Questions tagged [combinatorics]

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

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10
votes
2answers
846 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
943 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
860 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
652 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
902 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 edges ...
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 ...
13
votes
1answer
768 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
951 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 ...
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 ...
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
572 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
431 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
8k 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
2k 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
4k 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
3k 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 ...
24
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
11k 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 ...
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 ...
-2
votes
1answer
568 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
3k 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
704 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
868 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
347 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....
32
votes
18answers
46k views

Draw a line through all doors

I saw the following problem on 4chan and couldn't solve it: It's very likely to be some kind of troll (no solution). I'm hoping to see some rigorous proofs that disprove the existence of such a line....
1
vote
1answer
1k views

Minimum number of weights you need to define any integer weight from 1 to N

Following this question What's the fewest weights you need to balance any weight from 1 to 40 pounds? I am interested what is the minimum number M of weights you need to define any integer weight ...
8
votes
1answer
904 views

Number of tries to guess M-1 letters from M-letters-code

There are $N$ letters in an alphabet. There is a combination lock, the code to it consists from $M$ different letters. You can input $M$ letters combination and try to open the lock. (But you can't ...
4
votes
2answers
1k views

Is there a brute-force solution of this puzzle?

There is a puzzle Suppose a standard $8x8$ chessboard has two (arbitrary) squares removed. The only thing known is that these two squares have different colours. Is it always possible to place 31 ...
30
votes
5answers
6k views

Two spies throwing stones into a river

There is a puzzle about two spies: Two spies must pass each other two secret numbers (one number per spy), unnoticed by their enemies. They have agreed on a method for doing this using only 26 ...
10
votes
4answers
10k views

N balls and a scale

The question of twelve balls and a scale is probably the best-known example of the "find the ball of a different weight" problem. But does it generalize? Is there a general way to find a weighing ...
16
votes
3answers
23k views

What's the fewest weights you need to balance any weight from 1 to 40 pounds?

Suppose you want to create a set of weights so that any object with an integer weight from 1 to 40 pounds can be balanced on a two-sided scale by placing a certain combination of these weights onto ...
60
votes
9answers
95k views

Twelve balls and a scale

You are given twelve identical-looking balls and a two-sided scale. One of the balls is of a different weight, although you don't know whether it's lighter or heavier. How can you use just three ...
5
votes
1answer
251 views

MPire coloring game basic strategy

The MPire coloring game is a game where several "empires" are laid out touching each other. Your task is to color each empire a different color, and not let two of the same color touch. You also have ...

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