Problem 790

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:
$\begin{aligned} S_{0} & = 290797 \\ S_{t} & = S_{t - 1}^{2} mod 50515093 & t > 0 \end{aligned}$
The four numbers $N_{t} = (S_{4 t - 4}, S_{4 t - 3}, S_{4 t - 2}, S_{4 t - 1})$ 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 $3 \leq x \leq 9$ and $20 \leq y \leq 47$ , 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 $12 \to 1 \to 2 \to \dots$ .

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 $C (10^{5})$ .

时钟格阵

考虑一个长和宽均包含 $50515093$ 个点的格阵，每个格点上都放置有一台时钟。所有的时钟盘面上都只包含一根时针，且在开始时时针均指向 $12$ 。

序列 $S_{t}$ 按如下方式定义：
$\begin{aligned} S_{0} & = 290797 \\ S_{t} & = S_{t - 1}^{2} mod 50515093 & t > 0 \end{aligned}$
四个数 $N_{t} = (S_{4 t - 4}, S_{4 t - 3}, S_{4 t - 2}, S_{4 t - 1})$ 代表了格阵的一部分，其中第一对数表示横轴的范围，第二对数表示纵轴的范围。例如，若 $N_{t} = (3, 9, 47, 20)$ ，相应的格阵范围满足 $3 \leq x \leq 9$ 和 $20 \leq y \leq 47$ ，其中包含 $196$ 个时钟。

对于每个 $t$ $(t > 0)$ ，包含在上述由 $N_{t}$ 描述的格阵范围内的所有时钟均往后拨一小时，也即 $12 \to 1 \to 2 \to \dots$ 。

记 $C (t)$ 为经过第 $t$ 步后，所有时钟时针所指向的数之和。
已知 $C (0) = 30621295449583788$ ， $C (1) = 30613048345941659$ ， $C (10) = 21808930308198471$ ， $C (100) = 16190667393984172$ 。

求 $C (10^{5})$ 。