# Questions tagged [combinatorics]

A puzzle based on combinatorics, which is the study of counting discrete structures. Use with [mathematics]

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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 ...
586 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 ...
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?
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 ...
728 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 ...
879 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 $S$-...
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 ...
350 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....
57k 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....
2k 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 ...
918 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 ...
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 ...
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 ...
11k 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 ...
28k 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 ...