Three people check into a hotel.
logic/29 Added material
Problem
Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totalling $27. The bellboy has $2, totalling $29. Where is the remaining dollar?
Show solution
The calculation of $27 + $2 contains two errors: the manager's $5 is ignored and the bellboy's $2 is counted in the wrong direction. The original payment of $30 is correctly accounted for as $27 finally paid by the three guests, plus $5 refunded by the manager, MINUS $2 kept by the bellboy (so that only $3 of refund was received).
Additional material
As some suggested when this puzzle recently resurfaced, it might be good
for the solution to be expanded to include suggested explanations of why
people get confused by the puzzle, when simply changing the numbers seems
to eliminate the confusion.
My explanation for the confusion is this: normally, the amount that I pay
at a hotel is equal to the amount that the front desk receives. Thus if
someone steals some of the money that I've paid, it becomes tempting to
confuse the two quantities
(1) amount manager receives + amount stolen
and (2) amount I paid + amount stolen,
especially if you are misled into doing so by the person presenting the
puzzle. Now (1) is a meaningful quantity; it's a portion of the $30. So
the puzzle creates the illusion that what we're doing is adding everyone's
assets together to figure out how the $30 is distributed (since that's what
we're doing when we calculate (1)). Hence "Where is the remaining dollar?"
sounds like the right question to ask, and gets us thinking along the wrong
lines---we start trying to look for somebody with an asset of $1, instead
of trying to figure out why $27 + $2 is not a useful quantity. Finally,
the choice of numbers helps disguise the error since there are enough
possible combinations of +-$1, +-$2, +-$3, +-$5, etc., to keep us confused
for a while.
-- Tim Chow <tycchow@math.mit.edu>, September 1993
I like to reveal the mistake of the "missing dollar" by following with
a two-man version and then a (very transparent) one-man version.
Briefly:
Two guys pay $15 each. Clerk realizes he overcharged $3. Bellboy returns
$3. Each guy gets $1, bellboy gets $1 tip. 2x(15-1)=28. 28+1(tip)=29.
One guy pays $30. Clerk realize he overcharged $1. Bellboy returns $1,
which the man pockets giving the bellboy $0 (cheap bastard doesn't tip).
30-1=29. 29+0=29.
-- Al Stanger <S886680@UMSLVMA.UMSL.EDU>
[ Editor: a related puzzle was posted in June 1995: ]
> There are two rooms. Eight (8) guys goes into the first room and the other
> six (6) goes into the second room.
> Now, each guy buys a round of drinks for one another.
> (Say a drink costs $1)
> So:
> 8 guys buy 8 drinks = 8 x 8 = 64
> 6 guys buy 6 drinks = 6 x 6 = 36
> The waiter takes $64 + $36 = $100 to the counter.
> Later one of the guys decide to go visit next door.
> And every one drinks again
>
> 7 guys buy 7 drinks = 7 x 7 = 49
> 7 guys buy 7 drinks = 7 x 7 = 49
>
> The waiter takes $49 + $49 = $98 to the counter.
>
> The barman accuses him of stealing $2....
>
> What happened to the $2 ???
[ Editor: one reply restated the count in drinks (100 drinks in the
first round, 98 in the second, so $2 less was spent); the follow-up
said: ]
But that is just restating the problem, using drinks instead of dollars. It is
clear that the math is fine. But why does it look wrong?
Look at it this way. Suppose they went into 14 different rooms (1 guy per room)
and each guy bought a drink for everyone in his room. Total drinks would be 14.
As soon as 2 of those guys get together, the bar bill in their room is $4 for 4
drinks instead of the $2 for 2 drinks they were buying before. With 3 guys it's
9 drinks for $9. In other words, the cost goes up by the square of the number of
men in the room, not by the number of men in the room. So while it seems like a
constant number of men might generate a constant bar bill, the *square* rule
makes it not so. That 8^2 + 6^2 is so close to 7^2 + 7^2 is merely a
coincidence that makes the puzzle interesting.
-- <blask@gpx01.d39.lilly.com>
Once in a very rare while, just when you think you've heard everything
there is to be said about an ancient chestnut like this puzzle,
somebody comes up with a really nice twist which makes you think,
"Wow! I wish I'd thought of that!!"
Such was the case when this puzzle was posed on the NPR program
"Car Talk" several weeks ago. A math teacher from somewhere in the
Midwest (I wish I could remember his name to give him credit) sent in
a truly wonderful response to this puzzle. It went something like
this:
In order to find out where the missing dollar went, you have to
learn the rest of the story. You see, as the clerk was
descending the stairs on his way back to the desk, feeling very
pleased with himself after pocketing the $2, he noticed the
manager looking quite upset and calling him over. The clerk
momentarily panicked, thinking that he had been caught. But
realizing that there was no way the manager could have learned
about his underhanded thievery, he put on his most innocent face
and sauntered over to see what was the matter. "I can't believe
it," exclaimed the manager. "I made the very same mistake when
that couple came in a few minutes ago, and charged them $30 for
a $25 room as well. Please go upstairs and return these $5 to
them."
As the clerk takes the $5 and makes his way to the couple's
room, he can't believe his good fortune. Not only has he not
been caught, but he has a chance to pull the same trick again.
As before, he figures that it's hard to divide $5 between two
people, so he decides to give them $1 each, and pockets $3 for
himself.
So we have a situation where each of the two people ended up
paying only $14 each, for a total of $28. Plus the $3 that the
clerk kept for himself adds up to $31, not $30!
Remember that missing dollar you were asking about?!
THERE IT IS!!
Brilliant...Just brilliant...!
-- Clive Dawson <cdawson@acad.stedwards.edu>, April 1996
If all else fails (including a money-balance),
Just say this:
".......the manager took it. He'll probably take the boys tip, too!" :)
Still can't be blown off with a smile? Welllll....
I think the faulty math can easily be explained by exaggerating the error
of the manager.....
$30...each pay $10.....actual room was only $14 (cheap hotel, ehh?)
Bellboy goes up with $16, can't split it equally between the three, so he
gives $5 back to each and keeps $1.
Therefore, each man actually paid $5...and the bellhop kept $1....
This adds up to $5 + $5 + $5 + $1 = $16
My gosh, where did the 'other' $14 go?
When they try to figure out where the other $14 went on their own, they
usually can figure out what is going on. If you watch them closely, you
can actually see the faint glimmer of a light bulb above their head. Then,
they may rethink the original puzzle and the light bulb will burn
intensely...along with a beaming smile that will brighten both of your
days...
(How about if the manager realized that the room was free? The bellhop
gives each man $10 back and keeps $0! Each man paid 0, so 0+0+0+0=0!
Whoa!!! Who was the magician that made all $30 disappear?)
....If I had a dollar for every time that dollar seemed to be missing...
-- Chad Moore <csm@preferred.com>, November 1997