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 ...
SlowMagic's user avatar
  • 13.8k
5 votes
1 answer
341 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 ...
popcorn's user avatar
  • 263
0 votes
2 answers
281 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 ...
Dmitry Kamenetsky's user avatar
6 votes
1 answer
668 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 ...
popcorn's user avatar
  • 263
1 vote
1 answer
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 ...
Cong Chen's user avatar
  • 179
25 votes
1 answer
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 ...
SlowMagic's user avatar
  • 13.8k
-1 votes
1 answer
188 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 ...
hxksbq's user avatar
  • 11
3 votes
1 answer
372 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 ...
humn's user avatar
  • 21.9k
5 votes
1 answer
381 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 ...
imida k's user avatar
  • 153
5 votes
1 answer
361 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 ...
Will Octagon Gibson's user avatar
6 votes
2 answers
712 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 ...
thisIs4d's user avatar
  • 1,083
16 votes
1 answer
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 ...
Will Octagon Gibson's user avatar
3 votes
1 answer
230 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.
weissguy's user avatar
  • 195

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