The Extra Dollar in the Hotel Puzzle Solution

30.07.2015

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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