Eric I. said it best in a note attached to his solution:

I really enjoyed this quiz. On the surface it seems relatively

simple.

But there are some important subtleties.

This is the realization I came to as well. When Brian first suggested

the quiz,

we discussed the possible solutions. When I say discussed, I mean that

Brian

generated a nearly endless stream of excellent ideas while I watched it

all

happen. The more he generated, the more I realized how much I liked the

problem. It’s quite rich.

There are many ways to approach this quiz. Brian’s own solution

generates bit

map patterns representing blanks and non-blanks in a ticket, then uses

those

patterns to construct books filling in numbers as it works. Eric I.

used a

backtracking search, verifying that a solution is still possible as each

ticket

is added. Below I will examine a slightly simpler solution, that still

produces

pretty good results.

Andy Restrepo’s code is a random generation scheme. It builds random

books from

random tickets made out of random rows. If the book doesn’t end up

being valid,

it just tries again. However, it arranges the random generation such

that rows

are correct more often than you might expect.

Let’s jump into the code to see how it does that. The process begins

with this

trivial call:

TicketGenerator.new.print_book

That’s Ruby’s default new() and TicketGenerator doesn’t define an

initialize()

method, so let’s skip to print_book():

class TicketGenerator

def print_book

# Keep generating ticket books until a valid

# one is returned. Then, print out the tickets.

book = build_book until book

book.each { |t| t.print_ticket; puts “\n”}

end

end

This is the process I explained earlier. An attempt is made to build a

book.

If we get one, we’re done. Otherwise we try again. When we have them

all, the

tickets in the book are printed.

Let’s dig deeper into book creation:

class TicketGenerator

def build_book

# Generate 18 rows and divide them between six tickets

init_bins

all_rows = Array.new(18){ retrieve_row }

tickets = Array.new

0.step(15, 3) do |x|

ticket = Ticket.new(all_rows[x…x+3].sort_by { rand })

tickets.push(ticket)

# If an invalid ticket is found, indicate failure

# by setting the return value to false.

if not ticket.is_valid?

tickets = false; break

end

end

tickets

end

end

The first two lines here make use of other methods in the class to

create a set

of 18 rows. We will get to those methods in a moment, but take it on

faith for

now.

The rest of the method walks those rows in threes, making Ticket objects

out of

them. Each time a Ticket is made, the code ensures it is valid or bails

out. We

haven’t seen this class yet either, but it shouldn’t too hard to guess

what the

is_valid?() method does.

Here are the row making methods (with one minor edit by me):

class TicketGenerator

def init_bins

# Create and fill the 9 bins of numbers, corresponding to

# the allowed numbers for each column.

@bins = Array.new

# 1 through 9

@bins << (1…9).sort_by{ rand }

# 10 through 19, 20 through 29, etc.

10.step(70, 10) do |x|

@bins << (x…x+9).sort_by{ rand }

end

# 80 through 90

@bins << (80…90).sort_by{ rand }

end

def retrieve_row

# Create a row by pulling one number from each of five non-empty

bins.

row = Array.new(9, nil)

# Randomize which bins to choose from, but favor the most filled

# bins – so we don’t end up with less than 5 non-empty bins with

# still more rows to create.

bin_index_array = (removed_email_address@domain.invalid).sort_by{ |b|

[@bins[b].length, rand]

}

5.times do

bin_index = bin_index_array.pop

row[bin_index] = @bins[bin_index].pop

end

row

end

end

First, init_bins() fills an Array with Arrays of randomized numbers for

each

column. These columns span six tickets and are used to build the entire

book.

Now, retrieve_row() is used to create the 18 rows the code eventually

builds

books out of. It works by creating an empty row, choosing five of the

column

bins, and adding a random number to the row from each bin. The

selection of

bins is not-quite-random and that turns out to be the best and worst

feature of

this solution.

Bins are sorted first by length and then in random order before a pick

is made

from the end. This means that a selection will always be made from a

bin with

the most members remaining. That bin is selected randomly from those

with the

same number of members remaining, but that doesn’t mean the choice is

random.

