Questions tagged [combinatorics]
A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]
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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 ...
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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.)
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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, ...
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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 ...
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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 ...
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How Many Magic Hexagons that use repeated digits?
There exists only 1 normal magic hexagon that uses non repeating consecutive digits for 1 to 19.
If We allow digits to repeat we can create something like this hexagon that is made up using ...
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Colour the positive integers without making a blue equation
This puzzle is related to How do we find the numbers? but has a slightly more striking solution in my opinion. It is also based on one of my MathsSE answers.
What is the least number of colours you ...
- 6,954
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7 mathematicians around the clock in prison
In a very strange kingdom, 7 mathematicians (let's call them Ann, Ben, Cid, Dan, Eve, Flo and Guy) were sent to prison because they did a calculation which was correct, but was not in favor of the ...
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Twenty-four coins
I have twenty-four identical-looking coins, but two are fake and weigh possibly different from each other, though definitely different from the remaining genuine coins. I have a weighing scale and a ...
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Good and bad numbers of remaining mines
You've been tasked with finishing solving this Minesweeper board:
"How many mines remain?", you ask. "I'm just choosing that now, actually. Tell you what: I was going to consider every ...
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How many Nonconsecutive Sudoku solutions are there?
Consecutive Sudoku is a variant with the additional rule that orthogonally adjacent numbers are consecutive if and only if there is a dot/bar on the line between them. A Nonconsecutive Sudoku is one ...
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Two arcs equal three arcs
(Gonna answer my own question, as is encouraged.)
To set the stage: an arc (or a Jordan arc) is a non-self-intersecting curve with two distinct endpoints. (For those who are familiar with topology, it'...
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Infected squares warmup: infect a 7x7 board with 21 squares
You can consider this a "warmup" to my other question about infected squares.
On a $7\times7$ square, some cells are infected; if a cell shares an edge with $3$ infected squares, it becomes ...
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Infected cubes puzzle in 3D with threshold 4
(This question was previously posted on Math SE, but received no answers.)
3D infected cubes puzzle with threshold $4$:
On an $n\times n\times n$ cube, some cells are infected; if a cell shares a ...
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Tiling a chessboard
Say I have an eleven by eleven chessboard, so there's 121 squares total. I remove the centermost piece so there's 120 pieces. I want to tile the chessboard with 1x4 or 4x1 pieces in a way that none of ...
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More Genuine and Fake Coins
I have 36 identical coins of which four, all weighing the same, are known to be fake. Fake coins are either all heavier than genuine coins, or all lighter.
At most how many weighings on a balance ...
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Impossible tiling of board using dominoes [duplicate]
prove that no matter how you tile a 6 x 6 board using 2 x 1 tiles, there would always be a vertical or horizontal line separating the board. Separation here means that no tile would be cutting across ...
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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 ...
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How can 8 , 10 or 12 teams rotate through 7 or 8 games without overlaps? [closed]
We are scheduling a big scout event with children, and we have 7 or 8 games organized for them to rotate and play with each other.
Set up a game schedule that follows these rules: There are 3 ...
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The Lufthansa Lottery
In order to pass free time while striking for better pay, some Lufthansa workers organise a lottery where
each ticket picks three distinct numbers from $1$ to $11$ inclusive
the draw picks five ...
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Number of 6-person events so all groups of 3/10 people have dined together [closed]
Assume 10 people numbered 1-10 have to be invited for dinner events. However, the hotel can accommodate only 6 at a time. Therefore, they will be invited in batches until all groups of 3 people have ...
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Generalization of twelve balls and scale problem
This problem is a generalization of Twelve balls and a scale problem. So I can solve and understand how things are going if we have 12 balls or 9 balls but how do I generalize? If say we have $3^n$ ...
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The universal ticket
I am submitting a very interesting problem from a French mathematical recreation site:
http://www.diophante.fr/problemes-par-themes/g-probabilites/g2-combinatoire-denombrements/1434-g248-le-billet-...
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Taking turns adding a number 1,2,3 to a 3x3 matrix without repeating numbers in the rows or columns: does the first player always win?
Alice and Bob are playing a game on an initially empty 3x3 matrix. They take turns, and each turn:
They add a number in {1,2,3} to an empty cell.
They are not allowed to repeat a number in a row or ...
