This riddle is a classic example of an informal fallacy. An informal fallacy is an argument that may employ valid logical reasoning, but the resulting conclusion is false because the premises of the argument are flawed. This contrasts with a formal fallacy, which involves an error in the arguments logical structure.
You can see the original question here.
The answer, of course, is that there is no “missing dollar.” The method of calculating the total is simply flawed.
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The premise that adding each of the three guest’s adjusted payment of $9, totaling $27, to the bellhop’s pocketed $2 will give you the total sum of money paid is incorrect. The difference is between the actual amount of money paid and the smaller amount of money that could have been paid, which cannot be used to accurately arrive at the total.
The correct way to add up the total is to look at where all the money actually is. There is $25 in the cash register, $2 in the bellhop’s pocket, and each of the three guests has $1.
$25 + $2 + $1 + $1 + $1 = $30.
To further illustrate how the original method of adding up the guests’ payments is flawed, let’s change the riddle a little. Let’s say that instead of realizing that the guests only owed $25, the clerk discovers that he should have only charged them $10. The clerk therefore gives the bellhop $20 to return. The bellhop gives each guest $6 and pockets the remaining $2, so each guest effectively only paid $4.
$4 + $4 + $4 + $2 = $14.
So where has the missing $16 gone? This version makes it more obvious that the question is flawed. The correct way to sum up the total would be $10 in the register, plus $2 in the bellhop’s pocket, plus $6 for each guest, which is indeed $30.
Another way to look at it is this: The guests didn’t each pay $9, they paid $8.333 dollars and then received $1 each. So $8.333 x 3 = $25, and $25 + $3 + $2 = $30.
There are no disappearing dollars, only flawed logic.
Come back next week for another riddle!
*See all of our riddles here.
Jay Bennett is the associate editor of PopularMechanics.com. He has also written for Smithsonian, Popular Science and Outside Magazine.