Rolling Project Euler on Ruby

I first heard about Project Euler last week on the stackoverflow podcast. Michael Pryor (fogcreek co-founder) makes a quick side reference in discussion with Joel Spolsky, Jeff Atwood and the rest of the SO team.

Well I checked it out last week, got hooked and spent most of the weekend earning my "level 1" badge;-)

Aside from dusting off some long-forgotten and arcane knowledge from my youth, I found it a fantastic opportunity to stretch my fundamental ruby chops. In fact, I'd recommend a few questions at Project Euler as a right-of-passage whenever you are learning a new programming language.

I've only been using ruby for a year or so, and in that time thought I had picked up a fair bit. But I was still able to amaze myself at just how many of the problems were knocked over in just 1 or 2 lines with a bit of duck punching and liberal use of blocks with Enumerables.

I'm late to the Project Euler craze, so you will already find many people posting hints for specific questions if you just google. I thought I'd share some of the "common code" I've been building up as I go through the questions.

I put a recent copy of the source up on github for anyone who is interested (MyMath.rb), but what follows a sampling of some of the more interesting pieces.

First thing you will note is that I have written all these "common" routines as extensions to some of the fundamental classes in the ruby library: Integer, Array, String.

It doesn't have to be this way, and for less trivial activities you might be right to be concerned about messing with the behaviour of the standard classes. Maybe I'm still enjoying my ruby honeymoon period, but I do get a thrill out of being able to write things like

1551.palindrome?
=> true

Integer Extensions

It's just so easy to code up simple calculation and introspection routines..

class Integer
  # @see project euler #15,20,34
  def factorial
    (2..self).inject(1) { |prod, n| prod * n }
  end

  # sum of digits in the number, expressed as a decimal
  # @see project euler #16, 20
  def sum_digits
    self.to_s.split('').inject(0) { |memo, c| memo + c.to_i }
  end

  # num of digits in the number, expressed as a decimal
  # @see project euler #25
  def num_digits
    self.to_s.length
  end
  
  # tests if all the base10 digits in the number are odd
  # @see project euler #35
  def all_digits_odd?
    self.to_s.split('').inject(0) { |memo, s| memo + ( s.to_i%2==0 ? 1 : 0 ) } == 0
  end
  
  # generates triangle number for this integer
  # http://en.wikipedia.org/wiki/Triangle_number
  # @see project euler #42
  def triangle
    self * ( self + 1 ) / 2
  end
end

Prime numbers feature heavily on Project Euler, and I think calculating a prime series was my first lesson on why you can't brute-force everything;-) Enter the Sieve of Eratosthenes and related goodness..

class Integer 
  # http://en.wikipedia.org/wiki/Prime_factor
  # @see project euler #12
  def prime_factors
    primes = Array.new
    d = 2  
    n = self      
    while n > 1
     if n%d==0
        primes << d
        n/=d
      else
        d+=1
      end
    end
    primes
  end
  
  # http://en.wikipedia.org/wiki/Divisor_function
  # @see project euler #12
  def divisor_count
    primes = self.prime_factors
    primes.uniq.inject(1) { |memo, p| memo * ( ( primes.find_all {|i| i == p} ).length + 1) }
  end
  
  #
  # @see project euler #12, 21, 23
  def divisors
    d = Array.new
    (1..self-1).each { |n| d << n if self % n == 0 }
    d
  end

  # @see project euler #
  def prime?
    divisors.length == 1 # this is a brute force check
  end
  
  # prime series up to this limit, using Sieve of Eratosthenes method
  # http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
  # @see project euler #7, 10, 35
  def prime_series
    t = self
    limit = Math.sqrt(t)
    a = (2..t).to_a
    n = 2
    while (n < limit) do
      x = n*2
      begin
        a[x-2]=2
        x+=n
      end until (x > t )
      begin
        n+=1
      end until ( a[n-2] != 2 )
    end
    a.uniq!
  end

  # @see project euler #23
  def perfect?
    self == divisors.sum
  end

  # @see project euler #23
  def deficient?
    self > divisors.sum
  end

  # @see project euler #23
  def abundant?
    self < divisors.sum
  end
end

Next we visit the Collatz conjecture and an interesting routine to make numbers "speak english"..

class Integer     
  # http://en.wikipedia.org/wiki/Collatz_conjecture
  # @see project euler #14
  def collatz_series
    a = Array.new
    a << n = self
    while n > 1
      if n % 2 == 0
        n /= 2
      else
        n = 3*n + 1
      end
      a << n
    end
    a  
  end

  # express integer as an english phrase
  # @see project euler #17
  def speak
    case
    when self <20
      ["zero", "one", "two", "three", "four", "five", "six", "seven", "eight", "nine", "ten",
       "eleven", "twelve", "thirteen", "fourteen", "fifteen", "sixteen", "seventeen", "eighteen", "nineteen" ][self]
    when self > 19 && self < 100 
      a = ["twenty", "thirty", "forty", "fifty", "sixty", "seventy", "eighty", "ninety"][self / 10 - 2]
      r = self % 10
      if r == 0
        a
      else
        a + "-" + r.speak
      end
    when self > 99 && self < 1000
      a = (self / 100).speak + " hundred"
      r = self % 100
      if r == 0
        a
      else
        a + " and " + r.speak
      end      
    when self > 999 && self < 10000
      a = (self / 1000).speak + " thousand"
      r = self % 1000
      if r == 0
        a
      else
        a + ( r <100 ? " and " : " " ) + r.speak
      end      
    else
      self
    end
  end
end

