Questions tagged [combinatorics]

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

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19
votes
2answers
2k views

“Legally” filling a 10x10 table with 10 of each digit

Professor Halfbrain has spent his entire weekend by filling $10\times10$ tables with the digits $0,1,2,3,4,5,6,7,8,9$ so that each digit occurs exactly $10$ times. According to the professor, such ...
9
votes
4answers
2k views

How many different non congruent polygons can you make on a 3x3 dot grid?

There is a $3\times3$ dot grid. How many different non-congruent polygons can you make on the grid? Rules: All vertices of the polygon must be on the grid Only non self intersecting polygons Only ...
4
votes
5answers
1k views

Coin flipping game

An $8\times8$ checkerboard is filled with two-sided coins (that are blue on one side and red on the other side). The following picture shows three examples of a cross (multiplication sign): the five ...
38
votes
3answers
6k views

The coolest checkerboard magic trick

In the small town of Terni (Italy), there's a couple of young friends named Marco and Leonardo, who like to perform magic tricks to a restricted audience of common friends and relatives. They like to ...
2
votes
1answer
219 views

How many games should be played to avoid tying?

There are $n\ge3$ players playing a game. In this game, one person will come out in first place, one in second, and so on. It's impossible to tie. The person in first place gets $n$ points, the ...
9
votes
3answers
5k views

How can 12 teams rotate through 6 games without overlaps?

Given the following: Six Number Teams (1 - 6) Six Letter Teams (A - F) Six Games (Basketball, Football, Baseball, Volleyball, Hockey, Rugby) Six Time Slots (1pm - 6pm) Set up a game schedule that ...
5
votes
0answers
572 views

Who can find the most efficient path counting algorithm? [closed]

This question was put on hold as off-topic because: "...it appears to be a mathematics problem, as opposed to a mathematical puzzle." However, it is not a mathematics problem. There is no known ...
13
votes
1answer
419 views

It's twelve o'clock!

The twelve numbers on a clock are each either colored red or colored black. You are allowed to make several moves, where a move consists in picking a black number, and in flipping the colors of its ...
11
votes
2answers
743 views

Musings at a chess tournament

Professor Halfbrain has spent his entire weekend watching the games at the local chess tournament. As usual, every two players played exactly one game against each other. A win yields 1 point, a draw ...
36
votes
5answers
5k views

My grandfather's socks

My grandfather has a big drawer where he keeps his socks. The drawer contains more than 900 but less than 1000 individual socks. Each of his socks is black or blue, and there are more blue socks than ...
3
votes
3answers
966 views

Knights on a 5x5 chess board

What is the maximum number of knights that can be positioned on a $5\times5$ chess board, so that each knight attacks exactly two other knights?
5
votes
1answer
617 views

Pythagorean coins

To make payments, the Pythagoreans use coins in no more than three denominations. The three denominations are in whole Oboloi amounts, and the sum of the squares of the two smaller denominations ...
1
vote
2answers
287 views

Warped magic squares

A $3\times3$ grid contains altogether six squares that are formed by its nine entries: there are five squares whose sides are parallel to the sides of the grid (four small ones and a big one), and ...
4
votes
3answers
1k views

Circle of numbers

The integers $1,2,3,\ldots,n$ are to be arranged clockwise around a circle, such that adjacent integers always share a common digit (in their decimal representations). (a) Find the smallest integer $...
-2
votes
2answers
510 views

Binary manipulation game

I made a (for now, 2 player) game once that deals with manipulating binary numbers on a single list. Here are the rules. Rules For now, take $n=5$. If possible, provide a solution for generalized $n$...
5
votes
1answer
259 views

Elections at the Fun-and-Nonsense-Club

The new board of the Fun-and-Nonsense-Club is to be elected. The voting procedure itself is fairly simple: The club has 24 members, and every pair of (distinct) members announces to the public ...
5
votes
1answer
410 views

From Kings to Knights

In From knights to kings and From Knights to Kings on a rectangle we tried to help knights be next to their friends after they were promoted to kings. What happens if we have kings who are deposed and ...
3
votes
3answers
16k views

Numbers with distinct digits

Consider the set of natural numbers from 0 to 999, inclusive. Suppose we select some subset of this set such that any pair of numbers from it share no more than one digit in the same positions. To ...
14
votes
6answers
1k views

Finding Doctor No

James Bond is invited to a party with altogether $128$ participants (including Bond himself, the host, and the hostess). At the beginning of the party the host takes James Bond aside and asks him to ...
10
votes
2answers
792 views

Sleeping students

Professor Kafka tells proudly: Today I gave a fascinating lecture. There were five students sitting in the first row of the lecture room, and each of them was awake as I started my lecture. Each ...
10
votes
2answers
811 views

Table tennis tournament

Each of the five teams A, B, C, D, E consists of five table tennis players. In their tournament last week, each player has played one match against each of the twenty players in the other four teams. ...
11
votes
3answers
1k views

The juggling magician

In the magical circus, today a magician juggles with balls, cones, rings and umbrellas. While performing his art, he sometimes applies a powerful magic word: Ebrecedebre transforms one ball ...
13
votes
1answer
924 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 ...
16
votes
1answer
836 views

Rotationpuzzle in hex - The journey beyond the tomb

This is the 3rd themed puzzle of the tomb. (See puzzle 1 and puzzle 2) It is fully independent of the other two and just linked by the common story. After you set the colour-puzzle dials to the right ...
16
votes
5answers
648 views

Secret within the Gossiping Partiers

Al and Jane are hosting a party tonight. Al plans on proposing to Jane, a secret only he knows. The party begins and all 8 guests arrive on time. Immediately and again every minute, everyone forms ...
14
votes
2answers
818 views

Does this grid puzzle with a symmetry-breaking condition have a solution?

