Hey, this is Presh Talwalkar, and you're watching Mind Your Decisions.
The psychologist Max Wertheimer was friends with Albert Einstein.
In 1934, he sent Einstein a letter with the following puzzle:
There's an old car that needs to go up and down a hill.
The hill is one mile going up, and
Because the car is old, it can only average 15 mph going up the hill.
The hill is also one mile going down,
But the car may be able to go faster ,because it's going down the hill.
The question is: how fast must the car go going down the hill so that it averages 30 miles per hour for the entire 2 mile trip?
Can you figure it out?
Give this problem a try, and when you're ready, keep watching the video for a solution.
So before I get to the solution I want to provide some context. This puzzle comes from a 1934 letter(we know)
Albert Einstein had already been recognized with the Nobel Prize in Physics in 1921.
In 1905,
he had produced four amazing results including the photoelectric effect,
Brownian motion, special relativity, and energy and mass equivalents summarized by the equation e = mc^2.
Any one of these works would have been a lifetime achievement and he produced all four in one year, so
you might think if there's anyone who would be able to solve puzzles in his head, it would be Albert Einstein.
but even he had trouble. This puzzle was not immediately obvious to Einstein, as he wrote
he didn't see the trick until he calculated the answer.
So I think there two lessons from this: first, even Einstein had to work carefully to solve math problems, and
second, he enjoyed this simple puzzle, just like the many simple puzzles that I post in my videos.
So I sure everyone will enjoy all of the puzzles that I post, and even if they want harder puzzles,
they'll appreciate some of the easier problems, just like Albert Einstein did.
This story is told in Gerd Gigerenzer's book: "Risk Savvy: How to Make Good Decisions".
So now let's solve the problem.
We'll work backwards.
First we'll ask: what's the time necessary to average 30 miles per hour for the entire two mile trip?
So we set up the equation 30 mph = 2 mi/ t trip.
We'll solve for the time and we get this is equal to 1/15 of an hour.
As there are 60 minutes in an hour, this is equal to four minutes.
Now let's solve for the time the car takes going up the hill.
Well, the car takes for one mile of 15 mph average.
This simplifies to be one fifteenth of an hour, which again is four minutes.
So we see the car takes four minutes just to go up the hill and it needs to take four minutes for the entire
trip to average 30 miles per hour, and
this leads to a surprising conclusion:
the car must take 0 minutes going down to get 30 miles per hour for the entire trip;
therefore, this is a trick question.
It's not possible for the car to average 30 miles per hour for the entire two mile trip.
Unless it takes zero minutes to get down the hill. That would only be possible if the car could travel infinitely fast,
and we know that's not possible,
because nothing can travel faster than the speed of light.
As Einstein wrote: "Not until calculating did I notice there is no time left for the way down."
So this is a delightful little brain teaser that even Albert Einstein
enjoyed. Did you figure out this problem?
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