Fun math problem that stumps everyone

[Sk08]

I stumbled upon this simple looking math problem last week and showed it to several people and have not found anyone that got it right, including my daughters 6th grade math teacher. It took Albert Einstein an hour to figure it out.

If you travel at a speed of 30mph in the 1st mile, how fast do you have to go in the second mile so you have an average speed of 60mph over the 2 mile stretch?

 

[braumeister]

It's not possible.

 

[Sunset]

Funny, it looks like it's relatively simple, with the wrong but simple answer to be 90 as 90 + 30 averaged is 60.
But that is the Wrong answer.

I think if Einstein was working on the faster than light makes time go backward, I can see why he spent an hour on it.

For me, there is no answer.

 

[Car-Guy]

^^^^^Agreed it seems easy that 90mph is the answer but I'll need to think on it,,,,, I guess.

 

[Gumby]

As Braumeister says, it can't be done. Averaging 60 mile/hour = 1 mile/minute, so it requires 2 minutes to go two miles. However, 30 mile/hour = 0.5 mile/minute, so covering the first mile already took you 2 minutes. It would require infinite speed to achieve zero time over the second mile, but nothing travels faster than light at ~670 million miles/hour. Ergo, it cannot be done.

It took me longer to type this than it did to figure it out.

 

[RunningBum]

It's not possible.

+1. In the first mile you've already used up your 2 minutes needed to avg 60 mph in 2 miles. I had the answer before seeing this post, so I guess it didn't stump everyone.

 

[Koolau]

My favorite (and I wish I had a copy of it) was a simple series of additions and subtractions on a strip of paper. As each operation was completed, the paper was rolled to the next operation that used the result from the previous operation. Seriously, there were no tricks within the operations. I never saw anyone in my Jr. Hi class get it right - including two math teachers.

It turned out to be one of those crazy things where your mind leaps ahead (perhaps making you end up with 4000 instead of 3100 or perhaps 3900. I forget just how it worked but I never saw it fail. - even when everyone had heard about it.) YMMV

 

[Dtail]

As Braumeister says, it can't be done. Averaging 60 mile/hour = 1 mile/minute, so it requires 2 minutes to go two miles. However, 30 mile/hour = 0.5 mile/minute, so covering the first mile already took you 2 minutes. It would require infinite speed to achieve zero time over the second mile, but nothing travels faster than light at ~670 million miles/hour. Ergo, it cannot be done.

It took me longer to type this than it did to figure it out.

Conceptually, this was my thought process too. Should have posted it sooner.

 

[Qs Laptop]

If you drive 2 miles at 60 mph it takes 2 minutes to cover that distance.

You've already driven 1 mile at 30 mph. That took 2 minutes.

Problem can't be solved.

 

[Car-Guy]

Completely missed the 2 minute component but I get it takes 2 minutes to go the first mile at 30mph. Word game...

 

[Sk08]

Looks like we are good with numbers on this forum, not surprising I guess since early retirement has a lot to do with math.
I worded it wrong in the title when I said stumps everyone, should have said almost everyone.

If you ask your family and friends, chances are they will say 90.

 

[Jerry1]

This turned out better than I thought. I get really irritated with FB type feeds where they’ll show you a math problem but never the answer. So they get numerous answers but no one is sure which one is the right answer or why. Even though I like those type of problems, I skip them because there is no answer given.

 

[RobbieB]

I did the time thing and found it impossible too.

But hey, I'm an engineer and that's what they pay me for -

Although an engineer would say "light speed". But the math guy would say, even light speed would add time.

And the engineer would reply the the difference was insignificant -

 

[zinger1457]

No way it took Einstein an hour to figure it out unless his first name was Joe.

 

[Out-to-Lunch]

And if you think it took Einstein an hour to solve, I have a just-over-one-mile-long bridge in Brooklyn that I can let go for cheap.

EDIT: cross-posted with Zinger.

 

[Fermion]

He could have been trying to apply special relativity and determining if time dilation effects could cause the observed clock inside the 30mph vehicle to be slightly under 2min, thus allowing a theoretical less than light speed second mile to get in at 2min for 2 miles.

 

[ERD50]

If you travel at a speed of 30mph in the 1st mile, how fast do you have to go in the second mile so you have an average speed of 60mph over the 2 mile stretch?

So I'll post before looking at the responses.

This isn't that hard, I doubt it took Einstein an hour (proof of that?).

60 mph is a mile a minute. So to average 60 mph over 2 miles will take 2 minutes.

You traveled the first mile at 30 mph. Well, if 60 mph is a mile a minute, 30 mph is a mile in two minutes. You already used up all the time, you can't average 60 mph with only one mile to go (or one foot to go!). Can't be done, at any speed.

