H022
Where did the missing dollar go?
Three guests pay $10 each for a $30 room. The room turns out to cost $25, so the clerk returns $1 to each guest and keeps $2. Each guest paid $9: 3 × $9 = $27, plus the clerk's $2 makes $29. Where's the last dollar?
No dollar is missing. The question is. The guests' $27 isn't separate from the clerk's $2; it already includes it. $25 went to the hotel and $2 went to the clerk. Adding the $2 again counts the same money twice. The sums that make sense are $27 − $2 = $25, the price of the room, or $27 + $3 = $30, the guests' $27 plus the $3 they got back.
Every number in the riddle is true. The trick is adding two numbers that overlap. Before adding two totals, check whether one is already inside the other.
Try it somewhere else
A school newsletter says 60% of students play a sport and 50% play an instrument, so 110% of students are active.
Why can't the total be 110%, and what would you need to know to fix it?
Change the discount and the “missing” money changes
Keep the $1 refund to each guest but change the room's price, and the riddle's sum stops making any sense at all. The clerk keeps whatever is left of the discount:
| Room costs | Clerk keeps | Riddle's sum ($27 + clerk) | The riddle says |
|---|
With a $22 room, the same reasoning “finds” an extra $2. A real gap wouldn't come and go like that. Money is followed by adding up where it ended up: hotel + clerk + guests, which is always $30.
The same trap in real numbers
Economists face this riddle every time they measure an economy. Suppose a farmer sells wheat for $1, a miller sells the flour for $2, and a baker sells the bread for $4. Adding all three sales gives $7, but the wheat is already inside the flour, and the flour is already inside the bread. That's why measures like gross domestic product count only the final product, the $4 loaf, and not every sale along the way.
The same overlap shows up in survey results that add to more than 100%, in “economic impact” claims that count the same dollar as it changes hands, and in budgets where one expense is listed under two headings.
Where does the riddle come from?
Nobody knows who first told it. Puzzles built on the same trick, adding numbers that don't belong together, were in print by the 1930s: Cecil B. Read's Mathematical Fallacies (1933) has a bank customer whose withdrawals seem to add up to $51 from a $50 deposit, and R. M. Abraham's Diversions and Pastimes (1933) has a version about a traveler and a money order. Similar misdirection puzzles go back to 18th-century arithmetic textbooks. In many tellings, a bellhop keeps the $2 instead of the clerk.
Related exhibits
Sources: the riddle is traditional; its arithmetic is shown in full above. History: Wikipedia, “Missing dollar riddle,” citing Cecil B. Read, Mathematical Fallacies (1933), and R. M. Abraham, Diversions and Pastimes (1933). The wheat-flour-bread example is a standard illustration of why gross domestic product counts only final goods and services (U.S. Bureau of Economic Analysis).