Since challenge questions are up in the air, what better time to pose a puzzle-building question that's intrigued me for some time. Preamble Consider an infinite square grid made up of nodes and ...
12
votes
4answers
2k views

Random Walk on Ring City

Ring City has 100 houses arranged in a circle. Alice starts at her own house, and every day she randomly moves to one of the two adjacent houses, each with 50% probability. She repeats this until she ...
12
votes
1answer
710 views

Counting numbers with 3 dice

Yesterday I saw a pair of calendar dice that can be rotated and swapped to show all days of a month between 01 and 31. The dice have a digit painted on each face. First die: [0, 1, 2, 3, 4, 5] ...
4
votes
2answers
855 views

Counting triangle algorithm

There is a type of puzzle where one needs to count triangles in a figure, generally a large triangle full of lines which create smaller ones. One example Is there, or can someone develop ;), an ...
32
votes
11answers
3k views

Coin weighing with a single weighing device

You have 12 coins which each weigh either 20 grams or 10 grams. Each is labelled from 1 to 12 so you can tell the coins apart. You have one weighing device as well. At each turn you can put as many ...
12
votes
4answers
1k views

Nerds, Jocks, and Lockers

Here's an oldie but goodie from The Daily WTF; I paraphrase to avoid copyright issues: A middle school math teacher, who also happened to be the P.E. coach, made the following deal with the non-...
16
votes
2answers
1k views

Automatically a Knight, Knave, and Joker

Let M be a finite positive integer. It's exact value is not known. Suppose we have three classes of automaton, all of which accept a bit stream as input, produce a bit stream as output (one bit per ...
-3
votes
4answers
569 views

How many Jellybeans are in the bottle

There are $n$ jellybeans in a bottle with unknown dimensions. What is the only surefire way of knowing how many jellybeans are in the bottle?
4
votes
3answers
412 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 ...
15
votes
3answers
1k views

N-dimensional Tic-Tac-Toe variant

Consider the game surface to be an infinite N dimensional Cartesian lattice. The rules are X moves first, but O gets to move ...
0
votes
2answers
7k views

A King and his Servants [duplicate]

A king sent 10 of his servants out to buy him each 10 gold rings. Each ring should weigh 10 grams, and altogether there should be 100 gold rings. One of the King's spies told him that one servant had ...
13
votes
3answers
1k views

The Monster Garden

You are the praetor of Tri, a small triangular-shaped nation bordered on the northwest by the Kingdom of Sauria, on the northeast by the Sultanate of Avia, and on the south by the Republic of Cryosta, ...
0
votes
3answers
1k views

How Many Times will 1 Appear on the Broken Clock?

You just got a new clock, but it's broken. Your friend tells you he can fix it, but he needs a little bit of data from it. One thing he needs is how many times 1 appears in the run of 2 days. The ...
5
votes
3answers
3k views

The Money Puzzle: Maximum amount of change without a dollar

Bob goes to a vending machine to buy a can of soda. He opens his wallet, and, to his surprise, has no dollar bills or any bills for that matter. The soda costs exactly $1 USD. Bob also has change in ...
10
votes
5answers
2k views

What's the Maximum Moves Needed for this kind of Puzzle?

Imagine a 4x4 square with a picture you need to make with tiles in it, now imagine you can swap any 2 pieces next to eachother (horizontal, and vertical, not diagonal.) What's the maximum amount of ...
23
votes
8answers
4k views

How to choose at least half of everything

Some number of gold, silver, and copper coins are scattered in $N$ chests. You may look into each chest and count each type of coin in them, and then select $M$ of the chests. Your goal is to have at ...
13
votes
5answers
10k views

How many paths are there through a chess board? [closed]

A pawn is placed on the lower left corner square of a standard 8 by 8 chessboard. A 'move' involves moving the pawn, where possible, either: one square to the right, one square up, or diagonally one ...
19
votes
5answers
7k views

Flip a Fair Coin

I found this question and became curious, can anyone tell me the answer and prove it, i know it seems fairly simple but just thought an explanation of this would make an interesting case. Flip a fair ...
-2
votes
1answer
528 views

Partitioning a chessboard

You should see a standard chessboard - it is printed on paper. How many ways can you cut it up (around the squares) such that each piece has twice as many squares of one colour than of the other ...
6
votes
3answers
2k views

Amount of hair on the head

Can you prove in two different ways that at this moment, some two persons on Earth have exactly the same number of hairs on the head?
18
votes
6answers
1k views

The Ball Factory Worker

A woman works at a ball factory. She's instructed to open up a ball creation kit composed of one red rubber ball and two strips of digit stickers from $0$ through $9$. She's instructed to do the ...
6
votes
1answer
696 views

Unlocking a curiously-geared combination lock

The image shows numbers on ring-dials on a curiously-geared combination lock, where the current setting is (3,3,3,3) and ring sizes are (5,7,8,11), inner to outer. The setting (0,0,0,0) opens the ...
3
votes
2answers
849 views

How high a tower of tiles can be made?

An $S$-tileset is a collection of $n$ oriented tiles, where no two tiles have the same size, each tile is one unit thick, and its non-zero-integer length and width add up to $n+1$. (So, an $S$-...
18
votes
4answers
2k views

The 8-dimensional vegetable kebab

You are given two of each from the array of 8 vegetables numbered 1 to 8 as shown above. So in total you have 16 veggies(8 pairs). Your task is to make the longest kebab (sequence of vegetables ...
4
votes
3answers
342 views

How many subset intersections do you require?

Alice chooses a subset S of {A,B,C,D,E}. Bob makes a guess of any subset T and Alice tells him the number of elements in S$\cap$T. Bob has to continue making guesses until he can exactly determine S....