A similar problem is using mpg. Get 20 mpg on to a destination that is a run up a hill or against a head-wind, and 40 mpg on the way back does *not* give an average of 30 mpg. A "mpg" figure doesn't work that way, since the distances don't match the distance traveled on a gallon. You'd have to go 20 miles @ 20 mpg, and 40 miles @ 40 mpg to average 30 mpg. The distances are different.

It's also why going from a car with 20 mpg to one with 40 mpg is not the same fuel savings as going from 10 mpg to 20 mpg.

This leads people to make bad decisions.

Now, I hope I got all that right, or I'll be embarrassed! At least I avoided using affect and effect!

-ERD50

 

[RunningBum]

I was going to suggest it's just folklore that Einstein was presented with this problem, but maybe not. https://fs.blog/einstein-wertheimer-car-problem/ documents a similar problem. I doubt it took Einstein an hour, but if this story is true, did take him more than a moment. Perhaps being sent the question by a colleague made him think there must be an answer.

 

[REWahoo]

See my sig.

 

[ERD50]

I was going to suggest it's just folklore that Einstein was presented with this problem, but maybe not. https://fs.blog/einstein-wertheimer-car-problem/ documents a similar problem. I doubt it took Einstein an hour, but if this story is true, did take him more than a moment. Perhaps being sent the question by a colleague made him think there must be an answer.

I didn't get any sense it took him anywhere near an hour. The article states that he replied, “Not until calculating did I notice that there is no time left for the way down!”

You could say it wasn't immediately obvious to him that the solution would not be some reasonable speed. But "not until calculating it", could have been just a few seconds.

I'll also take exception to those using the phrasing that "there is no answer", or that it "can't be solved" (versus "it can't be done" - with "it" being "go fast enough to average 60 mph").

There is an answer to the problem, and the problem can be solved, and that answer is that there is no way to average 60 mph at that point. Engineers do calculations all the time which show that a thing cannot be done. That's the solution. It keeps people from wasting time and resources trying to do something which cannot be done (no, not even if you just "try hard enough").

-ERD50

 

[RobbieB]

Engineers make stuff happen, not just sit and say can't be done.

The first mile takes 2 minutes, the last mile takes 5 micro seconds -

 

[CindyBlue]

I used to assign this one in my math classes. It's based on the harmonic mean formula, which is one of the three Pythagorean means (the other two are arithmetic mean and geometric mean.)

If the distances are the same, then the average speed will equal the harmonic mean of the two speeds, or (sorry, this is hard to write the formula on my computer!)
Average speed = 2/(1/speed 1 + 1/speed 2) (and this is the harmonic mean formula already simplified to make it easier to directly sub in specifically for average speed.)
When you sub in 60 for ave. speed, 30 for speed 1, and "x" for speed 2, and work it out with algebra, it produces a "nonsense answer" of 1800=0, so it cannot be solved.

Then, of course, we had the discussion about logically why it cannot be solved, as Gumby described!

 

[Sunset]

I didn't get any sense it took him anywhere near an hour. The article states that he replied, “Not until calculating did I notice that there is no time left for the way down!”

You could say it wasn't immediately obvious to him that the solution would not be some reasonable speed. But "not until calculating it", could have been just a few seconds.

I'll also take exception to those using the phrasing that "there is no answer", or that it "can't be solved" (versus "it can't be done" - with "it" being "go fast enough to average 60 mph").

There is an answer to the problem, and the problem can be solved, and that answer is that there is no way to average 60 mph at that point. Engineers do calculations all the time which show that a thing cannot be done. That's the solution. It keeps people from wasting time and resources trying to do something which cannot be done (no, not even if you just "try hard enough").

-ERD50

I answered: "For me, there is no answer. " with the wink, Rather than give away the answer directly, so others would have a chance to try it.

It's all just for fun here.

 

[Ronstar]

Interesting. The problem doesn't say that the car averages 30 mph in the first mile - only that the car travels "at a speed of 30 mph in the first mile". So the 30 mph in the first mile is not necessarily the average.

Same for the question "How fast do you have to go in the second mile" doesn't ask for the average speed in the second mile- only asks what speed you need to get up to to average 60 mph for the 2 miles.

This is going to take some thought.

 

[Ronstar]

Ok. I take it this way. Driver traveled at 30 mph in the first mile. Let's assume that is the average.

So that means his/her speed at the end of the first mile is 60 mph since the starting speed is 0 mph.

Problem states that the driver averages 60 mph over 2 miles. Since the driver averaged 30 mph in the first mile, he/she needs to average 90 mph in the 2nd mile in order to average 60 mph for the whole 2 miles. (30 ave mph mile 1 + 90 mph ave mile 2) / 2.

Since the driver was driving 60 mph at the beginning of mile 2, the driver needs to get going 120 mph at the end of mile 2 to average 90 mph in the 2nd mile.

The answer is 120 mph.