Missing dollar paradox
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The missing dollar paradox is a famous problem which plays with confusion and misdirection.
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[edit] The Problem
Three ladies go to a restaurant for a meal. They receive a bill for $30. They each put $10 on the table, which the waiter collects and takes to the till. The cashier informs the waiter that the bill should only have been for $25 and returns $5 to the waiter in $1 coins. On the way back to the table the waiter realizes that he cannot divide the coins equally between the ladies. As they didn’t know the total of the revised bill, he decides to put $2 in his own pocket and give each of the ladies $1.
Now that each lady has been given a dollar back, each of the ladies has paid $9. Three times 9 is 27. The waiter has $2 in his pocket. Two plus 27 is $29. The ladies originally handed over $30. Where is the missing dollar?
[edit] Solution
We unravel this "paradox" 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 ladies is a reduction in the amount that the ladies paid, so it should be subtracted from the total. The waiter returned $3 ($1 each), making their total payment $27 (mathematically, $30 - $3). Note that the $3 is subtracted from the total. If the waiter then changed his mind and returned the additional $2 to the ladies, 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 waiter's pocket (totalling $27), plus the $3 in change that the ladies 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 ladies have paid. Of that $27, $25 went into the till and $2 went to the waiter. The other $3 is returned to the ladies.
This "paradox" provides a means to understand how misdirection, and irrelevent 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 ladies paid $9
- Three times 9 is 27.
- The waiter has $2 in his pocket.
- Two plus 27 is $29.
- The ladies 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 waiter kept ($2), and what the ladies think they spent ($27), thus mixing up pockets.
[edit] Conservation
For anyone, what one takes less 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 Waiter's Pocket
The waiter takes 30 dollars from the ladies, gives 25 to the cashier, and then gives back 3 to the ladies.
- 30 - 25 - 3 = 2 (The waiter ends up with 2 dollars.)
[edit] The Ladies Pockets
You can make a similar circle around one lady's pocket. You can also make a bigger circle around the three ladies (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 waiter, and get back 3.
- 30 - 30 + 3 = 3 (The ladies 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 Waiter and the Ladies
If you wanted, you could draw a circle collectively around the ladies and the waiter, together -- the Ladies/Waiter start with 30; they give 30 to the cashier, who returns 5.
- 30 - 30 + 5 = 5 (Between the ladies and the waiter, they end up with $5.)
This makes sense, as between the cashier on one hand, and the Ladies/Waiter on the other hand, there is still a total of $30 ($25 for the cashier and $5 with the Ladies/Waiter -- Of that 5 dollars, the waiter has 2, and the ladies have 3).
[edit] Everyone
Finally, a circle around all of them, the universe of people in the problem (the waiter, the ladies, 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, just like in this problem the cash is 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 waiter takes the money, but before handing it over to the cashier.
| Lady 1 | Lady 2 | Lady 3 | Cashier | Waiter | Total | |
|---|---|---|---|---|---|---|
| Before The Meal | $10 | $10 | $10 | $0 | $0 | $30 |
| When The Cash Is Given to Cashier | $0 | $0 | $0 | $30 | $0 | $30 |
| After They Leave | $1 | $1 | $1 | $25 | $2 | $30 |
| Difference [Before - After] | -$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 [Before - After]" 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 ladies brought to the restaurant. It does not show the content of the cashier's drawer, after the ladies leave. This table draws a circle around these five people and the ladies cash, only; if you want to know how the waiter or the restaurant fared this evening, you must ask different questions.
