Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game. The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains m dots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability . Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.

这道题目其实是一个经典的概率论问题，即求 n 次投掷 m 面骰子的最大点数的期望。

我们可以用以下公式计算最大点数的期望：

E = sum(k=1 to m, k * ((k/m)^n - ((k-1)/m)^n))

其中，^ 表示指数运算，sum 表示求和。

这个公式的含义是，我们枚举最大点数 k，然后计算 k 在 n 次投掷中出现的概率，即 (k/m)^n - ((k-1)/m)^n。然后将 k 乘上这个概率，再将所有的 k 的期望值加起来，即为最大点数的期望。

时间复杂度为 O(m)，可以通过此题。

参考代码如下：

相关问题

cv2.COLORMAP_TWILIGHT

cv2.COLORMAP_TWILIGHT是OpenCV中的一种颜色映射。根据引用，COLORMAP_TWILIGHT_SHIFTED是COLORMAP_TWILIGHT的一个变体，它是一种从蓝色到红色的渐变色映射。这种颜色映射可以用于将灰度图像转换为彩色图像，以增强图像的可视化效果。

以下是使用cv2.COLORMAP_TWILIGHT_SHIFTED的示例代码[^1]：

import cv2
import numpy as np

# 生成一个灰度图像
gray_image = np.ones((100, 100), dtype=np.uint8) * 128

# 将灰度图像转换为彩色图像
color_image = cv2.applyColorMap(gray_image, cv2.COLORMAP_TWILIGHT_SHIFTED)

# 显示彩色图像
cv2.imshow("Color Image", color_image)
cv2.waitKey(0)
cv2.destroyAllWindows()

运行上述代码后，将显示一个从蓝色到红色的渐变色图像。

Maghrib In the Sunni's point of view, the time for Maghrib prayer begins once the Sun has completely set beneath the horizon, that is, Maghrib = Sunset (some calculators suggest 1 to 3 minutes after Sunset for precaution). In the Shia's view, however, the dominant opinion is that as long as the redness in the eastern sky appearing after sunset has not passed overhead, Maghrib prayer should not be performed. It is usually taken into consideration by assuming a twilight angle like Maghrib = Dhuhr + T(4).使用jiava演示

抱歉，作为一个语言模型AI，我没有办法演示Java代码。但我可以给你提供一个Java方法来计算Maghrib时间，你可以在你的程序中使用它：

public static Date calculateMaghribTime(Date date, double latitude, double longitude) {
    Calendar calendar = Calendar.getInstance();
    calendar.setTimeZone(TimeZone.getTimeZone("UTC"));
    calendar.setTime(date);

    double sunsetTime = PrayerTimesUtils.calculateSunsetTime(calendar, latitude, longitude);
    double maghribTime = sunsetTime + 1.5 / 60.0;

    calendar.set(Calendar.HOUR_OF_DAY, (int) maghribTime);
    calendar.set(Calendar.MINUTE, (int) ((maghribTime - (int) maghribTime) * 60));
    calendar.set(Calendar.SECOND, 0);
    calendar.set(Calendar.MILLISECOND, 0);

    return calendar.getTime();
}

这个方法使用了另一个工具类 `PrayerTimesUtils` 来计算日出和日落时间，你需要自行实现这个工具类。在这个方法中，我们先计算出日落时间，然后加上一个预留的时间（这里是1.5分钟），就得到了Maghrib时间。最后，我们将时间设定为当地的小时和分钟，并返回一个 `Date` 对象。