Missing dollar riddle
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The missing dollar riddle is a famous problem which plays with confusion and misdirection.
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[edit] 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?
[edit] 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.
[edit] 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.
[edit] Conservation
For anyone, what one takes minus what one gives away is what one keeps. Draw a mental circle around a pocket, and count what goes in and out of that circle. Correctly done, the thinking goes like this:
[edit] The Bellhop's Pocket
The bellhop takes 30 dollars from the guests, gives 25 to the cashier, and then gives back 3 to the guests.
- 30 - 25 - 3 = 2 (The bellhop ends up with 2 dollars.)
[edit] The Guests' Pockets
You can make a similar circle around one lady's pocket. You can also make a bigger circle around the three guests (collectively, looking at their three pockets). However, the equations must derive from what goes in and out of that bigger circle: they start with 30, give 30 to the bellhop, and get back 3.
- 30 - 30 + 3 = 3 (The guests end up with 3 dollars.)
[edit] The Cashier
Or you could calculate the cashier's cash draw:
- 30 - 5 = 25 (The cashier ends up with 25 dollars.)
If you notice, carefully, the total of what everyone ends up with is 30 dollars (2+3+25). This is exactly the total of what they all started with, collectively. (See the Table, below.)
[edit] The Bellhop and the Guests
If you wanted, you could draw a circle collectively around the guests and the bellhop, together -- the Guests/Bellhop start with 30; they give 30 to the cashier, who returns 5.
- 30 - 30 + 5 = 5 (Between the guests and the bellhop, they end up with $5.)
This makes sense, as between the cashier on one hand, and the Guests/Bellhop on the other hand, there is still a total of $30 ($25 for the cashier and $5 with the Guests/Bellhop -- Of that 5 dollars, the bellhop has 2, and the guests have 3).
[edit] Everyone
Finally, a circle around all of them, the universe of people in the problem (the bellhop, the guests, and the cashier): they start with a total of $30, and end that way, and nothing in the problem gives money to another person.
[edit] Other uses for Conservation
Engineers use the same thinking process to solve basic problems in thermodynamics. In thermodynamics, energy is conserved; the cash in this problem is likewise conserved.
In general, to analyze these sorts of problems, where conservation of anything could be at issue, draw a theoretical circular boundary around some collection of things, and measure the amount of "stuff" that crosses that boundary. The number of things within the boundary can include everything in the universe, or only a couple of things. The skill is to choose the right collection of things, and then realizing that the math applies to that collection, only. Just a few examples (and areas of application) for this thought process include:
- Mass transport across a boundary layer (Oxygen transport across lung cellular membranes.)
- Accounting (Misappropriation or embezzlement of cash or accounts in a business.)
- Newtonian physics (Using vector equations for F=ma, to determine the movement of an object when shoved by various forces over time.)
- Fluid transport (Water flow in an open or closed system.)
- Heat transport across a surface (Cooking a turkey in an oven.)
- Astrophysics (Does the universe expand forever, or will it collapse, eventually.)
[edit] A Financial Cash Flow Analysis
The following table demonstrates the movement of cash, stating (in successive rows) where cash has moved over time. Each row represents an instant in time. Additional rows could have been added; as one example: just after the bellhop takes the money, but before handing it over to the cashier.
| Guest 1 | Guest 2 | Guest 3 | Cashier | Bellhop | Total | |
|---|---|---|---|---|---|---|
| Before Check In | $10 | $10 | $10 | $0 | $0 | $30 |
| When Cashier is Paid | $0 | $0 | $0 | $30 | $0 | $30 |
| After the Bellhop | $1 | $1 | $1 | $25 | $2 | $30 |
| Difference [After-Before] | -$9 | -$9 | -$9 | $25 | $2 | $0 |
The right-hand, "Total" column is the sum of all cash in everyone's hand; as expected, it is always $30. The bottom row, "Difference [After - Before]" is a calculation derived from two other rows. The designer of the table chooses which rows (and moments in time) to display, and also the actual means for deriving the "Difference" row. These choices can be the source for error or obfuscation.
For example, this table demonstrates what happens to the cash the guests brought to the restaurant. It does not show the content of the cashier's drawer, after the guests leave. This table draws a circle around these five people and the guests’ cash only; if you want to know how the bellhop or the restaurant fared this evening, you must ask different questions.
