This is my 25th week participating into the weekly challenge.

## TASK #1

### Flip Binary

You are given a binary number **B**, consisting of **N** binary digits `0`

or `1`

: **s0, s1, …, s(N-1)**.

Choose two indices **L** and **R** such that **0 ≤ L ≤ R < N** and flip the digits **s(L), s(L+1), …, s(R)**. By flipping, we mean change **0** to **1** and vice-versa.

For example, given the binary number `010`

, the possible flip pair results are listed below:

- L=0, R=0 the result binary:
`110`

- L=0, R=1 the result binary:
`100`

- L=0, R=2 the result binary:
`101`

- L=1, R=1 the result binary:
`000`

- L=1, R=2 the result binary:
`001`

- L=2, R=2 the result binary:
`011`

Write a script to find the indices (**L,R**) that results in a binary number with maximum number of **1**s. If you find more than one maximal pair **L,R** then print all of them.

Continuing our example, note that we had three pairs **(L=0, R=0)**, **(L=0, R=2)**, and **(L=2, R=2)** that resulted in a binary number with two **1**s, which was the maximum. So we would print all three pairs.

This was a bit of a challenging one as I don’t usually work with bit operators. I didn’t want to “cheat” and work with strings that represent binary and decided to use purely binary arithmetic for this challenge.

The hard part of those problem was flipping the correct bits. I solved this by splitting the binary number into two parts using masks. The first part being the bits to flip and the second part being the bits to keep.

So lets say we want to flip 010 where L = 0 and R = 1, we would need to seperate 010 into two parts 01 and 0, We use the masks which are 110 and 001 respectively.

Then add the flipped bits and the unflipped bits to get the answer.

Here is a simple example for 010 where L = 0 and R = 1

Split into two parts:

flip_bits: 010 & 110 = 010

kept_bits: 010 & 001 = 000

we then flip the “flip_bits”

~ 010 = 101

Reapply the mask to the flip_bits

101 & 110 = 100

Then add both parts

101 + 000 = 100

So 010 becomes 100, for L = 0 and R = 1

It’s cognitively heavy but it works, hopefully the code explains it better than i can.

To test: remember to use the integer representation of the bit string and the length of the bit string.

For example, to test 010:

./ch-1.pl 2 3

or

perl6 ./ch-1.p6 2 3

if the length of the bitstring isn’t supplied

like ./ch-1.pl 2, then the length of the bitstring will be calculated dynamically. In this case the bit string will look like 10, giving different results that the example.

#### Perl 5 solution

```
#!/usr/bin/perl
# Test: ./ch-1.pl 2 3
use strict;
use warnings;
use feature qw /say/;
my $b = $ARGV[0];
my $n = $ARGV[1];
# Calculate $n if not supplied
$n = int(log($b) / log(2)) + 1
unless($n);
# Lets keep it to 1 to 32 bits and
# throw out bits outside of our range
$n = 32 if ($n > 32);
$n = 1 if ($n < 1);
my $mask = (2 ** $n - 1);
$b = $b & $mask;
# Print the original number
say "ORIGINAL: " . sprintf("%0${n}B", $b) . "\n";
# Store the solution;
my %solutions;
my $longest_soluton = 0;
# flip algorithm
for my $l (0 .. $n -1) {
for my $r ($l .. $n - 1) {
# We create two bit masks.
# The first mask is what we use to flip the bits
# and the second mask is what we use to keep the
# original builts then add up the flipped bits
# with the kept bits
# Flip Mask
my $flip_mask = 0;
for my $i ($l .. $r) {
$flip_mask += (2 ** ($n - $i - 1) );
}
# Calculate the keep mask
my $keep_mask = (~ $flip_mask) & $mask;
# Flip the relevant bits and calculate kept bits
my $flipped_bits = ~ ($b & $flip_mask) & $flip_mask;
my $kept_bits = $b & $keep_mask;
# Add the bits outside the flipped bit
my $flipped_number = $flipped_bits + $kept_bits;
# Now store the number of ones
my $length = calculate_true_bits($flipped_number);
# Store the solutions
$solutions{$length} = []
unless ($solutions{$length});
push @{$solutions{$length}}, {
L => $l,
R => $r,
number => $flipped_number
};
# Length of the longest solution
$longest_soluton = $length
if ($longest_soluton < $length);
}
}
# Print the solutions
say "SOLUTIONS length($longest_soluton):";
for my $solution (@{$solutions{$longest_soluton}}) {
say 'L: ' . $solution->{L} .
' R: ' . $solution->{R} .
' number: ' . sprintf("%0${n}B", $solution->{number});
}
# Calculate the number of true bits
sub calculate_true_bits {
my $number = shift;
my $count = 0;
do {
$count++ if ($number & 1);
} while ($number = $number >> 1);
return $count;
}
```

