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April 26, 2021

Where Did the Other Dollar Go, Jeff?

There’s a brain teaser that shows up from time to time, in various forms. I first encountered it in a book of puzzles I read when I was a child. It’s called Where Did the Other Dollar Go? and it goes something like this:

Three tourists stop at a hotel, and the manager tells them that a shared room will cost $30. Finding the price agreeable, they pony up $10 each and retire to the room. Later that afternoon, the manager, who is honest, realizes that the room was meant to be priced at $25. The manager orders the bellhop to return the excess $5 to their guests. The bellhop, who is not honest, takes $5 from the register and return only $1 to each tourist, pocketing the remaining $2—​the guests don’t have to fuss over uneven change that way.

Now, each of the three tourists has spent $9, for a total of $27. The bellhop has retained $2, which brings the total to $29. Where did the other dollar go?

The puzzle uses mathematical sleight of hand to put together an equation, 3 × 9 + 2 = 29, that appears to model the situation described, but really it does not. By juxtaposing dollars spent with dollars held, the equation manages to double-count some dollars while failing to represent others, leaving a total that is just slightly off what we would expect.

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