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#!/Users/colincrain/perl5/perlbrew/perls/perl-5.32.0/bin/perl
#
# happy-happy-joy-joy.pl
#
# Happy Numbers
# Submitted by: Robert DiCicco
# Write a script to find the first 8 Happy Numbers in base 10. For
# more information, please check out Wikipedia.
#
# Starting with any positive integer, replace the number by the sum
# of the squares of its digits, and repeat the process until the
# number equals 1 (where it will stay), or it loops endlessly in a
# cycle which does not include 1.
#
# Those numbers for which this process end in 1 are happy numbers,
# while those numbers that do not end in 1 are unhappy numbers.
#
# Example:
#
# 19 is Happy Number in base 10, as shown:
#
# 19 => 1^2 + 9^2
# => 1 + 81
# => 82 => 8^2 + 2^2
# => 64 + 4
# => 68 => 6^2 + 8^2
# => 36 + 64
# => 100 => 1^2 + 0^2 + 0^2
# => 1 + 0 + 0
# => 1
#
# commentary:
# What makes a number happy? According to one theory, whose
# origins are lost to history, it's the presence of a numeric
# imperative; of a timeline; of eventual closure to its
# existence.
#
# In short, as we travel through life, we end up repeating
# similar actions again and again. The particulars my change,
# but the framework is what we know and how we know, and that
# remains consistent. If that resolves then we get closure and
# ultimately from this a sense of purpose, but if we find
# ourselves in the same situation again and again, so much that
# we know the outcome before we start, than we are unhappy,
# stuck in a loop, going nowhere, without meaning for the
# future and by extension the past.
#
# This is a pretty heady philosophical position for a number,
# unexpected as we generally consider the mathematical realm
# removed from the internal conflicts the physical plane.
#
# I personally have always considered the numbers to be above
# such emotional tethers, otherwise engaged in cosmic
# connections that we on Earth are only just beginning to
# comprehend.
#
# This in turn leads to questions of anthromophication of these
# entities, and whether the idea itself of happiness transcends
# the human condition, somehow wrapped in a larger concept of
# unity and truth to a broader nature: that we experience what
# we call happiness because we are somehow engaging in some
# larger process without conflict. This brings to mind
# questions abouut the apparent singular directionality of
# time, and with that movement direction, of action combined
# with data.
#
# This last idea is modeled in container theory, providing
# another bridge between the worlds, so the fundamental premise
# seems somewhat less far-fetched.
#
# As stated before, the source of this philosophical premise
# about happy, and by extension unhappy, numbers is itself
# unknown, which is somehow consistent and ultimately fitting.
# The nunbers themselves are notoriously tight-lipped on the
# subject.
# background:
#
# At the intersection of dynamical systems and number theory
# lies the field of arithmetic dynamics, examining the evolving
# state of parameters in an transformation equation repeatedly
# applied to the same mutating data. This process of
# re-applying a mapping to a field of data points is known as
# an iterated function. In arithmetic dynamics these methods
# are used on number-theoretical functions, looking for
# underlying structures behind these equation classes.
#
# The transformation of squaring and summing the coefficients
# of a polynomial expansion of a representational number,
# repeatedly applied until the result stabilizes into either a
# set result of a known cyclic pattern is one such system, and
# the determiniation of a number's happiness an analysis of
# that system.
#
# The most familiar application of dynamical systems analysis
# if the preparation of the Mandelbrot set, which is constucted
# from points on the complex plane repeatedly iterated through
# the transformation
#
# *z* = *z*² + *c*
#
# The set, visualized for a given constant *c*, is those points
# where the reperated application of the iterated function is
# determined to converge, versus those where the result is
# unbounded, with emphasis placed on the properties of the edge
# between the two possible outcomes and its particular fractal
# nature.
#
# In general, the actual parameter ranges observed in a
# dynamical system are known as the state space, and the
# meta-analytical analysis and mapping of the possible
# parameter states for a given equation is known as the phase
# space. Constrained
#
# self-referential feedback systems metastable
# method:
# To actually accompish the task we will revisit a technique we
# developed back in PWC106, where we had to recognize a
# repeating cycle of digits in a decimal representation of a
# fraction. In identifying a reptend, we modeled long division,
# and recognized that should a given remainder repeat once we
# were carrying down 0s, then that cycle would continue
# indefinitely. In a similar way, if one of our arthimetic
# transformations produces a previously seen result then we
# know that some repeating cycle will bring us around to that
# point again, with no variance is allowed in the process.
#
# Thus we establish a "seen" hash of all results caluculated,
# and after a transormation compare the result to the keys in
# this hash; if the key is present we are trapped in a
# repeating cycle an cannot escape.
#
# Alternately, if our reduction reaches 1 then we have won and
# can stop the transformation loop.
#
#
#
# © 2022 colin crain
## ## ## ## ## ## ## ## ## ## ## ## ## ## ## ## ## ## ## ##
use warnings;
use strict;
use utf8;
use feature ":5.26";
use feature qw(signatures);
no warnings 'experimental::signatures';
use List::Util qw( sum );
my $q = shift // 8;
my @out;
while ( @out < $q and ++$_ ){
push @out, $_ if happy($_);
}
{
local $" = ', ';
$_ = "@out";
say;
}
sub happy ($num) {
my %seen = ( $num => 1 );
while ( $num != 1 ) {
$num = sum map { $_ ** 2 } split //, $num;
return 0 if $seen{$num};
$seen{$num} = 1;
}
return $num;
}
__END__
the 10-happy number sequence up to 100:
1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100
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