This program solves addition problems with any number of terms. It
finds and displays all solutions to the problem.
The solving process is broken up into a sequence of simple steps all
derived from class Step. A Step can be something such as 1) choosing
an available digit for a given letter or 2) summing up a column and
seeing if the result matches an already-assigned letter. As steps
succeed the process continues with the following steps. But if a step
fails (i.e., there’s a contradiction) then the system backs up to a
point where another choice can be made. This is handled by recursing
through the sequence of steps. In fact, even when a solution is
found, the program still backtracks to find other solutions.
The expectation is that by testing for contradictions as early as
possible in the process we’ll tend to avoid dead ends and the result
will be much better than an exhaustive search.
For example, here are the steps for a sample equation:
send
+more
money
- Choose a digit for “d”.
- Choose a digit for “e”.
- Sum the column using letters “d”, “e” (and include carry).
- Set the digit for “y” based on last column summed.
- Choose a digit for “n”.
- Choose a digit for “r”.
- Sum the column using letters “n”, “r” (and include carry).
- Verify that last column summed matches current digit for “e”.
- Choose a digit for “o”.
- Sum the column using letters “e”, “o” (and include carry).
- Verify that last column summed matches current digit for “n”.
- Choose a digit for “s”.
- Verify that “s” has not been assigned to zero.
- Choose a digit for “m”.
- Verify that “m” has not been assigned to zero.
- Sum the column using letters “s”, “m” (and include carry).
- Verify that last column summed matches current digit for “o”.
- Sum the column using letters (and include carry).
- Verify that last column summed matches current digit for “m”.
- Display a solution (provided carry is zero)!
Eric
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====
This is a solution to Ruby Q. #128. As input it takes a "word
equation" such as “send+more=money” and determines all possible
mappings of letters to digits that yield a correct result.
The constraints are: 1) a given digit can only be mapped to a single
letter, 2) the first digit in any term cannot be zero.
The solving process is broken up into a sequence of simple steps all
derived from class Step. A Step can be something such as 1)
choosing an available digit for a given letter or 2) summing up a
column and seeing if the result matches an already-assigned letter.
As steps succeed the process continues with the following steps.
But if a step fails (i.e., there’s a contradiction) then the system
backs up to a point where another choice can be made. This is
handled by recursing through the sequence of steps. In fact, even
when a solution is found, the program still backtracks to find other
solutions.
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
@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
Step is an “abstract” base level class from which all the “concrete”
steps can be deriveds. 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 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 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 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 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
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 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
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
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
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)
How did you figured out that 2 = y in your