Solving the Diner Dollar Riddle

 From the Solver's Guild Instagram. 

You are a waiter at a popular diner near the business district of a busy city. One day, three customers are seated in your section in the middle of the lunch rush.⠀
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You fill their drinks and take their lunch orders. Unfortunately, the kitchen is slammed and it takes a long time for their food to arrive. After they’ve eaten, you present their check for $30. Each diner hands you a $10 bill.⠀
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Before you cash them out, your manager agrees to take $5 off their bill for the slow service. Afraid the diners won’t tip you after the long wait, you put $2 in your pocket and give each diner $1 back in change after explaining the situation.⠀
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Just as you suspected, they stiff you on the tip! Little do they know you already helped yourself! As you bus their table, something—other than your guilty conscious—troubles you. Each diner paid $9 for a total of $27, and you secretly took $2 for a total of $29.⠀
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Where did the last dollar go?

If you think you can solve this tricky puzzler, skip to the bottom and comment your answer. Otherwise, for the solution, read on!

SPOILERS AHEAD!

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So where did that missing dollar go? The answer is that it didn't go anywhere! All the money that changed hands during the scenario is present and accounted for. This riddle throws the solver a curve-ball by intentionally providing faulty math that the solver has to work backwards to uncover.

There's a few things at play here, and I'll cover them one by one.

First, the riddle states that the diner's total for the meal came to $30 exactly and that each diner paid with a $10 bill. Easy math there: 10 x 3 = 30. What the riddle doesn't do is tell you how much the total is after the diner manager applies a discount for slow service. He applies a $5 credit to their tab, but this is never explicitly stated as making their new total $25. This is done intentionally, because $30 is a round number evenly divisible by 3 that sticks in the solver's mind.

Second, the waiter takes the $5 discount and divides it among himself and the three diners unevenly. He keeps $2 and gives each diner one dollar back as a refund on their meal. This further attempts to skew the solver's attention to how much money is actually changing hands and where it should all be accounted for.

Finally, the waiter's math at the end of the riddle is just plain wrong. He thinks that they diners each paid $9 for their meal for a total of $27, which is correct. But adding in his clandestine tip is where the confusing mistake occurs. The diner's did not pay $29 for their meal and tip. They only paid $27 and that number includes the tip the waiter kept for himself. Let's look at it.

• Each diner pays $10, even splitting a check of $30... $30 / 3 diners = $10/diner.

• The manager removes $5 from their total check. $30 - $5 = $25 check.

• At this point there is $25 in the register--the total of their meal--and $5 in the waiter's hand. The total $30 is still accounted for. 

• The waiter keeps $2 for himself and gives $1 each back to the diners. Again, all the money is accounted for (if not where it belongs!). $25 in the till + $2 tip + $3 'change' = $30.

• The diner's did not pay $27 for their meal. In reality they paid $25 / 3 = ~$8.333... each.

• Their change from a $10 bill should have been $10 - $8.333... = $1.666... each.

• The change they actually received was $1, a discrepancy of roughtly $0.67 each.

• $0.67 or 0.666... x 3 = 2... or the $2 tip the waiter kept from himself!

Imagine it this way. Instead of a busy lunch hour, the diners come in for breakfast. They're the first customers of the day. The cash drawer is empty, the waiter hasn't received any tips, and for the purposes of example, there is no money in the diner except for a single $10 bill in each patron's wallet. When they enter the diner, there is only $30 in the building.

When they get their check, all $30 in their wallets is transferred to the cash till. After the manager takes a discount, there is $25 in the register and $5 with the waiter. Still only $30 in the diner. After the waiter takes his 'cut' he gives $1 back to each guest. The money in the till hasn't moved, and the remaining $5 has been distributed, but the total amount of money across all hands is still $30.

When the diner's leave, how much money is left in the diner? $27... not the $29 the waiter incorrectly assumed when he did the math while busing their table!

And that's the answer to today's riddle. The 'missing dollar' didn't exist in the first place, and the moral of the story is to pay attention when it comes to math. And don't steal... yeah, let's go with don't steal!