Certainly! This is a classic problem related to the Pigeonhole Principle.
Problem Statement
Show that if you select any 32 days from a year, then at least three of these days must fall in the same month.
Explanation Using the Pigeonhole Principle
Step 1: Understand the problem
- There are 12 months in a year.
- We choose 32 days from the year.
- We want to prove that among these 32 days, at least three days lie in the same month.
Step 2: Apply the Pigeonhole Principle
- Think of the 12 months as 12 pigeonholes.
- The 32 chosen days are the pigeons.
- We want to distribute 32 pigeons into 12 pigeonholes.
Step 3: Use the principle
- If we want to avoid having 3 days in the same month, the maximum number of days we could place in each month is 2.
- So, if each month has at most 2 days, the total number of days chosen would be at most:
12×2=24 days12\times 2=24\text{ days}12×2=24 days
Step 4: Compare with the actual number of days chosen
- We have chosen 32 days, which is greater than 24.
- This means it is impossible to place 32 days into 12 months without having at least one month with 3 or more days.
Conclusion
By the Pigeonhole Principle, when choosing 32 days from a year, at least three of these days must fall in the same month. If you'd like, I can also provide a similar proof for other numbers or variations!
