Scientists are able to predict the phase of the moon for every day of the year by understanding the mechanics of the Earth-Moon-Sun system. The relative positions of these celestial bodies determine the amount of sunlight that reflects off the Moon, resulting in different lunar phases. By calculating these positions using astronomical observations and mathematical models, scientists can accurately predict the phase of the Moon for any given date and time. Factors considered in these predictions include the Moon's orbit around Earth, Earth's orbit around the Sun, and the angle between the Earth, Moon, and Sun. These calculations allow astronomers to create lunar calendars and make long-term predictions about the Moon's appearance.