Photo courtesy of Rick Hall

On Monday, I posted the following travel- and math-related riddle. I’m guessing everyone was too scared to post their answers — or perhaps you’re all celebrating Independence Day a little early — because no one chimed in. But no worries, my feelings aren’t hurt in the least. Still, I promised the answer, so here it is.

First the riddle itself:

Three friends are traveling to their high school reunion together. They arrive at their hotel late at night, only to find that their reservations were lost.  There is only one room with three beds available. They have no choice but to share the room, which the hotel has discounted to \$30. Each of them takes out a 10 dollar bill, which the clerk collects.

After the friends are settled into their room, the manager reconsiders the discount. (He feels terrible!) He decides to offer the room at only \$25 and sends a porter upstairs with \$5 for the three friends.

The porter starts thinking about how to divide the \$5 into three equal parts. When he can’t figure it out, he decides to give \$1 to each friend, and pocket the rest. The friends accept the \$3 refund, and the porter heads back to his post, with the remaining \$2.

Given their \$3 refund, each of the three friends paid \$9 for the room (3 • 9 = \$27). The porter has \$2 in his pocket, making the total \$29 (\$27 + \$2 = \$29). But the friends originally paid \$30!

What happened to the \$1?

If you’ve been around the block a few times, you’ve probably heard this riddle. And if you google “missing dollar riddle,” you’ll find thousands of results that outline where that dollar actually is. (Heck, there’s a Wikipedia entry about it!) Most of these talk about a logical fallacy, which is a perfectly reasonable way to describe things. In my mathy brain, there’s another way to explain it, using equations.

This is what we know:

In other words, the friends originally paid \$30, but the manager decided to discount the room by \$5. That meant that the clerk took \$25 from the original \$30 and the porter took \$5 from the original \$30.

\$30 = \$25 + \$5

Then the \$5 was split up — \$3 for the friends and \$2 that the porter pocketed.

\$30 = \$25 + \$3 + \$2

Clearly there is no missing \$1. Here’s another equation to prove why. If you subtract \$3 from each side of the equation, you get this:

\$27 = \$25 + \$2

This works, because with their discount of \$1, each friend paid \$9 for the room, rather than the original \$10. Another way to look at it is this:

3 • \$9 = \$25 (the cost of the room) + \$2 (the amount the porter pocketed)
\$27 = \$25 + \$2
Get it? If not, take another look. It is confusing at first, but once you see it, it does make sense.
Now if you subtract the \$2 from both sides of the equation, you can see how the amount that the friends paid minus the amount that the porter pocketed equals the cost of the room itself.
\$27 – \$2 = \$25
\$25 = \$25
Make sense? Sometimes it does to me, and then my understanding floats away! But I do think it can be fun to look at these problems mathematically. I hope you did, too.
Did you come up with the correct reasoning before reading this? Did you use math? If not, how would you explain that the dollar is not missing at all? Share your ideas in the comments section!