Question

Suppose someone (say, Aesop) is marking days in some leap year (say, 2948). You do not know which days he marks, only how many. Use this to answer the following questions. (Warning: Some, but not all, of these questions use the Pigeon-Hole Principle.) (a) How many days would Aesop have to mark before you can conclude that he marked two days in January? (b) How many days would Aesop have to mark before you can conclude that he marked two days in February? (c) How many days would Aesop have to mark before you can conclude that he marked two days in the same month? (d) How many days would Aesop have to mark before you can conclude that he marked three days in the same month? (e) How many days would Aesop have to mark before you can conclude that he marked three days with the same date (for example, the third of three different months, or the 3 ist of three different months)? (f) How many days would Aesop have to mark before you can conclude that he marked two consecutive days (for example, January 31 and February 1 )? (g) How many days would Aesop have to mark before you can conclude that he marked three consecutive days?

Short answer

(a) 32 days, (b) 30 days, (c) 13 days, (d) 25 days, (e) 32 days, (f) 367 days, (g) 368 days.

Step-by-step solution

Determine Days in January

January has 31 days. According to the Pigeonhole Principle, if Aesop marks 32 days, then there must be at least two of these in January.

Determine Days in February (Leap Year)

February in a leap year has 29 days. Using the Pigeonhole Principle, Aesop must mark 30 days to ensure that at least two are marked in February.

Apply Pigeonhole Principle for Any Month

To ensure two days are marked in the same month, Aesop must mark at least 13 days. Since there are 12 months, marking 13 days ensures at least 2 in one month.

Ensure Three Days in a Month

For three days in one month, divide Aesop's total days by 12 and add 1 to ensure one month has at least 3 days. Thus, he must mark 25 days.

Marking Three Days on the Same Day Across Different Months

Since a month can have at most 31 days, and there are 12 months, marking 32 relieves you of having the same date in at least 3 different months.

Marking Two Consecutive Days

To ensure that Aesop has marked two consecutive days, he should mark at least 367 days, since there are 366 pairs of consecutive days in a leap year.

Marking Three Consecutive Days

To ensure three consecutive days, Aesop should mark at least 368 days since, including every possible sequence, a leap year contains 366 triplets of consecutively marked days.

Key concepts

Discrete Mathematics

Discrete mathematics may sound complex, but it's mainly about dealing with distinct, separate items or events that can be counted. It's used in computer science, logic, and set theory. A core concept within discrete mathematics is understanding how different principles can optimize or predict outcomes. One such principle is the Pigeonhole Principle. This principle states that if you put more pigeons into pigeonholes than you have holes, then at least one hole will contain more than one pigeon. In mathematical terms, if you have more items than containers, at least one container must hold more than one item. This simple yet powerful idea helps solve many real-world problems in discrete mathematics by making predictions about group distribution. For example, in the case of marking days on a calendar, if Aesop marks a number of days, the principle helps us determine when at least two marked days fall within the same month or period.

Leap Year Calculation

Leap year calculation is fundamental in understanding how our calendar works. Typically, a leap year occurs every 4 years to help synchronize our calendar year with the solar year, or the time it takes the Earth to complete its orbit around the Sun. Normally, a year is 365 days, but a complete orbit takes approximately 365.25 days. To address this discrepancy, an extra day is added to the calendar every four years making it 366 days. In February, this extra day is added resulting in 29 days instead of the usual 28. The rules for a leap year include that it must be divisible by 4. However, if the year can be evenly divided by 100, it is NOT a leap year, unless it can be evenly divided by 400, in which case it is a leap year. These rules precisely control the leap year calculation ensuring it accounts accurately for the Earth's orbit.

Consecutive Days

When tackling the idea of consecutive days, especially in the context of marking days, it is crucial to understand their underlying significance. Consecutive days occur one after another within a timeline, maintaining an unbroken progression. In a leap year, which has 366 days, there are 365 possible pairs of consecutive days (like March 1 and March 2), and 364 triplets (like March 1, 2, and 3). If Aesop wishes to ensure that he marks two or three consecutive days, the Pigeonhole Principle comes in handy. Using this principle means requiring that he mark more days than the total number of pairs or triplets within the year, ensuring overlap. For example, marking 367 days guarantees at least two consecutive days because there are only 366 pairs possible in a leap year. Similarly, marking 368 days ensures at least three consecutive days.