Here's my solution. Writing a method that can determine whether a number is

happy, and how happy it is, is fairly simple. Once the method exists, it's

fairly trivial to write short programs that use the method to determine the

happiness of a number, or to find the highest happy number or the happiest

number in a range, as requested by the quiz. But the Ruby Quiz generally

requests that you write a program, not a single method, so I decided to write a

program that can perform all of these tasks, depending on the input parameter.

And handle bases other than ten if desired, to boot.

Once I decided to do that, it made sense to optimize the method over multiple

runs, by memoizing results, and taking algorithmic shortcuts based on previously

memoized results. So my get_happy method is a bit more complicated than it was

originally, due to the optimizations.

Of course, once I added optimizations, I introduced bugs. They were subtle and a

bit tricky to track down. But they basically boiled down to this question: Is 1

a happy number, and if so, how happy? It's obvious that when you perform one

iteration of the happiness algorithm, the next number in the sequence has a

happiness value of one less than the current number. For example, take the

sequence used in the quiz description. It starts with 7, which is finally stated

to have a happiness value of 4. The remaining numbers in the sequence, 49, 97,

130, and 10, thus have happiness values of 3, 2, 1, and 0 respectively.

So, is 1 happy? If the definition of a happy number is that the algorithm

evenually reaches 1, then yes it is. What is its happiness value? It could

arguably be 0, because the algorithm goes to 1 right away, without generating

any other happy numbers. But, any number with a happiness value of 0, such as

10, has 1 as its next iterated value, which means that, according to the

paragraph above, 1 should have a happiness value of 1 less than 0, which would

be -1. So is 1's happiness value 0 or -1?

I guess it's an arbitrary choice. But until I realized what was going on, I had

a bug in which a number's happiness value would either be correct, or 1 higher

than correct, depending on whether or not it was being calculated based on a

previous memoized intermediate value. I had originally set 1's happiness value

to 0, but this caused 10's value to be calculated as 1 instead of 0, because it

was 1 higher than the happiness value of the next number in the sequence, which

was of course 1, whose happiness is 0. This happened only when 10's happiness

value was memoized during another number's sequence, but not when 10 itself had

been passed into the get_happy method. Then I naively changed 1's happiness

value to -1, to try to fix this. But this didn't work either, since -1 is my

magic value meaning "unhappy", so all numbers were determined to be unhappy

since 1's memoized value returned as -1 in the first optimization. So I changed

1's happiness value back to 0, and unconditionally decreased all numbers'

happiness values by 1, which also turned out to be wrong.

When I finally understood what was going on, I realized that the correct fix was

to add the "(temp != 1)" conditional in the first "if" statement, and the "ret

-= 1" line if the algorithm has iterated all the way to 1. At this point, 1's

happiness value isn't actually used in the algorithm for computing any other

number's happiness. It's only ever used if get_happy is called on the value 1

itself. And at last, the program works! (At least, I'm pretty sure it does )

#!/usr/bin/env ruby

## ···

#

# =Description

#

# This program determines the happiness of a number, or the happiest number and

# highest happy number in a range of numbers.

#

# A number's happiness is determined as follows: Sum the squares of the number's

# individual digits. Repeat this process with the result, until a value of 1 is

# reached, or until a value is repeated, thus indicating a loop that will never

# reach 1. A number for which 1 is reached is "happy". The number of other

# numbers generated besides 1 and the original number is its happiness value.

#

# =Usage

#

# happy.rb num1[-num2][:base]

#

# happy.rb takes a single argument. If the argument is a single number, that

# number's happiness value is displayed, or the number is said to be unhappy.

# If the argument is a range of numbers, such as "1-400", the happiness value of

# the happiest number (lowest number breaking ties) in that range is returned.

# If the argument ends with a colon and a number, such as "50:8" or "1-100:2",

# the number after the colon specifies the base of the first number(s). An

# unspecified base implies base ten.

require 'rdoc/usage'

