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!