## 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:

def next_hour(hour):

return (hour + 1) % 12

def calculate_range(S_t):

x_start = S_t[0] % 50515093 + 1

x_end = S_t[1] % 50515093 + 1

y_start = S_t[2] % 50515093 + 1

y_end = S_t[3] % 50515093 + 1

return (x_start, x_end, y_start, y_end)

def calculate_C(t):

S_t = [290797]

for i in range(1, 4 * t):

S_t.append((S_t[i – 1] ** 2) % 50515093)

hours = [0] * (50515093 + 1)

current_hour = 0

for i in range(1, t + 1):

x_start, x_end, y_start, y_end = calculate_range(S_t[4 * i – 4 : 4 * i])

for x in range(x_start, x_end + 1):

for y in range(y_start, y_end + 1):

hours[x * 50515093 + y] = current_hour

current_hour = next_hour(current_hour)

return sum(hours)

C_100000 = calculate_C(10**5)

print(C_100000)

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

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