Calculating integer partitions is one of my favourites ... a nice, super-fast recursive algorithm. For problems like "how many ways to make $2 in change?"

class Integer 

  # calculates integer partitions for given number using array of elements 
  # http://en.wikipedia.org/wiki/Integer_partition
  # @see project euler #31
  def integer_partitions(pArray, p=0)
    if p==pArray.length-1
      1
    else
      self >= 0 ? (self - pArray[p]).integer_partitions(pArray ,p) + self.integer_partitions(pArray,p+1) : 0
    end
  end   
end

Finally, rotations and palindromes (base 2 or 10): methods that rely on some underlying String routines that come later...

class Integer 
  # returns an array of all the base10 digit rotations of the number
  # @see project euler #35
  def rotations
    self.to_s.rotations.collect { |s| s.to_i }
  end

  # @see project euler #4, 36, 91
  def palindrome?(base = 10)
    case base 
    when 2
      sprintf("%0b",self).palindrome?
    else
      self.to_s.palindrome?
    end
  end
end

Array Manipulations

Array handling is particularly important. Start with some simple helpers, then move onto greatest common factor and a couple of least-common multiple implementations. My favourite here - lexicographic permutations.

class Array

  # sum elements in the array
  def sum
    self.inject(0) { |sum, n| sum + n }
  end
  
  # sum of squares for elements in the array
  # @see project euler #6
  def sum_of_squares
    self.inject(0) { |sos, n| sos + n**2 }
  end
  
  # @see project euler #17
  def square_of_sum
    ( self.inject(0) { |sum, n| sum + n } ) ** 2
  end
    
  # index of the smallest item in the array
  def index_of_smallest
    value, index  = self.first, 0
    self.each_with_index {| obj, i | value, index = obj, i if obj<value  }
    index
  end

  # removes numbers from the array that are factors of other elements in the array
  # @see project euler #5
  def remove_factors
    a=Array.new
    self.each do | x | 
      a << x if 0 == ( self.inject(0) { | memo, y | memo + (x!=y && y%x==0 ? 1 : 0)  } )
    end
    a
  end

  # http://utilitymill.com/edit/GCF_and_LCM_Calculator
  # @see project euler #5
  def GCF
    t_val = self[0]
    for cnt in 0...self.length-1
      num1 = t_val
      num2 = self[cnt+1]
      num1,num2=num2,num1 if num1 < num2
      while num1 - num2 > 0
        num3 = num1 - num2 
        num1 = [num2,num3].max
        num2 = [num2,num3].min
      end
      t_val = num1
    end
    t_val
  end

  # http://utilitymill.com/edit/GCF_and_LCM_Calculator
  # @see project euler #5
  def LCM
    a=self.remove_factors
    t_val = a[0]
    for cnt in 0...a.length-1
      num1 = t_val
      num2 = a[cnt+1]
      tmp = [num1,num2].GCF
      t_val = tmp * num1/tmp * num2/tmp
    end
    t_val  
  end

  # brute force method:
  # http://www.cut-the-knot.org/Curriculum/Arithmetic/LCM.shtml
  # @see project euler #5
  def lcm2
    a=self.remove_factors
    c=a.dup
    while c.uniq.length>1
      index  = c.index_of_smallest  
      c[index]+=a[index]
    end
    c.first
  end

  # returns the kth Lexicographical permutation of the elements in the array
  # http://en.wikipedia.org/wiki/Permutation#Lexicographical_order_generation
  # @see project euler #24
  def lexicographic_permutation(k)
    k -= 1
    s = self.dup
    n = s.length
    n_less_1_factorial = (n - 1).factorial # compute (n - 1)!
    
    (1..n-1).each do |j|
      tempj = (k / n_less_1_factorial) % (n + 1 - j)

      s[j-1..j+tempj-1]=s[j+tempj-1,1]+s[j-1..j+tempj-2] unless tempj==0
      n_less_1_factorial = n_less_1_factorial / (n- j)
    end
    s
  end
  
  # returns ordered array of all the lexicographic permutations of the elements in the array
  # http://en.wikipedia.org/wiki/Permutation#Lexicographical_order_generation
  # @see project euler #24
  def lexicographic_permutations
    a=Array.new
    (1..self.length.factorial).each { |i| a << self.lexicographic_permutation(i) }
    a
  end
    
end

String Helpers

Last but not least, some String methods that just make things so much easier...

class String

  # sum of digits in the number
  # @see project euler #16, 20
  def sum_digits
    self.split('').inject(0) { |memo, c| memo + c.to_i }
  end

  # product of digits in the number
  # @see project euler #8
  def product_digits
    self.split('').inject(1) { |memo, c| memo * c.to_i }
  end
  
  #
  # @see project euler #4, 36, 91
  def palindrome?
    self==self.reverse
  end 

  # returns an array of all the character rotations of the string
  # @see project euler #35
  def rotations
    s = self
    rots = Array[s]
    (1..s.length-1).each do |i|
      s=s[1..s.length-1]+s[0,1]
      rots << s
    end
    rots 
  end
    
end

With all the above in place - and with the aid of a few brain cells - some deceptively complicated questions (like "How many different ways can £2 be made using any number of coins?") are essentially one-liners:

require 'benchmark'
require 'MyMath'

Benchmark.bm do |r|
  r.report {
    answer = 200.integer_partitions([200,100,50,20,10,5,2,1])
  }
end

Love it;-)