#==============================================================================

# ----- Global variables -----

#==============================================================================

$hap_map = {} # Hash for memoization of happiness values

#==============================================================================

# ----- Instance methods -----

#==============================================================================

class String

# Indicates whether the string is a valid number of the specified base.

def is_num_of_base?(base)

# sub() call removes leading zeros for comparison

self.sub(/\A0+/, '').downcase == self.to_i(base).to_s(base).downcase

end

end

class Integer

# Pretty-print string including base, if base is not 10

def pretty(base)

self.to_s(base) + ((base == 10) ? "" : " (base #{base})")

end

end

#==============================================================================

# ----- Global methods -----

#==============================================================================

# This method returns the happiness value of the given number. A value of -1

# indicates that the number is unhappy.

def get_happy(num, base=10)

$hap_map[num] = 0 if num == 1 # Handles trivial case

return $hap_map[num] if $hap_map[num]

ret = 0

done = false

inters = []

temp = num

until done

digits = temp.to_s(base).split(//).map{|c| c.to_i(base)}

temp = digits.inject(0) {|sum, d| sum + d**2}

ret += 1

if (temp != 1) && $hap_map[temp]

# Optimization; use knowledge stored in $hap_map

if $hap_map[temp] == -1

ret = -1

done = true

else

ret += $hap_map[temp]

done = true

end

else

if temp == 1

ret -= 1 # Don't count 1 as an intermediate happy number

done = true

elsif inters.include?(temp)

ret = -1

done = true

else

inters << temp

end

end

end

$hap_map[num] = ret

# Optimization

if ret == -1

# If num is not happy, none of the intermediates are happy either

inters.each {|n| $hap_map[n] = -1}

else

# If num is happy, each intermediate has a happiness value determined by

# its position in the array

inters.each_index {|idx| $hap_map[inters[idx]] = (ret - 1) - idx}

end

return ret

end

# nums is a range of integers. This method returns two values: the happiest

# number, and the highest happy number, in the range. Two nil values will be

# returned if there are no happy numbers in the range.

def get_superlatives(nums, base)

happiest_num = nil

happiest_ness = nil

highest = nil

nums.each do |n|

happy = get_happy(n, base)

next if happy == -1

highest = n

if (!happiest_ness) || (happy > happiest_ness)

happiest_num = n

happiest_ness = happy

end

end

return happiest_num, highest

end

#==============================================================================

# ----- Script start -----

#==============================================================================

if ARGV.size != 1

RDoc.usage('Usage', 'Description')

end

# Parse arg

ARGV[0] =~ /\A([\d\w]+)(?:\-([\d\w]+))?(?::(\d+))?\Z/

num1, num2, base = $1, $2, $3

# Ensure legal arg

RDoc.usage('Usage', 'Description') unless num1

# Fill in defaults

base = 10 unless base

num2 = num1 unless num2

# Convert numbers from strings to numeric values

base = base.to_i

[num1, num2].each do |s|

unless s.is_num_of_base?(base)

puts "Error: #{s} is not a valid base #{base} number"

exit

end

end

num1 = num1.to_i(base)

num2 = num2.to_i(base)

# Calculate and print results

if num1 == num2

happiness = get_happy(num1, base)

print num1.pretty(base)

if happiness == -1

print " is not happy.\n"

else

print " has a happiness of #{happiness}\n"

end

else

if num1 > num2

num1, num2 = num2, num1

end

happiest, highest = get_superlatives((num1..num2), base)

if !highest

puts "None of those numbers are happy."

else

puts "The happiest number is " + happiest.pretty(base) +

", with a happiness of #{get_happy(happiest, base)}"

puts "The highest happy number is " + highest.pretty(base) +

", with a happiness of #{get_happy(highest, base)}"

end

end

--

Karl von Laudermann - karlvonl(a)rcn.com - http://www.geocities.com/~karlvonl

#!/usr/bin/env ruby

require "complex";w=39;m=2.0;w.times{|y|w.times{|x|c=Complex.new((m*x/w)-1.5,

(2.0*y/w)-1.0);z=c;e=false;49.times{z=z*z+c;if z.abs>m then e=true;break;end}

print(e ?" ":"@@");puts if x==w-1;}}