When I was a kid, my grandpa (a trained biochemist) used to teach me how to burn things and blow things up with my little home chemistry set. But that’s a story for another time. He also used to tell me a story that I never could quite figure out when I was young. It went something like this:

Three men rent a room together at a motel, and the clerk at the desk charges them \$30 for it. They split the cost, paying ten dollars each. Later the manager tells the clerk that he overcharged the men, and that the actual cost should have been \$25. The manager gives the bellhop \$5 and tells him to take it up to the men.

However, the bellhop, deciding that the men would be none the wiser, only gives the men \$3 back (\$1 dollar each) and pockets the \$2 difference. With the \$1 refund, each man has now paid \$9 to stay in the room. 3 x \$9 = \$27. The bellhop has pocketed \$2. \$27 + \$2 = \$29.00…where is the missing dollar?

The problem is that the question is designed to confuse you about where the money went. There are two ways to think about how that \$30 moves around, and the “trick” in this question is that it combines both ways and tries to fool you. You can think about this in terms of how much money the men originally paid OR how much money they ended up paying, but not both. In the former, we started with \$30. The owner kept \$25, the bellhop pocketed \$2, and the men got \$3 back. No missing dollar. In the second method, the men ended up paying \$27, \$25 of which the owner kept and \$2 the bellhop pocketed. Again, no missing dollar. The problem is that the question leads you to double count the bellhop’s \$2. The \$27 that the men pay already includes the bellhop’s \$2, so adding another \$2 makes no sense.

That leads me to the two types of questions from payroll departments and employees that have overflowed my inbox since I started writing about the Pay Period Leap Year.

To briefly recap, employers have three options in a Pay Period Leap Year:

1. Pay the same amount in each pay period as you did in the non-Pay Period Leap Year.Under Option 1, employees will receive an effective increase of approximately 2% (weekly pay periods) or 4% (bi-weekly pay periods).
2. Divide the total salary by 53 (or 27) pay periods rather than 52 (or 26). This ensures that employees get the same compensation as in non-Pay Period Leap Years, but it also means employees would get slightly less per paycheck.
3. Adjust only the last paycheck of the year.

Many employers have chosen Option 2, which is a sensible middle ground.

Like we did last week, let’s look at a hypothetical employee who we’ll call “Carol Concerned.” Carol is an exempt employee who earns an annual salary of \$100,000, paid bi-weekly. Carol started in mid-2014 and barges into your office to tell you that Option 2 is costing her a paycheck. She worked exactly half of 2014 and will work all of 2015, and by her calculation, she should have made \$150,000. Waiving a printout ofmy last post about John Newbie, she is convinced that your decision is going to leave her short, too. You check Carol’s W-2 from 2014 and panic, since her W-2 does not show \$50,000 in gross earnings.

We pay in arrears, meaning that the last pay period worked in 2014 is not paid until 2015. By choosing Option 2, are we actually taking away an extra paycheck from employees that they are earned in 2014?

Have you made a big mistake with Carol Concerned? Not at all. This is just the story of the missing dollar, with a twist!

First, before we get to the missing dollar, let’s deal with that twist. Looking at the Leap Year Pay Period Spreadsheet.xlsx I have attached might help you here. To make the math easier (so we can use full pay periods), let’s assume that Carol started on June 27, 2014, as opposed to July 1. Carol has told you something that seems logical at first glance: “I worked ½ of 2014. I should make \$50,000.” Carol would be correct if a year was 48 weeks long, not 52, or if you had actually promised to pay her exactly \$50,000 or the equivalent per diem rate (\$50,000 divided by the number of days left in 2014). Most likely, though, you just prorated her annual salary of \$100,000. That’s important because, in reality, Carol worked only 12 pay periods in 2014 out of 26 (less than half), and not 13 (exactly half):