Vsauce!
Kevin here and you just lost a dollar.
I mean not literally -- obviously this is a fictional scenario that examines an unexpected
slice of numerical cognition.
And in our scenario, a dollar is mysteriously missing.
You, and your friends Michael and Jake went out to lunch and split the $30 bill evenly.
You paid $10 bucks each.
But the server accidentally overcharged you and the bill should’ve been only $25 dollars.
So she gives the busboy your $5 dollar refund but he can’t figure out how to evenly divide
it amongst three people, and frankly, you don’t even know that you've been overcharged,
so he just takes a $2 tip for his efforts and gives you, Michael and Jake one dollar
back each.
So now you’ve only paid 9 dollars each for lunch.
9 + 9 + 9 = 27 plus the busboy kept $2 for himself for a total of $29.
But you originally paid $30.
So... a dollar is missing.
How does a dollar disappear?
Where is the missing dollar?
Before we breakdown this riddle let’s try this with different numbers and just see what
happens.
Let’s say you still pay a $30 dollar bill but this time the server realizes there’s
been a huge mistake, and the total should’ve been only $10 bucks.
So now the busboy gets $20 to return to you, Michael and Jake, which he also can’t divide
equally.
You don’t know about the exact price change, so the busboy takes his $2 tip, leaving him
with $18 to cleanly divide three ways.
He refunds each of you $6 bucks.
Reducing your effective price to 4 dollars each.
So you paid $4 + $4 + $4 + plus the busboy tip of $2 = $14.
And just like that $16 is gone!
Because you only paid $14 for a $30 bill?
No.
No!
Something is clearly completely wrong with this setup.
This is actually a pretty popular and rather old riddle.
It seems to have originated in the mid 1700s with mathematician Francis Walkingame’s
“Tutor’s Assistant,” a sort of arithmetic textbook.
"If 48 taken from 120 leaves 72, and 72 taken from 91 leaves 19, and 7 taken from thence
leaves 12, what number is that, out of which, when you have taken 48, 72, 19, and 7, leaves
12?"
Part of what makes the Missing Dollar Riddle confusing is that it bombards you with numbers.
That’s okay, though because your abilities with numerical cognition are actually pretty
good.
Yale cognitive scientist Karen Wynn found that infants can understand simple arithmetic
like 1 + 1 = 2.
If I bombard you with numbers, like 2 + 5 + 3 + 1 + 7 + 12, you aren’t confused about
what’s happening and you can work through it to an answer of 30.
But when context is added to those numbers, it’s easy to get sidetracked.
We get so wrapped up in what’s happening with the lunch, the refund, and the bus boy
that we gloss over the real math and hear 9 + 9 + 9 + 2 = 29 and wonder where the dollar
went?
But it’s not missing.
We just played a mathematical sleight of hand.
Here’s the breakdown.
We started with $30.
$25 of that is still in the restaurant’s cash register.
The other $5 has been given back -- $3 have been returned to you and $2 for the busboy’s
tip.
25 + 3 + 2 = 30, so where’s the disconnect?
Really?
Where exactly is the disconnect?
Let’s find it.
There are really two huge problems in this riddle.
One is how we think about that $3 refunded dollars-- we’re only considering it once
when we really need to consider it twice.
And the second problem is how we’re incorrectly factoring in the bus boy’s tip.
Let’s talk about that tip first.
Because the bus boy takes his $2 tip, that's money that’s never returned to you, Michael
or Jake, and therefore must now be considered as part of the total cost of the lunch.
Which means that the new cost of the meal is the $25 still in the cash register, plus
the bus boy’s $2 tip equaling $27.
When each of the three of you pays $10 bucks and then gets $1 dollar back, your effective
cost of the meal -- the food plus the waiter’s tip -- is $9 each, for $27.
That leaves $3 dollars missing.
The $3 dollars returned to you by the bus boy.
So now we’re left with the $3 problem.
The 9 plus 9 plus 9 plus 2 equation of the riddle is simply the wrong equation.
Actually, as we just learned, the 9’s are right but the 2 is wrong.
The 2 is already factored into the 9’s.
What we need to do is add back the 3.
Which is kinda confusing and part of that sleight of hand.
The riddle uses the 3 to subtract from the customers’ new final, total cost of the
meal, but we have to add it back in at the end to get to the original $30 that changed
hands.
Look. You paid ten dollars each.
Refund your $3 taking the amount you paid down to $27.
And then give the busboy his $2 tip and the restaurant keeps $25.
There is no missing dollar.
The riddle is just answering the right question the wrong way.
Legendary statistician John Tukey once said, “An approximate answer to the right question
is worth a great deal more than a precise answer to the wrong question.”
When it comes to the Missing Dollar Riddle, the precise answer to the right question is
worth at least... a buck.
And always -- thanks for watching.