Day 11: Plutonian Pebbles

Let $\ell$ be a list where for $e\in\ell$ we have $e\in\mathbb{N}$.

Part A

Algorithm 1:

• let $n\leftarrow$ invoke Algorithm 2 with argument $k=25$;
• return $n$.

Algorithm 2 with argument $k\in\mathbb{N}$:

• let $\Sigma\leftarrow 0$;
• for $e\in\ell$:
  • let $a\leftarrow$ invoke Algorithm 3 with arguments $n=e$ and $k$;
  • assign $\Sigma\leftarrow\Sigma+a$;
• return $\Sigma$.

Algorithm 3 with arguments $n,k\in\mathbb{N}$:

• if $k=0$, then return $0$;
• if $n=0$, then:
  • let $a\leftarrow$ invoke Algorithm 3 with arguments $n=1$ and $k-1$;
  • return $a$;
• let $d\leftarrow\lfloor\log_{10}{n}+1\rfloor$;
• if $2\mid d$, then:
  • let $a\leftarrow$ invoke Algorithm 3 with arguments $n\mod 10^{\frac{d}{2}}$ and $k-1$;
  • let $b\leftarrow$ invoke Algorithm 3 with arguments $\left\lfloor\frac{n}{10^{\frac{d}{2}}}\right\rfloor$ and $k-1$;
  • return $a+b$;
• let $a\leftarrow$ invoke Algorithm 3 with arguments $2024\cdot n$ and $k-1$;
• return $a$.

Part B

Algorithm 3:

• let $n\leftarrow$ invoke Algorithm 2 with argument $k=75$;
• return $n$.