It’s 2012, which means it’s leap year. And leap day, Feb. 29, is upon us.

For this occasion, we bring you 10 ways to learn about and work with the calendar and to make leap day a teachable moment.

#### 1. Pick a Date, any Date …

You’ve probably noticed that your birthday falls on a different day of the week each year. This is because a calendar year does not consist of a whole number of weeks.

Each normal calendar year has 365 days, which means there are 52 full weeks and one extra day. If there were exactly 52 weeks between birthdays, your birthday would be on the same day of the week every year, but that extra day knocks the cycle out of sync.

For example, under normal circumstances, if your birthday was a Monday one year, it would fall on a Tuesday the next year. Leap years change the equation. If there is a leap day between your birthdays, then the cycle is off by two days. Thus, if your last birthday before this leap day was on a Monday, your first birthday after the leap year will be on a Wednesday.

Make sense? Test yourself by trying to calculate the day of the week for specific days, like March 15, 2009, or July 21, 2007. Look at the front page of the New York Times from that day to see if you calculated correctly. Hint: The further back in time you go, the more leap years, and thus, the more extra days there are to consider.

Go back more than 100 years and the math gets even more complicated. Usually a year is a leap year if it is divisible by four, like 2012, or 1992. But leap year gets canceled every hundred years or so to keep the calendar in balance with Earth’s orbit.

So the rule, in reality, works like this: If the year is divisible by four, it’s usually a leap year.

But there are exceptions. If the year is divisible by 100 but not by 400, it is not a leap year, and if the year is divisible by 400, then it is a leap year. You can read more about determining leap years on TimeandDate.com’s leap year page.

In fact, this kind of calendar math is so challenging, a question on the 2012 American Mathematics Competition asked for day of the week Charles Dickens was born, the date being Feb. 7, 1812. See if you can figure out the answer, and check your answer on Wolfram Alpha. Then explore the many formulas and algorithms for computing the day of the week for any date in history.

#### 2. Calendar Reform

Suffering from calendar confusion? Tired of leap years, leap days and leap seconds? You’re not alone. The astronomer Richard Conn Henry and the economist Steve Hanke have proposed a new calendar that stays the same every year. The Hanke-Henry calendar employs a 364-day year and a “mini-month” every few years that balances the cosmic equation.

And this isn’t the first try at calendar reform. Check out these articles from The New York Times’s archives: from 1922, a proposal of a 364-day year and “Freaks of the Calendar,” which shows that calendar confusion has been around since at least 1892. Now, write an Op-Ed piece proposing your own solution to calendar confusion.

#### 3. Happy Birthdays

How rare is a leap day birthday? If we assume that every day is equally likely to be a birthday, the probability that a person’s birthday is a non-leap-day (say, Aug. 18) would be 1 out of 365, or about 0.27 percent. We can calculate the probability of a leap day birthday thus:

- Assume one leap day every four years.
- In four years, there are 365 * 4 + 1 = 1,461 total days.
- The probability of a leap day birthday would be around 1 /1,461, or 0.068 percent.

Take a poll in your school to find out everyone’s birth date; collect the data and make a chart. What are the most common birthdays in your school? The least common? Does each day represent 0.27 percent of the total school population? Are there any general trends? And does anyone have an elusive leap day birthday?

#### 4. The Final Frontier

Leap year on Earth arises because of the strange relationship between the length of our year (Earth’s rotation around the sun) and the length of our day (Earth’s rotation about its axis). But what do the days and years look like on other worlds in our galaxy?

Get more information about the orbits of the planets, and their days and years. Explore the calendar of Mars and try to figure out a suitable calendar for the Martian year. And if you think the year and day on Mars is strange, be sure to check out Mercury.

You can even calculate your age on Jupiter and other planets.

#### 5. One Small Leap

Recently, 700 delegates from 70 nations met in Geneva to decide whether to keep or abolish the “leap second,” an extra second tacked on once every few years to synchronize atomic clocks with Earth’s rotational cycle.

Explore the reasons for and against this time-keeping convention and how the U.S. Atomic Clock works to keep standard time and consider your own position on the leap second. (The result of the international meeting was to postpone the decision for three years, so you’ve got time to make up your mind.)

On a somewhat related note, you can also take on the question of daylight saving time and various nations’ policies on the subject as well as how it plays a role in our understanding of time.

#### 6. Calendar Calculations

Leap day is just one of many mathematically interesting opportunities to explore the calendar. Another is palindrome days, which occur when the date, written as a series of numbers, can be read the same backward and forward. For example, Nov. 2, 2011, was a palindrome day, as the numeric date 11/02/2011 (or 11022011) reads the same forward and backward. How rare are palindrome days? When will the next palindrome day occur?

Another interesting oddity is the permutation day which occurs when the year can be rearranged to form the day and the month. For example, Feb. 12, 2011, is a permutation day, since 2012 can be rearranged to form 02 and 12. When is the next permutation day? How often do they occur?

Play with the calendar and invent your own mathematical holidays, like pi day, or explore other calendar math like the number of Friday the 13ths that occur each year.

#### 7. Show Me the Money

Many workers are paid yearly salaries, which means they earn a set amount of money per calendar year. Since salaries are typically not adjusted for leap year, this means that, during a leap year, an employee will probably end up working one extra day for the same yearly salary.

How much does this “free day” affect your daily wages? Check out average yearly salaries for positions like an architect in New York City, and calculate the average daily pay by dividing the yearly salary by 365. Divide the salary by 366 to compute the average daily pay for a leap year, and compare. How much does leap year cost workers?

It’s not all bad news, though. Rental payments are typically made on a monthly basis, too. Check out the monthly costs of real estate rentals in your area. How much is that rent-free day in February worth?

#### 8. What a Difference a Day Makes

Use this extra 24 hours to think about just how much happens every single day, and different ways to quantify a day.

For example, how much do companies earn on a daily basis? Check out the financials on companies like Apple, Netflix or Ford, and calculate their daily revenues, net income and labor costs.

Or think of a single day in terms of population. How many births, deaths, and divorces occur every day in the United States? Check out data at the Census Bureau and calculate how much the population changes from day to day.

Yet another way to parse a day is to quantify how people spend their time and analyze trends in the data.

#### 9. The More Things Change …

Look at The New York Times from the last leap day, Feb. 29, 2008, which included coverage of Barack Obama’s presidential aspirations, the increasing number of homeowners walking away from their mortgages and the uneasy relationship between the United States and Pakistan. Use Times Topics to do some research and get up to date on the issues, and and write a follow-up article to your chosen story. Then, find a story from this year’s leap day New York Times and look into the future to write a follow-up for next leap day.

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