The Extra Dollar in the Hotel Puzzle Solution
The key to understanding the riddle intuitively is to realize that, while each man did pay $9, he did not pay $9 to the hotel. Each one paid $8⅓ to the hotel ($8⅓ × 3 = $25), and $⅔ to the bellhop ($⅔ × 3 = $2), for a total of $27 spent by the group. $25 + $2 = $27, and so does $9 × 3.
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 (mixing cost and cash). Instead the $2 should be subtracted from the $27 to get the revised bill of $25.
This becomes clear 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)
This question is very simple when thought of in a different manner. The hotel receives $30 from the guests and gives $5 to the bellhop to return to the guests. $30 minus $5 equals $25. $3 is given to the guests and the bellhop keeps $2. Thus, $3 plus $2 equals $5, and the remaining $25 is in the possession of the hotel.
This is clearly not a paradox, and involves only the switching of subtraction for addition. Each patron has paid $9 for a total of $27. The storyteller adds the $2 that the bellhop pilfered, but he should have subtracted the $2 to make a total of $25 paid. So 3 × $9 = $27, which accounts for the $25 room and the $2 given to the bellhop.
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