# Questions tagged [combinatorics]

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

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### “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 ...
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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### 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?
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### 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 ...
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 ...
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 $S$-...
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....