# Verbal Arithmetic (#128)

**URL:** <https://www.ruby-forum.com/t/verbal-arithmetic-128/107410>\
**Category:** Ruby\
**Created:** [June 21, 2007, 2:51pm UTC](https://www.ruby-forum.com/t/verbal-arithmetic-128/107410 "2007-06-21T14:51:14Z")\
**Posts on this page:** 1\
**Page:** 1

<div class="post-metadata">

**Author:** ![bbazzarrakk](https://www.ruby-forum.com/letter_avatar_proxy/v4/letter/b/47e85d/32.png) [@bbazzarrakk](https://www.ruby-forum.com/u/bbazzarrakk)\
**Post date:** [June 21, 2007, 2:51pm UTC](https://www.ruby-forum.com/t/verbal-arithmetic-128/107410/1 "2007-06-21T14:51:14Z")

</div>

I’m always impressed by the creativity of the solutions, but I think  
this week  
stands out even more than usual. I literally spent hours going through  
the  
solutions and learned some really great tricks from them. I wish I  
could take  
you on the same tour of the code, but that would take this summary into  
the  
range of a full text book in length.

Because I’m going to miss all of the following, let me point out some  
highlights  
for your own explorations:

- Though the brute-force solutions are slow, most of them handle any  
math equations Ruby can. That is an interesting advantage.
- Andreas L. sent in a fun preview of his Google Summer of Code  
project that looks to simplify many of these search problems we  
commonly use as quizzes.
- Glen’s solution is a nifty metaprogramming solution that customizes  
itself to the equation entered. It’s lightning quick too.
- Morton G. solved the quiz with some genetic programming and  
that code is still quite a bit zippier than a brute-force search.

The solution I will show is from Eric I. It has an interesting state  
machine  
design that tries to fail fast in an attempt to aggressively prune the  
search  
space. It too finds solutions quite rapidly, though it only works for  
addition  
problems.

Eric’s code breaks the equation down into a small series of steps.  
Instead of  
searching for a match for all numbers and then checking the result, this  
solution checks as many little sub-criteria as possible. Does just this  
column  
add up correctly, given what we know at this point? Is this digit a  
zero,  
because it starts a term somewhere else?

These smaller tests lead to failures that allow the search to skip large  
groups  
in the set of possible solutions. For example, if S can’t be seven in  
just one  
column, it’s impossible to have any scenario where S is seven and all  
such  
attempts can be safely skipped. That allows the code to zoom in on a  
correct  
answer faster.

Now that we understand the logic, let’s start tackling the code:

require ‘set’

# State represents the stage of a partially solved word equation. It

# keeps track of what digits letters map to, which digits have not yet

# been assigned to letters, and the results of the last summed column,

# including the resulting digit and any carry if there is one.

class State  
attr\_accessor :sum, :carry  
attr\_reader :letters

```
def initialize()
  @available_digits = Set.new(0..9)
  @letters = Hash.new
  @sum, @carry = 0, 0
end

# Return digit for letter.
def [](letter)
  @letters[letter]
end

# The the digit for a letter.
def []=(letter, digit)
  # if the letter is currently assigned, return its digit to the
  # available set
  @available_digits.add @letters[letter] if @letters[letter]

  @letters[letter] = digit
  @available_digits.delete digit
end

# Clear the digit for a letter.
def clear(letter)
  @available_digits.add @letters[letter]
  @letters[letter] = nil
end

# Return the available digits as an array copied from the set.
def available_digits
  @available_digits.to_a
end

# Tests whether a given digit is still available.
def available?(digit)
  @available_digits.member? digit
end

# Receives the total for a column and keeps track of it as the
# summed-to digit and any carry.
def column_total=(total)
  @sum = total % 10
  @carry = total / 10
end

```

end

# …

This State object tracks progress through the equation, which will be  
solved  
column by column. It has operations to track what each letter is  
currently  
assigned to, assign letters as they are determined, examine which digits  
have  
and have not been used, and track the sum of the last column plus any  
value  
carried over to the next column. There’s nothing too tricky in this  
data  
structure code.

What we need to go with this, is an algorithm that drives this State  
object to a  
solution. That code begins here:

# …

# Step is an “abstract” base level class from which all the “concrete”

# steps can be derived. It simply handles the storage of the next

# step in the sequence. Subclasses should provide 1) a to\_s method to

# describe the step being performed and 2) a perform method to

# actually perform the step.

class Step  
attr\_writer :next\_step  
end

# …

This base Step is about as simple as things get. I merely provides a  
means of  
storing the next step in the process.

