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The missing dollar riddle is a famous problem which plays with confusion and misdirection. more...
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The problem
Three guests check into a hotel. The Cashier says the bill is $30 so each pays $10. Later the cashier realizes the bill should only be $25. To rectify he gives the bellhop five singles to return to the guests. On the way back to the room the bellhop realizes that he cannot divide the money evenly. As they didn’t know the total of the revised bill, he decides to give each guest a dollar and keep two for himself.
Now that the guests have been given a dollar back, each has paid $9. Three times nine is 27 and the bellhop has $2. Two plus 27 is 29. If the guests originally handed over $30, what happened to the remaining dollar?
Solution
We unravel this confusion by recognizing that there is no reason to add $2 to $27. It should be subtracted.
The $3 amount that has been returned to the guests is a reduction in the amount that the guests paid, so it should be subtracted from the total. The bellhop returned $3 ($1 each), making their total payment $27 (mathematically, $30 - $3). Note that the $3 is subtracted from the total. If the bellhop then changed his mind and returned the additional $2 to the guests, it would also be subtracted from the total. The mistake is made in trying to add this $2 instead of subtracting it. Simple math demonstrates what readers intuitively sense, that there is no missing money. The sum of their payments is $25 in the till, $2 in the bellhop's pocket (totaling $27), plus the $3 in change that the guests now have, which brings the total up to $30.
The incorrect solution is: ($10 - $1) x 3 + $2 = $29. This equation is not meaningful: the number 29 is not significant to the problem, i.e. there is no "missing $1".
The correct solution is: ($10 - $1) x 3 - $2 = $25. In this case the solution is the bill amount, which is also the amount of money left in the till.
In other words, $27 is the amount that the guests have paid. Of that $27, $25 went into the till and $2 went to the bellhop. The other $3 is returned to the guests.
This problem provides a means to understand how misdirection, and irrelevant facts and questions, can foil clear analysis. Additionally, the tools used to resolve this paradox are used in the analysis of a wide range of financial and scientific areas.
Misdirection
The problem's second paragraph states five truths:
Each of the guests paid $9;
Three times 9 is 27.;
The bellhop has $2 in his pocket.;
Two plus 27 is $29.;
The guests originally handed over $30.;
Unfortunately, No. 4 is a misdirection. In this problem, we wonder about what is going in and out of folks' pockets, and how much is staying there. However, to think about pockets correctly (and to write sensible math), you must mentally draw a circle around each pocket, and count everything that goes in and out of that single pocket. Thus, the equation for one pocket must be derived from what goes in and out of that (same) one pocket. No. 4 confuses what the bellhop kept ($2), and what the guests think they spent ($27), thus mixing up pockets.
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