A Bargain Hotel?

Three Schlumberger employees, after a day of work in a remote location, arrived at a hotel to pass the night. They approached the concierge and asked for three rooms. The concierge gave them the three rooms at $10 per room. Each Schlumberger mate paid $10 ($30 in total) and went to sleep.

Later the concierge approached the hotel manager and told her about the guests who had just arrived. The manager reminded the concierge that the hotel had a special rate for Schlumberger employees and he should charge only $25 in total for the three rooms.

The next morning when the guests were checking out, the concierge remembered the change in rate and returned five dollar bills to the employees; each one of them took $1, and they left $2 as a tip for the concierge.

Summarizing

Each employee paid $10.
Each employee received $1 as a refund, so in the end, each one paid only $9.
In total they paid $27; that is, 3 times $9.
Adding the $2 tip, the total is $29

But the Schlumberger employees paid $30 in total, so where’s the missing dollar?

Course:

Math [1]

Result/Solution(s)

Solution: A Bargain Hotel?

There is no missing dollar. The puzzle is presented in a way that is designed to confuse you into thinking that a dollar has disappeared, when actually all the money can be accounted for.

Here are two ways of looking at the situation to see that nothing is missing. Then, by examining these two viewpoints together, we can see how the puzzle maker misled us.

1. Follow the original $30.

The guests come to the hotel with $30. Let's see where that money goes at each step in the puzzle. There are three places where money can be: in the guests' hands, in the hotel till, or in the concierge's pocket. This chart tracks the money movements:

Where Is the Money?

What the guests have

Amount in the hotel till

Amount in the concierge's pocket

1. At the start

$30

2. Upon check in

$30

3. After the $5 refund is made

$25

4. After the guests tip the concierge

$25

At each step, add up what the guests have plus the amount in the hotel till plus the amount in the concierge's pocket. It's always $30. Nothing is missing.

2. Keep track of what the guests have spent at each step.

What the guests have spent

1. At the start

2. Upon check in

$30

3. After the $5 refund is made

$25

4. After the guests tip the concierge

$27

This adds up perfectly. The guests spent $10 each when they checked in, but since they each got a $1 refund, in the end they spent a total of $27. Of this total, $25 went to the hotel and $2 to the concierge.

The puzzle is presented in such a way as to lead you to focus on the original $30—our first approach—but then shifts to keeping track of what is spent, as in our second approach. But the total spent is $27, and the puzzle presents the total as $29. What's the explanation for the difference? The puzzle implies that the $2 tip came from the guests, when, in fact, it came from the $5 refund. So the total spent is really $27, of which $25 went to the hotel and $2 to the concierge.

Beware of tricky puzzles!

math [2]
Math Puzzle [3]