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The famous missing dollar riddle and Solutions

The missing dollar riddle is a famous riddle that involves an informal fallacy. It dates back to at least the 1930's, although similar puzzles are much older.

Although the wording and specifics can alter, the puzzle runs along these lines:

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 $25. To rectify this, he gives the bellhop $5 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 $1 and keep $2 as a tip for himself. Each guest got $1 back, so now each guest only paid $9, bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?

Solution
The misdirection in this riddle is at the end of the description, where a bunch of unrelated totals are added together, and the listener assumes these numbers should add to 30. There is, in fact, no reason this sum should add 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, totalling $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 that we really desire is this one:

$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, we can alter the riddle 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 bellboy $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 silly. One cannot simply add a bunch 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.
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).

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The famous missing dollar riddle and Solutions

The missing dollar riddle is a famous riddle that involves an informal fallacy. It dates back to at least the 1930's, although similar puzzles are much older.

Although the wording and specifics can alter, the puzzle runs along these lines:

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 $25. To rectify this, he gives the bellhop $5 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 $1 and keep $2 as a tip for himself. Each guest got $1 back, so now each guest only paid $9, bringing the total paid to $27. The bellhop has $2. And $27 + $2 = $29 so, if the guests originally handed over $30, what happened to the remaining $1?

Solution
The misdirection in this riddle is at the end of the description, where a bunch of unrelated totals are added together, and the listener assumes these numbers should add to 30. There is, in fact, no reason this sum should add 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, totalling $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 that we really desire is this one:

$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, we can alter the riddle 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 bellboy $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 silly. One cannot simply add a bunch 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.
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).