August 12, 2011

Missing dollar riddle

I was asked the following question last thursday:

You want to buy a T-shirt that costs RM97. You have no money.

You borrow RM50 each from your brother and your sister and with your current RM100, you purchase said T-shirt.
You hold in your hands the balance (RM3). You then decide to return RM1 to your brother, RM1 to your sister, and keep the remainder.

Now that each of them has RM1 back, the total owed is RM98. If you have RM1, what happened to the remaining RM1?

This riddle illustrates problems of confusion and misdirection can foil a person's clear understanding of the problem.

The original problem is as follows:

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 shall attempt to solve the original question.

Explanation 1:

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)

Explanation 2(or the TL;DR version):

The question attempts to confuse you by adding up loaned money. Simply put, the $2 is contained within the $27, and should not be added together.

An analogous situation would be adding 99 empty pails and 1 full pail of water. The two quantities are distinct from each other and it is meaningless to call the total "100 full pails of water" nor "100 empty pails of water".

The wikipedia entry elaborates further:


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.

Have a nice day.

See also:



  1. Pretty awesome stuff, I like breaking my head on these things... thanks!

  2. Honestly: I stopped reading in one particular moment... The payment examples resembles the case of arrow and the turtle. It's too much philosophy for me, yet I have to admit: misdirection is a very interesting idea/problem/case/ whatever one could call it. It's me who simply isn't into that kind of philosophical debates.

  3. Pretty cool I got caught by this one

  4. Portugal should hire you :D

  5. That was intense, lol... great brain exercises... +followed

  6. I'm no good at math, and this proved it haha

  7. Mindfuck, now in a text-mode for your pleasure.
    Nah but honestly, great subject :). Original and fun !

  8. I got it! Its a simmple but good illusion :D
    Nice post! want more of these!


  9. There is no dollar missing. The question is itself mis leading us.
    The answer to this misleading question is pretty simple rather very simple if you understand the question.

    As the question misleads us asking
    " Now that each of them has RM1 back, the total owed is RM98. If you have RM1, what happened to the remaining RM1? "

    If each of them have $49 back with them then it combining their money becomes $98. Here $98 includes after deducting the paying the amount for T-shirt and $1 kept by himself without giving to his brother or sister. Remaining $2 are distributed equally between them.

    Similar is the case follows the 2nd question.

    As it says

    " 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? "

    Now after examining it we can say the total paid $27 includes the bill paid and tips paid to the waiter. Then comes the other statement " The bellhop has $2" why it should be added again even after paying to waiter instead it should be said " The remaining $3 has been received by each of them.


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