For challenges involving combinatorics.
28
votes
7answers
387 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 [4|5]. Two tiles can be joined together if they contain a ...
23
votes
10answers
610 views
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 ...
17
votes
8answers
296 views
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 ...
8
votes
1answer
141 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, ...
4
votes
0answers
116 views
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 ...
22
votes
19answers
1k views
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 ...
8
votes
15answers
192 views
Evaluate the Binomial Coefficient [duplicate]
Given two nonnegative integers n,k such that 0 <= k <= n, return the binomial coefficient
c(n,k) := (n!) / (k! * (n-k)!)
Test cases
Most languages will probably have a built in function.
c(...
11
votes
7answers
393 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).
Repeat until this value is (y - 1) ^ ...
12
votes
12answers
667 views
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:
00
01
10
11
Your ...
12
votes
3answers
231 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.
0 0 0 1 or
0 0 1 0 or
0 1 0 0 or
1 0 0 0
We need a ...
9
votes
1answer
136 views
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 ...
9
votes
0answers
116 views
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 by that file.
An example filesystem is expressed like this:
15 ALPHA=3,...
21
votes
20answers
5k views
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 ...
16
votes
15answers
2k views
Compute the Eulerian number
The Eulerian number A(n, m) is the number of permutations of [1, 2, ..., n] in which exactly m elements are greater than the previous element. These are also called rises. For example, if n = 3, there ...
43
votes
8answers
2k 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 ...
15
votes
13answers
852 views
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 (1,2,3), (1,3,2), (2,1,3), (2,3,1), (...
22
votes
27answers
2k views
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 1 to n whose sum is n. For example, given input n = 4, ...
23
votes
1answer
970 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 ...
12
votes
3answers
296 views
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 ...
6
votes
1answer
225 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 whose each subarray is of length W. Example: a 2x3 grid (. character used as a blank):
..
..
..
A unit is a ...
21
votes
12answers
1k views
Number of cycles of a permutation
Consider a permutation of the integers 1, ..., n, such as this one for n = 6:
[5,2,4,3,6,1]
If you view the permutation as a mapping from [1,2,3,4,5,6] to [5,2,4,3,6,1], the permutation can be ...
11
votes
10answers
544 views
Verify Wolstenholme's theorem
Definition
Wolstenholme's theorem states that:
where a and b are positive integers and p is prime, and the big parentheses thingy is Binomial coefficient.
Task
To verify that, you will be given ...
11
votes
4answers
245 views
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 the number of combinations which add up to y. For example, given x = {1,...
23
votes
5answers
734 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
d_1 * d_2 * ... * d_k = n
Each positive integer has a unique prime factorisation, but in general they ...
6
votes
6answers
787 views
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 ...
13
votes
11answers
2k views
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, ...
2
votes
2answers
218 views
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 (...
20
votes
15answers
1k views
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
[1, 1, 1, 1, 1]
[1, 1, 1, 2]
[1, 1, 3]
[1, 2, 2]...
5
votes
11answers
895 views
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 ...
10
votes
3answers
279 views
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: +1,+2 ...
2
votes
29answers
4k views
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 ...
12
votes
3answers
369 views
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 ...
35
votes
16answers
3k views
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 [1, 2, 3, 4].
The numbers in the ...
8
votes
7answers
297 views
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, |A| ≥ 3, and
2) its distinct self-sum A + A ...
5
votes
3answers
142 views
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:
[1,1,1,1,1]
[1,1,1,2,2]
[1,1,2,2,2]
[2,2,1,1,1]
[2,2,2,1,1]
[2,2,2,2,2]
which is 6.
If n is 4 and r is 3:
[1,1,1,1]
[1,1,...
12
votes
10answers
626 views
Permutations with Indistinguishable Items
Given a list of integers, output the number of permutations of the integers, with indistinguishable permutations counted once. If there are n integers, and each group of indistinguishable numbers has ...
7
votes
8answers
364 views
m-nomial coefficient
While binomial coefficient is the coefficient of (1+x)**n, m-nomial coefficient is the coefficient of (1+x+x**2+...+x**(m-1))**n.
For example, m(3,5,6) is the coefficient of x**6 in the expansion of (...
10
votes
2answers
440 views
Tic-tac-toe with only crosses as quick as possible
As per Luke's request and Peter Taylor's addition to this challenge.
Introduction
Everyone knows the game tic-tac-toe, but in this challenge, we are going to introduce a little twist. We are only ...
28
votes
3answers
861 views
Stuffing primes in a box
Your task is write a program or function that can fill a given rectangle with
prime numbers. The width and height of the rectangle will be the input. The
output must be a list of height strings ...
10
votes
13answers
799 views
Generate combinations with replacement
List all of the combinations with replacement (or combinations with repetition) of size k from a set of n elements.
A combination with replacement is an unordered multiset that every element in it ...
11
votes
20answers
2k views
Output the Military Order [duplicate]
In almost every row nowadays, the people tends to order themselves as militaries would do.
Challenge
Suppose 4 people where:
person 1 has priority against person 2, 3 and 4,
person 2 has priority ...
17
votes
5answers
1k views
Chocolate numbers
Given an m by n chocolate bar, m,n positive, output the number of ways to break the bar into mn 1 by 1 pieces where each break occurs on a gridline.
Order is important. Pieces are also ...
15
votes
1answer
376 views
Same-color arithmetic progressions
Van der Waerden's theorem says that
For any given positive integers r and k, there is some number N such
that if the integers {1, 2, ..., N} are colored, each with one of r
different colors, ...
21
votes
19answers
1k views
Rearranging the sequence
Introduction
Let's observe the following sequence (non-negative integers):
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...
For example, let's take the first three numbers. These are 0, 1, 2. ...
24
votes
6answers
751 views
Enumerate rhyme schemes
A "rhyme scheme" is a string of letters a to z, such that the first occurrences of the characters are in ascending order (without gaps), starting from a. For example (with first occurrences marked):
...
7
votes
5answers
751 views
Hypercube sides
Your goal is to output all the "sides" (corners, edges, faces, etc.) of an N-dimensional unit hypercube, where N is non-negative. A "side" is defined as any (N-M)-dimension surface embedded in N-...
22
votes
11answers
952 views
Compute the multinomial coefficient
Time for another easy challenge in which all can participate!
The multinomial theorem states:
The expression in parentheses is the multinomial coefficient, defined as:
Allowing the terms ki to ...
0
votes
0answers
81 views
How many ways are there to achieve a winning score in American Football? [duplicate]
[I decided to salvage Ben Reich's question How many unique ways are there to achieve a score in Football? but it ended up being so different that it's only suitable as its own question not spliced ...
31
votes
6answers
1k views
Tic-tac-toe with only crosses
Introduction
Everyone knows the game tic-tac-toe, but in this challenge, we are going to introduce a little twist. We are only going to use crosses. The first person who places three crosses in a row ...
17
votes
9answers
438 views
Counting Fountains
A fountain is arrangement of coins in rows so that each coin touches two coins in the row below it, or is in the bottom row, and the bottom row is connected. Here's a 21 coin fountain:
Your challenge ...