Note that this class’s abstract status and the required implementation  
for  
subclasses are all handled through the documentation. That’s perfectly  
reasonable in a dynamic language like Ruby where we can count on duck  
typing to  
resolve to the proper methods when the search is actually being  
performed.

Let’s advance to a concrete implementation of the Step class:

# …

# This step tries assigning each available digit to a given letter and

# continuing from there.

class ChooseStep \< Step  
def initialize(letter)  
@letter = letter  
end

```
def to_s
  "Choose a digit for \"#{@letter}\"."
end

def perform(state)
  state.available_digits.each do |v|
    state[@letter] = v
    @next_step.perform(state)
  end
  state.clear(@letter)
end

```

end

# …

This ChooseStep handles the digit guessing. It is created for some  
letter and  
when perform() is triggered, it will try each unused in turn digit in  
that  
position. After a new guess is set, the ChooseStep just hands off to a  
later  
step to verify that the current guess works.

Here’s another Step subclass:

# …

# This step sums up the given letters and changes to state to reflect

# the sum. Because we may have to backtrack, it stores the previous

# saved sum and carry for later restoration.

class SumColumnStep \< Step  
def initialize(letters)  
@letters = letters  
end

```
def to_s
  list = @letters.map { |l| "\"#{l}\"" }.join(', ')
  "Sum the column using letters #{list} (and include carry)."
end

def perform(state)
  # save sum and carry
  saved_sum, saved_carry = state.sum, state.carry

  state.column_total =
    state.carry +
    @letters.inject(0) { |sum, letter| sum + state[letter] }
  @next_step.perform(state)

  # restore sum and carry
  state.sum, state.carry = saved_sum, saved_carry
end

```

end

# …

This SumColumnStep will be added whenever guesses had been made for an  
entire  
column. It’s job is to add up that column and update the State with  
this new  
total. You can see that it must save old State values and restore them  
when  
backtracking.

Once we know a column total, we can use that to set a letter from the  
solution  
side of the equation:

# …

# This step determines the digit for a letter given the last column

# summed. If the digit is not available, then we cannot continue.

class AssignOnSumStep \< Step  
def initialize(letter)  
@letter = letter  
end

```
def to_s
  "Set the digit for \"#{@letter}\" based on last column summed."
end

def perform(state)
  if state.available? state.sum
    state[@letter] = state.sum
    @next_step.perform(state)
    state.clear(@letter)
  end
end

```

end

# …

This AssignOnSumStep is added for letters in the solution of the  
equation. It  
will set the value of that letter to the calculated sum of the column,  
provided  
that is a legal non-duplicate digit choice.

When we have assigned that letter, we need to verify that the whole  
column makes  
sense mathematically:

# …

# This step will occur after a column is summed, and the result must

# match a letter that’s already been assigned.

class CheckOnSumStep \< Step  
def initialize(letter)  
@letter = letter  
end

```
def to_s
  "Verify that last column summed matches current " +
    "digit for \"#{@letter}\"."
end

def perform(state)
  @next_step.perform(state) if state[@letter] == state.sum
end

```

end

# …

Now, if we did all the guessing, summing, and assigning everything  
probably adds  
up. But as we continue through the equation, some numbers will already  
be  
filled in. Sums created using those may not balance with the total  
digit. This  
CheckOnSumStep watches for such a case.

If the sum doesn’t check out, this class causes backtracking. Note that  
all it  
has to do is not forward to the following steps which will cause  
recursion to  
unwind the stack until it has another option.

One last check can trim the search space further:

# …

# This step will occur after a letter is assigned to a digit if the

# letter is not allowed to be a zero, because one or more terms begins

# with that letter.

class CheckNotZeroStep \< Step  
def initialize(letter)  
@letter = letter  
end

```
def to_s
  "Verify that \"#{@letter}\" has not been assigned to zero."
end

def perform(state)
  @next_step.perform(state) unless state[@letter] == 0
end

```

end

# …

This CheckNotZeroStep is used to ensure that a leading letter in a term  
is  
non-zero. Again, it fails to forward calls when this is not the case.

One more step is needed to catch correct solutions:

# …

# This step represents finishing the equation. The carry must be zero

# for the perform to have found an actual result, so check that and

# display a digit -\> letter conversion table and dispaly the equation

# with the digits substituted in for the letters.

class FinishStep \< Step  
def initialize(equation)  
@equation = equation  
end

```
def to_s
  "Display a solution (provided carry is zero)!"
end

def perform(state)
  # we're supposedly done, so there can't be anything left in carry
  return unless state.carry == 0

  # display a letter to digit table on a single line
  table = state.letters.invert
  puts
  puts table.keys.sort.map { |k| "#{table[k]}=#{k}" }.join(' ')

  # display the equation with digits substituted for the letters
  equation = @equation.dup
  state.letters.each { |k, v| equation.gsub!(k, v.to_s) }
  puts
  puts equation
end

```

end

# …

This method first ensures that we are successful by validating that we  
have no  
remaining carry value. If that’s true, our equation balanced out.

