Missing dollar riddle: a famous problem that illustrates problems of confusion and misdirection in conversation

The missing dollar riddle is a famous problem that illustrates problems of confusion and misdirection in conversation. It illustrates how misdirection and irrelevant facts and questions can foil a person's clear understanding of a problem.

 

The problem

Three guests 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 for himself.

Now that each of the guests has been given $1 back, each has paid $9, bringing the total paid to $27. The bellhop has $2. If the guests originally handed over $30, what happened to the remaining $1?

(I usually use the mamak stall as the reference!)

 

Solution

The initial payment of $30 is accounted for as the clerk takes $25, the bellhop takes $2, and the guests get a $3 refund. It adds up. After the refund has been applied, we only have to account for a payment of $27. Again, the clerk keeps $25 and the bellhop gets $2. This also adds up.

There is no reason to add the $2 and $27 – the $2 is contained within the $27 already. Thus the addition is meaningless. Instead the $2 should be subtracted from the $27 to get the revised bill of $25.
This becomes clearer when the initial and net payments are written as simple equations. The first equation shows what happened to the initial payment of $30:

$30 (initial payment) = $25 (to clerk) + $2 (to bellhop) +$3 (refund)

The second equation shows the net payment after the refund is applied (subtracted from both sides):

$27 (net payment) = $25 (to clerk) + $2 (to bellhop)

Both equations make sense, with equal totals on either side of the equal sign. The correct way to get the bellhop's $2 and the guests $27 on the same side of the equal sign ("The bellhop has $2, and the guests paid $27, how does that add up?") is to subtract, not add:

$27 (final payment) - $2 (to bellhop) = $25 (to clerk)

Misdirection

The "paradox" cleverly sets its room rates so that when we add the two terms $27 and $2, we nearly get $30. If not for this "near-miss", we would be more inclined to ask if those two terms have to add up to $30 when we break down the situation this way (and to realize that they do not).

With different prices, the illusion would vanish. Say the clerk initially accepted $30 but then learned that rooms are only $10 no matter how many people are in them, and sends back a refund of $20 via the bellhop. Again, the bellhop, seeing that $20 doesn't evenly divide, gives each guest $6 (for a total of $18) and keeps the leftover $2 for himself. Therefore each of the three guests paid $4, bringing the total paid to $12; add that to the bellhop's 2 dollars to get a total of $14. So where did the other $16 go?

With this setup it is more clear that the guests' new total amount paid ($12) is only the bellhop's $2 away from the actual room price of $10, not the original room price of $30. The target price to account for is the new $10 bill, not the old $30 one. In the original riddle it is only the "near-miss" with $30 that makes $30 seem like the correct target of the operation.

The riddle involves the phenomenon of 'suspension of disbelief' inherent in storytelling and its power over the human imagination. If one were to make the story a bit more complex and compelling the illusion is almost guaranteed to work in the moment of its telling and can be a good illustration for the explanation of the anomaly, although not a perfect one because there is an explanation. The more points added to the story cause the listener to pause and try to compute what each element may signify.

There are dozens of variations to the riddle.

Cash flow analysis

The following table demonstrates the movement of cash, stating (in successive rows) where cash has moved over time. Each row represents an instance in time. Additional rows could have been added; as one example: just after the bellhop takes the money, but before handing it over to the cashier.

Cash Flow Analysis

	Guest 1	Guest 2	Guest 3	Cashier	Bellhop	Total
Before Check In	$10	$10	$10	$0	$0	$30
After Check In	$0	$0	$0	$30	$0	$30
After the Bellhop	$1	$1	$1	$25	$2	$30

 

Follow-up

A follow-up is often mentioned as a mock resolution to the problem.

A few months later, two of the original three guests check into a hotel room in the same hotel. The clerk says the bill is $20, so each guest pays $10. Later the clerk realizes the bill should only be $15. 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 $3 for himself.

Now that each of the guests has been given $1 back, each has paid $9, bringing the total paid to $18. The bellhop has $3, so $18 + $3 = $21, and the guests originally handed over $20, so that's where the missing dollar from the original problem is!

Moral of this riddle is that you must analyse the words used in any conversation. You cannot take all at face value!