The Missing Dollar Riddle

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! 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..

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