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:

- Values are selected for the numbers in each column as needed.
- Columns are summed
- 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…