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;-)
