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Trick Question of the Week

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Post December 27th, 2010, 7:29 pm

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Hey everybody! Here is this week's question:

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?

Dont forget, the first person to post the right answer gets 50 CC points!

Good luck!

Post December 27th, 2010, 7:32 pm

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Post December 27th, 2010, 7:32 pm
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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 guest's 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.

For instance, the story might be better told if 3 men were seeking a private venue to play a game of poker. They find a cheap motel in town with a $30/per night rate. They would play at home but none of their wives approve. So, each of them pays out $10 for the private space. They go to the room and commence to gambling.

Two of the guys do not realize that they are playing with a card shark and after a couple of hours find themselves completely wiped out. In anger they decide to head back home. One of them calls the concierge and notifies him of their intention. After hanging up, the concierge, feeling bad, decides to offer a rebate of $5, which he gives to the maid to return to the gentlemen.

Although a hard worker and a steady employee, the maid is not that honest and she can't count that well, so, to make it easier on herself, she decides to pocket two bucks and just give them each a dollar back. Things have been somewhat difficult at home and she needs all the help she can get.

After giving them the three dollars, the men leave the hotel. The shark disapears in the shadows, hustling off with his small fortune and chuckling at the incompetence of his peers. The other two men talk amongst themselves. They concur that at least they came out of the game with some money. After all, they each ultimately paid $9 for the room. That is a fact. And 9 x 3 = 27

Post December 27th, 2010, 7:37 pm
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A new thread every week is a little too much.

Just continue in one topic, you can edit the title each week to let folks know there's a new question,

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