A total of 37 full Moons happen during 3 years in a solar calendar. Show that at least two of the 37 full moons must happen during the same month of the solar calendar.
This can be proven using the pigeonhole principle.
Three years in a solar calendar have 36 months (12 months × 3 years, ignoring leap years for this calculation).
If we have 37 full Moons distributed across 36 months, and each month could have at most one full Moon, we would account for only 36 full Moons. Since there are 37 full Moons, by the pigeonhole principle, at least one month must contain 2 full Moons.
Therefore, at least two of the 37 full Moons must happen during the same month of the solar calendar.
Explanation
This is an application of the pigeonhole principle. The chapter explains that the lunar cycle is about 29.5 days, while solar calendar months have 30 or 31 days. This means there are slightly more than 12 lunar cycles in a solar year (about 12.37), leading to approximately 37 full Moons in 3 years. Since there are only 36 months in 3 years, and 37 full Moons must be distributed among these 36 months, at least one month must have 2 full Moons.
Solution Steps
Step 1: Calculate the number of months in 3 years of solar calendar = = 36 months
Step 2: Given: Number of full Moons = 37
Step 3: Apply pigeonhole principle: If 37 items (full Moons) are placed in 36 containers (months), at least one container must have more than one item
Step 4: Therefore, at least one month must have 2 full Moons, proving that at least two full Moons occur in the same month