# Perl Weekly Challenge 108

Task 1 this week is kind of a joke, but task 2 was interesting!

Write a script to declare a variable or constant and print its location in the memory.

### Solution

See below for explanation and any implementation-specific comments.

sub challenge(Any $variable) returns Int {$variable.WHERE;
}

sub MAIN {
my $variable = rand; #  say challenge($variable);
}


This program runs as such:

$raku ch-1.raku 140444494947864  ### Explanation Raku has the WHERE method built into it that “Returns an Int representing the memory address of the object,” we simply need to utilize that. #### Specific comments 1. rand is a built-in subroutine that returns a random integer. 2. Following the Scala style, I would normally use parentheses for subroutines that are not “pure” and write this as rand(), but Raku complains with the following error: Unsupported use of rand(). In Raku please use: rand. ## Task 2: Bell Numbers Write a script to display top 10 Bell Numbers. Please refer to the Wikipedia page for more information. ### Example • B0: 1, as you can only have one partition of zero element set. • B1: 1, as you can only have one partition of one element set {a}. • B2: 2 {a}{b} {a,b}  • B3: 5 {a}{b}{c} {a,b}{c} {a}{b,c} {a,c}{b} {a,b,c}  • B4: 15 {a}{b}{c}{d} {a,b,c,d} {a,b}{c,d} {a,c}{b,d} {a,d}{b,c} {a,b}{c}{d} {a,c}{b}{d} {a,d}{b}{c} {b,c}{a}{d} {b,d}{a}{c} {c,d}{a}{b} {a}{b,c,d} {b}{a,c,d} {c}{a,b,d} {d}{a,b,c}  ### Solution GitHub Link See below for explanation and any implementation-specific comments. use experimental :cached; sub challenge(Int$n where $n >= 0) is cached returns Int { #  given$n {
when 0|1 { 1 }
default {
my $n-minus-one =$n - 1;
gather for (0..$n-minus-one) ->$k {                       # 
take (^$n-minus-one).combinations($k) * challenge($k); #  }.sum } } } sub MAIN(Int$n = 10) {
say (^$n).map(&challenge); }  This program runs as such: $ raku ch-2.raku
(1 1 2 5 15 52 203 877 4140 21147)


### Explanation

Note: The question asks for the “top 10 Bell Numbers.” Since this is an ever-increasing sequence, there is no “top,” so I interpreted that to mean first 10 Bell numbers.

At first glance, this code has nothing to do with the input sequences, but digging into the above-linked Wikipedia entry, we find this equation:

$B_{n+1} = \sum_{k=0}^{n}{n \choose k}B_{k}$

This tells us that each Bell number is built upon by the previous Bell numbers. And we know the first two Bell numbers, so we can follow this.

Let’s re-write this to calculate $B_{2}$ and see how it works; remember, since we are calculating for $B_{n+1}$, we use $n = 1$.

$B_{2} = \sum_{k=0}^{1}{1 \choose k}B_{k}$

Which can be expanded as:

$B_{2} = {1 \choose 0}B_{0} + {1 \choose 1}B_{1}$

Which can be reduced to:

$B_{2} = (1)(1) + (1)(1) = 2$

So $B_{2} = 2$ and we can use that to calculate $B_{3}$ and so on.

Now that we have the algorithm figured out, and decided we are going to apply it to the first 10 terms, it’s a simple matter to map the function over the sequence from 0 (inclusive) to 10 (exclusive).

1. Since this is a recursive function, once we calculate $B_{2}$ we will use it in $B_{3}$, $B_{4}$ and so on. No use re-calculating it every time, so we memoize this function using the is cached trait.
2. gather/take is a construct to build up a sequence using a Supply (in this case, the for loop).