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, totaling $27. The bellboy has $2, totaling $29.
Where is the remaining dollar?
There’s a nice post on the answer now at Partially Derivative, but I want to discuss my meta-question:
Can you generalize the errors made in the puzzle? Can you give a textbook, not-designed-as-a-puzzle example where this happens?
That is, is there some general principle at work here, perhaps something to do with psychology, revealed by the puzzle?
The operations in the first paragraph use (more or less) a number line.
While I drew this as in the negative direction, I expect mentally most people would be thinking in positive numbers (as they are given in the problem) so I colored them accountant-style rather than put negative signs in front.
The operations in the second paragraph hew closer to an allocation model:
Dissecting it this way, it’s fairly clear the “manager” line is missing, and the bellboy’s take is in the wrong direction.
I would consider the psychology “trick” to be the jump between two mental models, made worse by the majority of people considering the first part of the problem on a positive number line. In an allocation model the idea of “negative allocation” doesn’t occur to most people.
While a $1 disparity would be difficult to find “in the wild” — the numbers and the omission of the $5 are carefully engineered for the puzzle — I would expect to find real textbook problems that ask students to transition from one mental model to another with potential confusion of signs. Here’s an example of what I mean:
Rick’s bank account is overdrawn by $219. What will the new balance be if he deposits $196?
— Elementary Algebra: Equations and Graphs by Yoshiwara, Yoshiwara, and Drooyan
This isn’t a perfect match, though. Does someone have a better example?