Questions tagged [combinatorics]

A puzzle based on combinatorics, which is the study of counting discrete structures.

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8
votes
1answer
445 views

How many friends does Tiffany have?

Tiffany has 14 classmates; all of her classmates have a different number of friends in the class. How many of them are friends with Tiffany? (If A is a friend of B, then B is a friend of A.)
7
votes
4answers
330 views

3 Colors of Chess Pieces Attacking Each Other Once Each

Yes, it's another "Chess Pieces Attacking Each Other" puzzle. This time we have 3 colors. Your goal is to place as many of the same type of chess piece (excluding pawns since you can't define the "...
7
votes
4answers
316 views

Savage Road Signs (Part 2)

Please read part 1 or this might be confusing Since part 1, you have replaced the stolen stickers and your daughter has forgiven you. The highway ended up being a full 700km long, so you are happy ...
5
votes
2answers
695 views

Knights covering a 10x10 chess board

What is the minimum number of knights you need to place on a 10x10 chess board, such that every empty cell is attacked by at least one knight? Good luck!
33
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 ...
53
votes
4answers
6k views

Is this chromatic puzzle always solvable?

I've created a new puzzle from an Alexey Nigin's idea. It consists of a 8x8 board where each square is randomly assigned one of three colors. A movement is defined by picking any two orthogonal ...
18
votes
4answers
2k views

Four buttons on a table

I was asked lately (in an interview) to solve this puzzle, which is similar to the 4 cups on table puzzle. In a certain room there is a rotating round table, with 4 symmetrically located ...
21
votes
5answers
2k views

Numerical Boggle

You are probably familiar with the word game Boggle, where you need to construct words by concatenating letters from a grid. Here we will play a numerical version of the game. The rules are as follows:...
16
votes
1answer
864 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 ...
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 ...
16
votes
5answers
827 views

Maximize the number of paths

You have exactly 990 edges. Assemble them into a simple undirected graph with two distinguished vertices A and B, such that the number of different simple paths from A to B is as large as you can make ...
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 ...
13
votes
2answers
924 views

The impossible digital sum

There are 10 digit numbers you are supposed to use shown as below; And there is a very special addition where every digit is used only once. As you see, most of the digital signals (blue squares) are ...
8
votes
3answers
603 views

How Many Squares on the Peg Solitaire

We have a well known peg solitaire which is not played yet as seen below: At most how many squares can you make by joining the points as exemplified below? Note: No ball (point) in the middle! so ...
57
votes
3answers
5k 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!
13
votes
5answers
1k views

A row of 2015 red and white chips

There is a row of 2015 chips, of which 2014 are white and one is red. You are allowed to make moves of the following type: "Choose one red chip, and flip the colors of its two neighboring chips (from ...
13
votes
2answers
2k views

A Guide to the Number Rotation Puzzle

This is an extension of What is the strategy to solve Simon Tatham's Twiddle? in that it explicitly goes beyond the default gamemodes of Twiddle The Number Rotation Puzzle (NRP) is a combination ...
25
votes
2answers
4k views

Two chessmasters at work

Viswanathan Anand plays a chess game against Magnus Carlsen. Anand plays white and Magnus plays black. They use a non-standard digital double chess clock that is counting up from zero (instead of the ...
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 ...
14
votes
3answers
799 views

How many possible starting positions are uniquely solvable for a nonogram puzzle?

This type of puzzle goes by many names: Nonogram, Picross, and Griddlers are all mentioned on the Wikipedia page, Simon Tatham calls it Pattern, I was introduced to it as Descartes Rainbow, ... The ...
9
votes
6answers
1k views

A closed path on the Rubik's cube

Is it possible to draw a closed path on the surface of a standard $3\times3\times3$ Rubik's cube such that the path traverses each of the $54$ little squares exactly once, and such that the path ...
9
votes
4answers
548 views

Breaking Balance (Part C)

For a starting number of otherwise identical coins there are among them TWO IDENTICAL counterfeit coins which are either heavier or lighter than the rest. Using a three-pan balance (described in ...
8
votes
2answers
698 views

Four Magic Ellipses

These four ellipses represent four sets and all the possible ways they can intersect (a Venn diagram, in other words). There are 8 regions inside each ellipse, and 15 regions altogether. Is it ...
8
votes
3answers
1k views

Possible pawn combinations

This may seem simple, but I have a problem calculating it. It may be because it's Monday morning. How may possible valid combinations of one color pawn (white or black, your choice) positions are ...
5
votes
4answers
926 views

The coolest checkerboard magic trick. Version 2

Version 1: The coolest checkerboard magic trick You and your friend are imprisoned. Your jailer offers a challenge. If you complete the challenge you are both free to go. The rules are The jailer ...
24
votes
7answers
6k views

Hacking an electronic keypad

You are a spy trying to break into an enemy facility. The back door is protected by an electronic keypad lock. You know that this particular lock is opened by a four digit code. Any stream of button ...
20
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 ...
18
votes
6answers
4k views

Guess five binary digits!

