Clock Grid

There is a grid of length and width 50515093 points. A clock is placed on each grid point. The clocks are all analogue showing a single hour hand initially pointing at 12.
A sequence $S_t$ is created where:
$$
\begin{align}
S_0 &= 290797\\
S_t &= S_{t-1}^2 \bmod 50515093 &t>0
\end{align}
$$
The four numbers $N_t = (S_{4t-4}, S_{4t-3}, S_{4t-2}, S_{4t-1})$ represent a range within the grid, with the first pair of numbers representing the x-bounds and the second pair representing the y-bounds. For example, if $N_t = (3,9,47,20)$, the range would be $3\le x\le 9$ and $20\le y\le47$, and would include 196 clocks.
For each $t$ $(t>0)$, the clocks within the range represented by $N_t$ are moved to the next hour $12\rightarrow 1\rightarrow 2\rightarrow \cdots $.
We define $C(t)$ to be the sum of the hours that the clock hands are pointing to after timestep $t$.
You are given $C(0) = 30621295449583788$, $C(1) = 30613048345941659$, $C(10) = 21808930308198471$ and $C(100) = 16190667393984172$.
Find $C(10^5)$.

This is a complex problem that involves number theory, sequences, and modular arithmetic.

To solve it, we first need to understand the problem and how the sequence of numbers is created. Here is a potential approach:

Step 1: Creation of Sequence $S_t$
Every single $S_t$ is obtained from the previous $S_{t-1}$ by squaring it and finding the remainder when it’s divided by 50515093. This is achieved through the formula $S_t=S_{t-1}^2 \bmod 50515093$

Step 2: Creation of $N_t$
The range for each $t$ is defined by four numbers $N_t = (S_{4t-4}, S_{4t-3}, S_{4t-2}, S_{4t-1})$. For each $t$, these four numbers are calculated using $S_t$.

Step 3: Moving the Clock Hands
Next, we move the clock hands within the range defined by $N_t$ to the next hour in the sequence 12, 1, 2, …, 11.

Step 4: Calculate $C(t)$
$C(t)$ is defined as the sum of the hours that the clock hands are pointing to after timestep $t$. This requires adding up the “hours” of all the clocks in the grid range defined by $N_t$.

Given the complexity of this problem and the large size of $t=10^5$, we would likely need to come up with an efficient algorithm or use a programming language to compute $C(10^5)$.

Writing code in Python or another language could allow us to iteratively calculate $S_t$, $N_t$, and $C(t)$ for the range $t=0$ to $t=10^5$.

Here’s the Python code implementation to solve the problem:

This problem, formatted as it is, appears from Project Euler (Problem 790).

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