Think of it this way, the bins don’t even start with the same number of

members.

The eighth bin has eleven and will always sort to be chosen first.

Likewise,

the zeroth bin always sorts first and thus cannot be selected by this

algorithm

during the creation of the first row. Even after the first row is

created the

eighth bin will still be tied for most entries and thus it will always

come up

again in the second row. That means the first ticket produced by this

solution

will always have two or three numbers in the final column. Never just

one.

The downside of all of this is that the tickets aren’t completely

random.

There’s a pattern to them and you can find it if you look for it. Given

that,

these books would be fine for casual play, but probably not for games

where

money is on the line.

The upside is speed. Even though tickets are randomly generated, the

sorting

keeps valid combination extremely likely. In a set of ten sample runs I

conducted the code only called build_book() a maximum of two times per

run and

it only needed one call eight out of ten times. A truly random pattern

would

need a lot more attempts to find a workable solution. You can simulate

this

scenario by removing the bin length sort condition from Andy’s code. It

still

finds answers, but it can take some time.

The last bit of TicketGenerator code sets method visibility:

class TicketGenerator

private :init_bins, :retrieve_row, :build_book

public :print_book

end

The other piece of this puzzle is the Ticket class, so we will look at

that now.

Tickets are just three rows that know how to validate and draw

themselves. The

initialization process for a Ticket is spread over three methods, so

let’s begin

with those:

class Ticket

def initialize(rows)

# A ticket consists of an array of three rows,

# with 5 numbers and 4 nil entries per row.

@rows = rows

@empty_column = false

validate_ticket

end

def validate_ticket

# Convert three rows of 9 numbers into 9 columns of three numbers,

# check that each column satisfies the ascending order constraint,

# and then convert back into rows.

columns = Array.new(9) { [] }

columns.each { |c| @rows.each { |r| c << r.shift }; rectify© }

@rows.each { |r| columns.each { |c| r << c.shift } }

end

def rectify(column)

# If there are 2 or 3 numbers in a column, they must

# appear in increasing order downward. If they don’t, then

# swap the numbers around while maintaining 5 numbers

# in each row.

case column.nitems

when 0 then @empty_column = true

when 1 then column # do nothing

when 2

nil_index = column.index(nil)

non_nils = [0,1,2] - [nil_index]

first_nn, last_nn = non_nils.first, non_nils.last

# Swap the two non-nil elements

if column[first_nn] > column[last_nn]

column[first_nn], column[last_nn] = column[last_nn],

column[first_nn]

end

when 3 then column.sort! # just sort the three numbers

end

end

end

You can see that initialize() just stores the rows and kicks off

validate_ticket(). That method does a bit of a dance to rearrange the

rows into

columns, clean and validate those columns, and change them back.

Array#transpose() probably would have simplified this process a bit.

The real work happens in rectify(). This method serves two purposes.

Its main

job is to reorder the numbers in a column to ensure they count down. As

it

works though, it watches for empty columns that would invalidate the

Ticket.

Once we know whether or not a Ticket is valid, we need to provide easy

access to

that information:

class Ticket

def is_valid?

not @empty_column

end

end

That’s the method we saw TicketGenerator using to determine when a book

build

needed to be restarted.

Tickets can also print themselves:

class Ticket

def print_ticket

puts “±—” * 9 + “+”

@rows.each do |row|

line = row.inject("|") do |str, x|

if not x

str + " |"

elsif x < 10

str + " #{x} |"

else

str + " #{x} |"

end

end

puts line

puts “±—” * 9 + “+”

end

end

end

That code just walks the rows, printing fields as it goes. The if chain

in the

middle could be simplified a bit to:

str + " %2s |" % x

Finally, the code again sets the visibility for these methods:

class Ticket

private :validate_ticket, :rectify

public :print_ticket, :is_valid?

end

My thanks to all the book builders. I didn’t expect to see so many

varied

approaches.

Tomorrow, I will put you to work solving a long-standing Ruby Q.

issue…