Person A thinks of a 5 digit binary number. Person B tries to guess the number. B can guess a 5 digit binary number and A will respond with the number of correct digits (digits in the right place). ...
14
votes
2answers
609 views

Sort 9 train cars on 3 paths

On the three paths of a station are A, B, and C types of train cars as shown in the figure. A locomotive driver (L) can attach from 1 to 9 train cars to a locomotive at any time, move them to the ...
11
votes
3answers
1k views

Ten distinct numbers in the table

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9
votes
3answers
1k views

How many hexagonal paths?

Here is a hexagonal tiling, borrowed from Wikipedia. I start in any hexagon on the left hand side. I end at any hexagon on the right hand side. I can only travel to the right, not up, down or ...
8
votes
2answers
4k views

Alice and Bob play a game

The rain was still falling and Alice and Bob were terribly bored of having to stay inside the caravan, so they decided to play a game. The game is that Alice chooses a number $x$ in the interval [1,n] ...
8
votes
4answers
943 views

Colored balls in a 4x4 grid

Colored balls are placed in a 4x4 grid. A move consists of swapping two adjacent (horizontally or vertically) balls. What is the least number of moves required to form 4 connected components*, one for ...
8
votes
1answer
202 views

Generalized color balls in a 4x4 grid

This is a generalization of the Colored balls in a 4x4 grid puzzle that was proposed by Darrel Hoffman. Colored balls from 4 different colors are placed in a 4x4 grid. There is at least one ball from ...
7
votes
2answers
283 views

Dissect a square into 3:2 non-congruent integer-sided rectangles

(Similar to the recent 3:1 rectangle question) Tile a square completely with rectangles which have aspect ratio 3:2, integral side lengths and all different sizes. In other words selected from 2x3, ...
6
votes
2answers
591 views

Labeling wires in a *damaged* bundle

Variant of: Labeling wires in a bundle At a remote location, you just finished trenching a data cable across a large plot of land. The cable has 64 individual wires that are not color-coded or ...
6
votes
3answers
1k views

A party puzzler

At a party, everybody is friend with exactly $22$ of the other persons present. Whenever two persons are friends, they do not have any friends in common. Whenever two persons are not friends, they ...
5
votes
1answer
300 views

Holo-puzzle (1)

A boundary of red and blue squares is given. Can you fill in the interior such that each 5-square pattern consisting of an interior cell plus its four nearest neighbours always contains an even number ...
5
votes
2answers
308 views

General orchard planting problem for circles

My previous puzzle asked for the maximum number of 4-point circles attainable from a configuration of $n=10$ points drawn on a plane. I am now interested in generalizations of this puzzle to arbitrary ...
20
votes
1answer
1k views

Dominos on a checkerboard

What's the maximal number of dominos (2x1 tiles) that can be placed on a checkerboard (8x8 square) so that every domino covers exactly 2 squares of the checkerboard and no two dominos form a 2x2 ...
16
votes
5answers
3k views

Knights attacking exactly three knights

Can you place 14 black and 14 white knights on a standard 8x8 chess board, such that each knight attacks exactly 3 opponent knights? Bonus question: can you do it with 15 black and 15 white knights? ...
15
votes
5answers
979 views

Dividing the first 20 numbers into 3 lists

Place every number from 1 to 20 into one of three lists $P$, $Q$ or $O$, such that any number from $P$ added to any number from $Q$ gives a prime. What is the fewest number of elements that can be in $...
15
votes
6answers
780 views

Place 4x12 detainees on a 7x7 grid of cells

You are a prison captain and you have got 4 groups of detainees, let's call them Red, Blue, Green and Yellow. You have got 12 of each. The prison is a square grid 7x7 cells. You need to place the 48 ...
14
votes
4answers
1k views

Professor Halfbrain's chessboard theorems

Professor Halfbrain has spent his last weekend with analyzing $n\times n$ chessboards. Halfbrain says that a subset $S$ of squares on such a chessboard is queen-connected, if a chess queen can move ...
14
votes
1answer
778 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] ...
14
votes
6answers
2k views

Covering an 8x8 grid with X pentominoes

What is the minimum number of X pentominoes you need to cover every cell of an 8x8 grid? Pentominoes may overlap each other and sit outside the boundary of the grid. An X pentomino looks like this:
13
votes
5answers
2k views

Generating numbers with cubes

I saw an interesting calendar in a shop. It is composed of two cubes with numbers written on their 6 sides. By placing these cubes side by side one can make any day of the month from 1 to 31 (even 32)....
13
votes
1answer
955 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 ...
12
votes
3answers
2k views

Professor Halfbrain and the 99x99 chessboard (Part 1)

Professor Halfbrain has spent the last weekend with filling the squares of a $99\times99$ chessboard with real numbers from the interval $[-1,+1]$. Whenever four squares form the corners of a ...
12
votes
4answers
2k views

Consecutive numbers that are Manhattan distance 3 apart

Can you place numbers from 1 to 16 on a 4x4 grid, such that the distance between any two consecutive numbers ($a$ and $a+1$) is Manhattan distance 3? Bonus question: can you also make 1 and 16 be ...