# Questions tagged [combinatorics]

For challenges involving combinatorics.

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### Count the number of Hamming distance sequences

The Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. Let P be a binary string of ...
975 views

### Count how many distance sequences are far from all others

The Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. Let P be a binary string of ...
1k views

### Number of string permutations that are palindromes

Your input will be a string consisting of small english letters. Your task is to determine the number of distinct permutations of the original string that are a palindrome. The input string has up ...
350 views

### Stable marriage problem

Background Suppose that there are 2*n people to be married, and suppose further that each person is attracted to exactly n ...
355 views

### Rafting Problem (Knapsack variant)

First puzzle from me, suggestions for improvement gladly received! The scenario is; You work as a manager for a whitewater rafting company. Every morning, you are given a list of bookings, and you ...
2k views

### Necklace splitting problem

Background I was inspired by 3Blue1Brown's recent video about the necklace splitting problem (or as he calls it, the stolen necklace problem) and its relationship to the Borsuk-Ulam theorem. In this ...
639 views

### Hold'em or Fold'em?

Your friend invited you to a high-stakes poker game at the last minute and, as a computer scientist, you've decided to use your skills to gain an edge on the game. Your task will be, given ...
388 views

### Total Derangement (Difficulty Level: Hard)

The problem: Write a function which, given a cycle length n and a number of cycles m, where 'm' is within ...
68 views

### Code Golf - Print all unique ways of writing a number as a sum with a lexical order [duplicate]

I found about Code Golf after writing this program and decided to post the rules here so you can try it out too. I saw this, which is a less restrictive version of my question. I've got a bit more ...
812 views

### Arranging Bubbles

Note, challenge copied from question asked at math.stackexchange. Recently, I attained quite some skill at blowing bubbles. At first I would blow bubbles like this: But then things started getting ...
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### Every word from babab to zyzyz

Your task is to write a program that will output a readable list of every five letter words with the structure: consonant - vowel - consonant - vowel - consonant The output should be sorted ...
2k views

### Longest domino chain

Challenge description Dominoes is a game played with tiles with two values on it - one on the left, one on the right, for example [2|4] or ...
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### List all multiplicative partitions of n

Given a positive number n, output all distinct multiplicative partitions of n in any convenient format. A multiplicative partition of n is a set of integers, all greater than one, such that their ...
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### Absolute Sums of Sidi Polynomial Coefficients

Background The Sidi polynomial of degree n – or the (n + 1)th Sidi polynomial – is defined as follows. The Sidi polynomials have several interesting properties, but so do their coefficients. The ...
224 views

### Number of prime knots with n crossings

A prime knot is: a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Explanation of a knot-sum: put the two knots adjacent, ... then draw two lines between them, ...
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### Straight Chain Alk*nes without Order

This is much like my earlier challenge, except, this time, order doesn't matter. A straight-chain alk*ne is defined as a sequence of carbon atoms connected by single (alkane), double (alkene), or ...
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### Number of Straight-Chain Alk*nes of given length

A straight-chain alk*ne is defined as a sequence of carbon atoms connected by single (alkane), double (alkene), or triple bonds (alkyne), (implicit hydrogens are used.) Carbon atoms can only form 4 ...
417 views

### Evaluate the Binomial Coefficient [duplicate]

Given two nonnegative integers n,k such that 0 <= k <= n, return the binomial coefficient ...
441 views

### Sum digits till Square

Given is any integer x > 0 and any base y > 3. Sum all digits of x (if written in the set base). Multiply this by the highest possible digit (is always base -1). ...
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### Generate all combinations of given list of elements, sorted

Make a code that takes a list and a number as input, and generates all possible combinations with the length of the number. For example, with the list {0,1} and the number 2: ...
254 views

### Convert a sample to an index

We are putting balls into a fixed number a bins. These bins begin empty. Empty bin (a=4): 0 0 0 0 And one by one we add balls to the bins. ...
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### Dobble/SpotIt card generator

Introduction Dobble/SpotIt is a card game, where people have to spot same symbol on pair of cards in shortest time, indicate it and move to next pair. Each card has multiple symbols (8 in normal ...
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### Fewest disk writes to defrag multiple files

Introduction A disk is a linear container with blocks indexed 0 through size-1. A file is a named list of block indexes used ...
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### Ordered list of binary sequences of multiple dimensions

