Standing on the shoulders of trolls

 Published on Saturday, 31st October 2020, 01:00 pm; Solved by 177;
Difficulty rating: 40%

Problem 732

$N$ trolls are in a hole that is $D_N$ cm deep. The $n$-th troll is characterized by:

  • the distance from his feet to his shoulders in cm, $h_n$
  • the length of his arms in cm, $l_n$
  • his IQ (Irascibility Quotient), $q_n$.

Trolls can pile up on top of each other, with each troll standing on the shoulders of the one below him. A troll can climb out of the hole and escape if his hands can reach to the surface. Once a troll escapes he cannot participate any further in the escaping effort.

The trolls execute an optimal strategy for maximizing the total IQ of the escaping trolls, defined as $Q(N)$.

$r_n = \left[ \left( 5^n \bmod (10^9 + 7) \right) \bmod 101 \right] + 50$
$h_n = r_{3n}$
$l_n = r_{3n+1}$
$q_n = r_{3n+2}$
$D_N = \frac{1}{\sqrt{2}} \sum_{n=0}^{N-1} h_n$.

For example, the first troll ($n=0$) is 51cm tall to his shoulders, has 55cm long arms, and has an IQ of 75.

You are given that $Q(5) = 401$ and $Q(15)=941$.

Find $Q(1000)$.