A math puzzle has been stuck in my head today like the chorus of a three-note pop song. Actually, it’s not so much the puzzle itself worming through the folds of my prefrontal cortex as it is the various ways of formulating its solution that occurred to me on encountering the puzzle yesterday.
Maybe if I share some of these formulations—thereby making the puzzle’s trick obvious to the world—it will head over to the hippocampus for transport back to wherever it had been hanging out since 1992, which was when I first encountered it (I ran into it yesterday while perusing some of the dustier regions of my extended mind: an old cassette tape on which I and some friends discuss the puzzle; hearing the tape called up a vague memory of that conversation and an even vaguer memory of having discussed the puzzle beforehand with my dad).
I’m sure there are already plenty of videos and writings out there explaining the Missing Dollar Riddle (as its usually called). I haven’t looked at those, aside from skimming its Wikipedia entry. I think the below solution is as intuitive as can be, or as need be.
The Missing Dollar Riddle
Scenario (1):
A party of three checks into a hotel. The party pays $30 to stay for the night. The hotel manager realizes, however, that the party has overpaid by $5. So the manager gives the bellhop $5 to return to the party. The bellhop decides to return just $3 in order to save the party’s members the hassle of trying to split $5 three ways. The bellhop keeps the remaining $2 for his trouble. All said and done, the party has paid $27 to stay for the night. But wait! The party paid $27 and the bellhop has $2, which comes out to only $27 + $2 = $29. Where’s the missing dollar?!?
To make it glaringly obvious where this scenario goes wrong, pop in some more extreme numbers.
Scenario (2):
A party of three checks into a hotel. The party pays $30 to stay for the night. The hotel manager realizes, however, that the party has overpaid: they were due a promotional discount and so should have paid $0, not $30. The manager gives the bellhop $30 to return to the party. The bellhop returns all $30 and keeps $0 for her trouble. The party, then, has paid $0 to stay for the night. But wait! The party paid $0 and the bellhop has $0, which comes out to only $0 + $0 = $0. Where’s the missing $30?
Obviously, there is no missing $30. The party has it. In other words, in Scenario (1), we should not add the amount the party pays (i.e., $30 – $3 = $27) to the bellhop’s $2. Rather, we should add the portion the party has of the original $30 (i.e., $30 – $27 = $3) to the portion the bellhop has (i.e., $2) and to the portion the manager has (i.e., $25). This gives $3 + $2+ $25 = $30.
There are other ways to represent this.
For example, the amount the party pays minus the bellhop’s holdings* should equal the amount the manager (or, more precisely, the hotel income ledger) has at any point in the scenario; e.g., $27 – $2 = $25. In other words, the amount the party pays should always equal the sum of the bellhop and manager’s holdings; e.g., $27 = $2 + $25. This makes sense, as those holdings should be precisely whatever the party pays to stay for the night.
[*By “holdings” and such, I of course refer to portions of the money originating with the party. The penny, for instance, that the bellhop comically swallowed the night before and that’s still making its way through the bellhop’s large intestine is not included in the bellhop’s holdings.]
Another approach is to imagine that the party has -$30 after paying at check-in. Then the sum of what the party, bellhop, and manager have at any point in the story must equal $0; e.g., -$27 + $2 + $25 = 0.
And so on.
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