Given a positive integer n, output the 2^n binary sequences of length n sorted in the ...
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### A penny saved is a penny

...counted! You will pass your program a variable which represents a quantity of money in dollars and/or cents and an array of coin values. Your challenge is to output the number of possible ...
2k views

### Compute the Eulerian number

The Eulerian number A(n, m) is the number of permutations of [1, 2, ..., n] in which exactly ...
4k views

### Determine if a coin system is Canonical

The Cashier's Algorithm is an algorithm for making change in the minimal number of coins that works quite well for most currency systems. However like most greedy algorithms it is not without its ...
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### Inverse permutation index

Introduction The lexicographical permutations of a list with n elements can be numbered from 0 to n! - 1. For example, the 3! = 6 permutations of (1,2,3) would be <...
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### List all ordered partitions of n

The challenge is to list all ordered partitions (composition (combinatorics)) of a given positive integer n. These are the lists of numbers from ...
2k views

### Figure Out the Android Lock Pattern

Lets say you saw your friend enter his or her password into their Android phone. You don't remember how they made the pattern but you remember what the pattern looks like. Being the concerned friend ...
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### Verify a ballot triangle

A ballot number, which we'll label B, is the number of ways of arranging the numbers from 1 through B(B+1)/2 into a triangle, such that each row and column is in any increasing order. The first four ...
272 views

### Count blocks in a 2D grid

Challenge description Let's define an W x H grid as a two-dimensional array of length H ...
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### Number of cycles of a permutation

Consider a permutation of the integers 1, ..., n, such as this one for n = 6: ...
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### Verify Wolstenholme's theorem

Definition Wolstenholme's theorem states that: where a and b are positive integers and p ...
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### Coin Change Problem Enumeration Using N Coins And Each Denomination

The coin change problem is very well documented. Given an infinite supply of coins of denominations x_1 to x_m you need to find ...
855 views

### Partial factorisations of a positive integer

A collection of positive integers d_1 d_2 ... d_k is a factorisation of a positive integer n if ...
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### Is case sensitivity important? Part II: reality check

In this question Tom learned that, in general, there are many motivations to choose to include case sensitivity in his programming language, because the possible combinations for a variable name are ...
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### Is case sensitivity important?

Tom is going to implement a new programming language of his invention. But before actually starting working on it, he wants to know whether his language should be case sensitive or not. On one hand, ...
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### Marching Through A City

There is a parade marching through a city! There are 3 main groups of marchers: the (B)and, Poster (C)arriers, and (F)lag Holders. Also, every (P)oliceman in the whole city is on duty. Flag holders (...
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### Calculate the partitions of N

Your challenge is simple: GIven an integer N, ouput every list of positive integers that sums to N. For example, if the input was 5, you should output ...
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### Recursion bracketed; or Dyck words generation

We already have challenges to check if a string of brackets is fully matched and to count the number of balanced strings. It remains for us to generate these strings, but it will not be so easy… A ...
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### Ways to get to the number

Given the input of the first number and the second number (both positive integers, zero exluded), determine in how many ways could you make the second out of the first, using following actions: ...
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### The handshake problem

The handshake problem is the classic problem that for n people in a room, if they all shake hands, what's the total number of handshakes that occur. You code should take an input of any number and ...
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### Generalized Birthday Problem

Tonight, my fiancée took me out to dinner to celebrate my birthday. While we were out, I heard Happy Birthday sung to 5 different guests (including myself), in a restaurant full of 50 people. This got ...
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### Has My Pie Been Bisected?

Write a program or function that takes in a nonempty list of positive integers. You may assume it is input in a reasonable convenient format such as "1 2 3 4" or <...
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### Case Permutation

Who needs to compare things case insensitively when you're able to generate every permutation of uppercase and lowercase? No one! That's the answer. No one does. Your task is to achieve this feat; ...
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### Largest Distinctly Sum-Free Partition

related and inspired by -- Finding sum-free partitions A set A is defined here as being distinctly sum-free if 1) it consists of at least three elements, ...
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### Find the number of ways to choose n objects from r objects with repetition so that every object has a matching neighbour

For example, if n is 5 and r is 2, it would mean the number of lists below: ...