Missing dollar riddle
The missing dollar riddle is a famous riddle that involves an informal fallacy. It is old, dating back to at least the 1930s, although similar puzzles are much older.[1]
Although the wording and specifics can vary, the puzzle runs along these lines:
Three guests check into a hotel room. The manager says the bill is $30, so each guest pays $10. Later the manager realizes the bill should only have been $25. To rectify this, he gives the bellhop $5 as five one-dollar bills to return to the guests.
On the way to the guests' room to refund the money, the bellhop realizes that he cannot equally divide the five one-dollar bills among the three guests. As the guests are not aware of the total of the revised bill, the bellhop decides to just give each guest $1 back and keep $2 as a tip for himself, and proceeds to do so.
As each guest got $1 back, each guest only paid $9, bringing the total paid to $27. The bellhop kept $2, which when added to the $27, comes to $29. So if the guests originally handed over $30, what happened to the remaining $1?
There seems to be a discrepancy, as there cannot be two answers ($29 and $30) to the math problem. On the one hand it is true that the $25 in the register, the $3 returned to the guests, and the $2 kept by the bellhop add up to $30, but on the other hand, the $27 paid by the guests and the $2 kept by the bellhop add up to only $29.
Solution
The misdirection in this riddle is in the second half of the description, where unrelated amounts are added together and the person to whom the riddle is posed assumes those amounts should add up to 30, and is then surprised when they do not — there is, in fact, no reason why the (10 − 1) × 3 + 2 = 29 sum should add up to 30.
The exact sum mentioned in the riddle is computed as:
SUM = $9 (payment by Guest 1) +
$9 (payment by Guest 2) +
$9 (payment by Guest 3) +
$2 (money in bellhop's pocket)
The trick here is to realize that this is not a sum of the money that the three people paid originally, as that would need to include the money the clerk has ($25). This is instead a sum of a smaller amount the people could have paid ($9 × 3 people = $27), added with the additional money that the clerk would not have needed had they paid that smaller amount ($27 paid - $25 actual cost = $2). Another way to say this is, the $27 already includes the bellhop's tip. To add the $2 to the $27 would be to double-count it. So, the three guests' cost of the room, including the bellhop's tip, is $27. Each of the 3 guests has $1 in his pocket, totaling $3. When added to the $27 revised cost of the room (including tip to the bellhop), the total is $30.
To obtain a sum that totals to the original $30, every dollar must be accounted for, regardless of its location.
Thus, the sensible sum can be expressed in this manner:
$30 = $1 (inside Guest pocket) +
$1 (inside Guest pocket) +
$1 (inside Guest pocket) +
$2 (inside bellhop's pocket) +
$25 (hotel cash register)
This sum does indeed come out to $30.
To further illustrate why the riddle's sum does not relate to the actual sum, the riddle can be altered so that the discount on the room is extremely large. Consider the riddle in this form:
Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $10. To rectify this, he gives the bellhop $20 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $6 and keep $2 as a tip for himself. Each guest got $6 back: so now each guest only paid $4; bringing the total paid to $12. The bellhop has $2. And $12 + $2 = $14 so, if the guests originally handed over $30, what happened to the remaining $16?
Now it is more obvious that the question is quite unreasonable. One cannot simply add a couple of payments together and expect them to total an original amount of circulated cash.
More economically, money is accounted by summing together all paid amounts (liabilities) with all money in one's possession (assets). That abstract formula holds regardless of the relative perspectives of the actors in this exchange.
- The guests of the hotel paid $27, but also have $3 among their pockets at the story's end. Their assets are $3, and their liabilities are $27 ($30 = 27 + 3). Thus, the original total is accounted for.
- From the perspective of the hotel clerk, the hotel has $25 in assets and lost $5 in liabilities ($30 = 25 + 5).
- From the perspective of the bellhop, his assets are $2, and his liabilities are $3 to guests and $25 to the register at the desk ($30 = 2 + 3 + 25).
To illustrate the issue through equations:
1) 10 + 10 + 10 = 30
2) 10 + 10 + 10 = 25 + 2 + 3
3) 10 + 10 + 10 - 3 = 25 + 2 + 3 - 3 (adding -3 to both sides of the equation to cancel out the +3 on the right side)
4) 10 - 1 + 10 - 1 + 10 - 1 = 25 + 2
5) 9 + 9 + 9 = 25 + 2 (obs: tip to bellhop has already been paid)
6) 27 = 27
How the riddle is deceptive comes in line 7:
7) 9 + 9 + 9 = 25 + 2
8) 9 + 9 + 9 + 2 = 25 (pushing +2 to the other side without inverting the sign)
9) 27 + 2 = 25
10) 29 ≠ 25
How it should be:
7) 9 + 9 + 9 = 25 + 2
8) 9 + 9 + 9 -2 = 25 + 2 -2 (adding -2 to both sides of the equation to cancel the +2 on the right side, which means the bellhop returned the tip or gave a discount of $2)
9) 9 + 9 + 9 - 2 = 25
10) 27 - 2 = 25
11) 25 = 25
The puzzle should subtract the bellhop's tip from the $27 rather than add it.