# Solution: Math Puzzle: A Bargain Hotel?

## Solution: A Bargain Hotel? Math Puzzle

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.

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 \$0 \$0 2. Upon check in \$0 \$30 \$0 3. After the \$5 refund is made \$5 \$25 \$0 4. After the guests tip the concierge \$3 \$25 \$2

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 \$0 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!