Missing dollar riddle

The so-called missing dollar paradox is a puzzle which seems at first glance to violate the laws of mathematics. In fact the story is neither a genuine paradox (since there no inherent contradiction once the solution is discovered) nor a mathematical puzzle (since it relies on trickery to achieve its effect).

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, 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. Where is the missing dollar?

Solution

The problem's second paragraph states five truths:

1. Each of the ladies paid $9
2. Three times 9 is 27.
3. The waiter has $2 in his pocket.
4. Two plus 27 is $29.
5. 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. The equation must apply to one "pocket", not two. No. 4 confuses what the waiter kept ($2), and what the ladies think they spent ($27), thus mixing up pockets.

For the waiter, to do it right, you draw a mental circle around his pocket, and count what goes in and out. For the waiter, if you correctly draw a mental circle around his pocket, the thinking goes like this:

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.)

You can make a similar circle around one lady's pocket, but you can make a similar circle around the ladies collective pockets, where they start with 30, give 30 to the waiter, and get back 3.

30 - 30 + 3 = 3 (The ladies end up with 3 dollars.)

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.

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).

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.

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 might 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.)

Another, prior explanation

The ladies initially paid $10 each, $30 in total. $10 x 3 (ladies) = $30 in total

They were each given $1 back. $3 (from the $5 change) divided by 3 = $1 each

Hence, they each paid a net $9. $10 - $1 = $9

That means they paid a total of $27. $9 x 3 (ladies) = $27

$2 of that $27 was taken by the waiter. They paid their bill of $25 and unwittingly gave him $2. $25 + $2 = $27

Therefore, there is no missing dollar.

Simply put, people get confused between the $3 returned to the ladies and the $2 in the waiter's pocket.