**Output**

```
ORIGINAL: 010
SOLUTIONS length(2):
L: 0 R: 0 number: 110
L: 0 R: 2 number: 101
L: 2 R: 2 number: 011
```

#### Raku solution

```
# Test: perl6 ch-1.p6 2 3
multi MAIN(Int $b where * > 0) {
my $n = ((log($b) / log(2)) + 1).Int;
$n = 32 if ($n > 32);
MAIN($b, $n);
}
multi MAIN(Int $a where * > 0 , Int $n where 0 < * < 32) {
# Throw out bits outside of our range
my $mask = (2 ** $n - 1);
my $b = $a +& $mask;
# Original binary
say "ORIGINAL: " ~ sprintf("%0*b", $n, $b) ~ "\n";
# Store the solution;
my %solutions;
my $longest_solution = 0;
# flip algorithm
for (0 .. $n -1) -> $l {
for ($l .. $n - 1) -> $r {
# Flip Mask
my $flip_mask = 0;
for ($l .. $r) -> $i {
$flip_mask += (2 ** ($n - $i - 1) );
}
# Calculate the keep mask
my $keep_mask = (+^ $flip_mask) +& $mask;
# Flip the relevant bits and calculate kept bits
my $flipped_bits = +^ ($b +& $flip_mask) +& $flip_mask;
my $kept_bits = $b +& $keep_mask;
# Add the bits outside the flipped bit
my $flipped_number = $flipped_bits + $kept_bits;
# Now store the number of ones
my $length = calculate-true-bits($flipped_number);
# Store the solutions
%solutions{$length} = []
unless (%solutions{$length});
my %solution = (
L => $l,
R => $r,
number => $flipped_number
);
%solutions{$length}.push(%solution);
# Length of the longest solution
$longest_solution = $length
if ($longest_solution < $length);
}
}
say "SOLUTIONS length($longest_solution):";
for (@(%solutions{$longest_solution})) -> $solution {
say 'L: ' ~ $solution{'L'} ~
' R: ' ~ $solution{'R'} ~
' Number: ' ~
sprintf("%0*b", $n, $solution{'number'});
}
}
# Calculate the number of true bits
sub calculate-true-bits(Int $n is copy) {
my $count = 0;
repeat {
$count++ if ($n +& 1);
} while ($n = $n +> 1);
return $count;
}
```

**Output**

```
ORIGINAL: 010
SOLUTIONS length(2):
L: 0 R: 0 number: 110
L: 0 R: 2 number: 101
L: 2 R: 2 number: 011
```

## TASK #2

### Wave Array

Any array **N** of non-unique, unsorted integers can be arranged into a wave-like array such that **n1 ≥ n2 ≤ n3 ≥ n4 ≤ n5** and so on.

For example, given the array **[1, 2, 3, 4]**, possible wave arrays include **[2, 1, 4, 3]** or **[4, 1, 3, 2]**, since 2 ≥ 1 ≤ 4 ≥ 3 and 4 ≥ 1 ≤ 3 ≥ 2. This is not a complete list.

Write a script to print all possible wave arrays for an integer array **N** of arbitrary length.

#### Notes:

When considering **N** of any length, note that the first element is always greater than or equal to the second, and then the ≤, ≥, ≤, … sequence alternates until the end of the array.

Brain was a bit exhausted from the last challenge so I just Brute forced this with https://metacpan.org/pod/Algorithm::Combinatorics and raku’s `permutations`

as we’re probably doing an exhaustive search.

I’m sure there is a mathematically better way to come up with a solution.

#### Perl 5 solution

```
#!/usr/bin/perl
# Test: ./ch-1.pl 4
use strict;
use warnings;
use feature qw /say/;
use Algorithm::Combinatorics qw(permutations);
my $n = $ARGV[0];
my @array = (1 .. $n);
# Permutations
my $iter = permutations(\@array);
# Loop through each combination
while (my $p = $iter->next) {
my $is_wave = 1;
for (my $i = 1; $i < scalar(@$p); $i++) {
if ( $i % 2 == 1 && $p->[$i] >= $p->[$i - 1] ||
$i % 2 == 0 && $p->[$i] <= $p->[$i - 1] ) {;
$is_wave = 0;
last;
}
}
say join ' ', @$p if ($is_wave);
}
```

**Output**

```
2 1 4 3
3 1 4 2
3 2 4 1
4 1 3 2
4 2 3 1
```

#### Raku solution

```
# Test: perl6 ch-2.p6 4
sub MAIN(Int $n where * > 0) {
[1 .. $n].permutations.grep({is-wave($_)}).join("\n").say;
}
# Is the array a wave
sub is-wave(@n) {
my $is_wave = True;
loop (my $i = 1; $i < @n.elems; $i++) {
if ( $i % 2 == 1 && @n[$i] >= @n[$i - 1] ||
$i % 2 == 0 && @n[$i] <= @n[$i - 1] ) {;
$is_wave = False;
last;
}
}
return $is_wave;
}
```

**Output**

```
2 1 4 3
3 1 4 2
3 2 4 1
4 1 3 2
4 2 3 1
```

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