#!/usr/bin/perl
use strict;
use warnings;
use 5.014;
=head1 DESCRIPTION
In a 3x2 cross-hatched grid, a total of 37 different rectangles could be
situated within that grid as indicated in the sketch.
There are 5 grids smaller than 3x2, vertical and horizontal dimensions being
important, i.e. 1x1, 2x1, 3x1, 1x2 and 2x2. If each of them is cross-hatched,
the following number of different rectangles could be situated within those
smaller grids:
1x1: 1 2x1: 4 3x1: 8 1x2: 4 2x2: 18
Adding those to the 37 of the 3x2 grid, a total of 72 different rectangles
could be situated within 3x2 and smaller grids.
How many different rectangles could be situated within 47x43 and smaller grids?
=cut
# A 2-dimensional cache.
# $Rects_Coefficients_Cache[$rect_x][$rect_y] = { min => $min, offset => $offset};
# Formula for step is calculated at 2*$board_dim + $offset
my @Rects_Coefficients_Cache;
# A 4-dimensional cache:
# 0 - board large dim.
# 1 - board small dim.
# 2 - rect large dim.
# 3 - rect large dim.
# Each value is:
# {
#
# Cache for the boards:
# $Boards_Cache[$board_x][$board_y] = { 'num_unique' => , 'rects' => \@rects, }
# rects is:
# $rects[$rect_x][$rect_y]
my @Boards_Cache;
sub get_unique_rects
{
# $x >= $y >= $rect_x , $rect_y
my ($x, $y, $rect_x, $rect_y) = @_;
my $board_struct = ($Boards_Cache[$x][$y] //= +{});
if (! defined($board_struct->{num_unique}))
{
# Calculate the unique rectangles.
$board_struct->{rects} //= [];
}
return $board_struct;
}
sub round_two_up
{
my ($n) = @_;
return (($n&0x1) ? $n+1 : $n);
}
sub get_total_rects
{
my ($x, $y) = @_;
my $sum = 0;
foreach my $xx (1 .. $x)
{
foreach my $yy (1 .. $y)
{
$sum += $xx * $yy * ($x - $xx + 1) * ($y - $yy + 1);
}
}
my $diag_sum = 0;
foreach my $xx (1 .. $x)
{
foreach my $yy (1 .. $y)
{
foreach my $rect_x (1 .. ($xx<<1))
{
foreach my $rect_y (1 .. ($yy<<1))
{
my $x_even_start = $rect_x;
my $x_even_end = (($xx<<1) - $rect_y);
my $y_even_end = (($yy<<1) - ($rect_x+$rect_y));
# For the even diagonals.
if ($x_even_end > $x_even_start and $y_even_end > 0)
{
$diag_sum +=
(((($x_even_end&(~0x1)) - round_two_up($x_even_start)
>> 1 ) * ($y_even_end >> 1)) <<
($rect_x == $rect_y ? 0 : 1)
);
$diag_sum +=
(((((($x_even_end&0x1)?$x_even_end:$x_even_end-1) - ($x_even_start|0x1))
>> 1 ) * (($y_even_end >> 1) - 1)) <<
($rect_x == $rect_y ? 0 : 1)
);
}
}
}
}
}
return {num_straight_rects => $sum, num_diag_rects => $diag_sum};
}
my ($x, $y) = @ARGV;
# my $num_rects = get_rects(47, 43, 47, 43);
my $rects_struct = get_total_rects($x, $y);
my $num_straight_rects = $rects_struct->{num_straight_rects};
my $num_diag_rects = $rects_struct->{num_diag_rects};
print "Straight Rects = $num_straight_rects\n";
print "Diag Rects = $num_diag_rects\n";
print "Total sum = " , $num_straight_rects + $num_diag_rects, "\n";