The rest of the work here is just in printing the found result. Nothing  
tricky  
there.

We’re now ready to get into the application code:

# …

# Do a basic test for the command-line arguments validity.

unless ARGV[0] =~ Regexp.new(’^[a-z]+(+[a-z]+)\*=[a-z]+$’)  
STDERR.puts “invalid argument”  
exit 1  
end

# Split the command-line argument into terms and figure out how many

# columns we’re dealing with.

terms = ARGV[0].split(/+|=/)  
column\_count = terms.map { |e| e.size }.max

# Build the display of the equation a line at a time. The line

# containing the final term of the sum has to have room for the plus

# sign.

display\_columns = [column\_count, terms[-2].size + 1].max  
display = []  
terms[0…-3].each do |term|  
display \<\< term.rjust(display\_columns)  
end  
display \<\< “+” + terms[-2].rjust(display\_columns - 1)  
display \<\< “-” \* display\_columns  
display \<\< terms[-1].rjust(display\_columns)  
display = display.join("\n")  
puts display

# AssignOnSumStep which letters cannot be zero since they’re the first

# letter of a term.

nonzero\_letters = Set.new  
terms.each { |e| nonzero\_letters.add(e[0, 1]) }

# A place to keep track of which letters have so-far been assigned.

chosen\_letters = Set.new

# …

This code validates the input and breaks it into terms. After that, the  
big  
chunk of code here displays the equation in a pretty format, like the  
examples  
from the quiz description.

The rest of the code begins to divide up the input as needed to build  
the proper  
steps. The first tactic is to locate and letters that must be nonzero,  
because  
they start a term. A set is also prepared to hold letters that have be  
given  
values at any point in the process.

Here’s the heart of the process code:

# …

# Build up the steps needed to solve the equation.

steps = []  
column\_count.times do |column|  
index = -column - 1  
letters = [] # letters for this column to be added

```
terms[0..-2].each do |term| # for each term that's being added...
  letter = term[index, 1]
  next if letter.nil? # skip term if no letter in column
  letters << letter # note that this letter is part of sum

  # if the letter does not have a digit, create a ChooseStep
  unless chosen_letters.member? letter
    steps << ChooseStep.new(letter)
    chosen_letters.add(letter)
    steps << CheckNotZeroStep.new(letter) if
      nonzero_letters.member? letter
  end
end

# create a SumColumnStep for the column
steps << SumColumnStep.new(letters)

summed_letter = terms[-1][index, 1] # the letter being summed to

# check whether the summed to letter should already have a digit
if chosen_letters.member? summed_letter
  # should already have a digit, check that summed digit matches it
  steps << CheckOnSumStep.new(summed_letter)
else
  # doesn't already have digit, so create a AssignOnSumStep for
  # letter
  steps << AssignOnSumStep.new(summed_letter)
  chosen_letters.add(summed_letter)

  # check whether this letter cannot be zero and if so add a
  # CheckNotZeroStep
  steps << CheckNotZeroStep.new(summed_letter) if
    nonzero_letters.member? summed_letter
end

```

end

# …

This code breaks down the provided equation into the steps we’ve seen  
defined up  
to this point. Though it’s a fair bit of code, it’s pretty  
straightforward and  
very well commented. In short:

1. Values are selected for the numbers in each column as needed.
2. Columns are summed
3. Sums are assigned and or validated as needed.

With the setup complete, here’s the code that kicks the solver into  
action:

# …

# should be done, so add a FinishStep

steps \<\< FinishStep.new(display)

# print out all the steps

# steps.each\_with\_index { |step, i| puts “#{i + 1}. #{step}” }

# let each step know about the one that follows it.

steps.each\_with\_index { |step, i| step.next\_step = steps[i + 1] }

# start performing with the first step.

steps.first.perform(State.new)

Here the FinishStep is added, all steps are linked, and the perform()  
call is  
made to get the ball rolling. You can uncomment the second chunk of  
code to  
have a human-readable explanation of the steps added to the output.

My thanks to all the super clever solvers who tackled this problem. I  
was blown  
away with the creativity.

Tomorrow we will put Ruby Q. to work helping some friends of ours…