# Time puzzle



## robert@fm (Sep 23, 2013)

This is a classic one. 

At 3pm on a fine summer's day, a hiker sets out from his inn to walk to the top of a nearby hill. He averages 3mph on the flat, 2mph going uphill, and 6mph going downhill. When he reaches his goal, he spends the briefest of moments admiring the view, before setting off back; he returns to the inn at 9pm.

1) How far away is the hilltop?

2) To the nearest half-hour, when did he reach it?


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## moonymama (Sep 23, 2013)

driving me mad..tell me.!!!!!


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## Royston46 (Sep 23, 2013)

driving me mad also


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## Redkite (Sep 23, 2013)

I'm assuming he didn't walk along the flat at all, but just went straight up and down.  In which case the hill top is 9 miles away, and he got to the top at 7.30pm.


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## Redkite (Sep 23, 2013)

For those who want the maths  :

Let distance to top of hill = x

Total time taken = 6 hours

At 2mph, do x miles in x/2 hours
At 6mph, do x miles in x/6 hours

So x/2 + x/6 = 6

So 3x/6 + x/6 = 6

So 4x/6 = 6

Ie. 4x = 36 

Ie. x = 9miles

Time to top =x/2 = 4.5 hours


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## Aoife (Sep 23, 2013)

even the maths doesn't help my tiny little mind Redkite!


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## Andy HB (Sep 23, 2013)

I hate this kind of puzzle. One has to make an 'unjustified' assumption to make it solvable.

Making the assumption that there is no part of the route that is flat is, in my opinion, 'unjustifiable' given the stated facts (unless stating that there IS a solution is a fact that implies that there is no flat part of the route .... Ok, I stand corrected!!).

Andy


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## Redkite (Sep 23, 2013)

Andy HB said:


> I hate this kind of puzzle. One has to make an 'unjustified' assumption to make it solvable.
> 
> Making the assumption that there is no part of the route that is flat is, in my opinion, 'unjustifiable' given the stated facts (unless stating that there IS a solution is a fact that implies that there is no flat part of the route .... Ok, I stand corrected!!).
> 
> Andy


The thing is, it doesn't matter if part of the route is on the flat, because he would be walking at the same speed in both directions, thus having the same thing on both sides of the equation, which would cancel each other out.  So the 3mph on the flat is a red herring.


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## Andy HB (Sep 23, 2013)

Redkite said:


> The thing is, it doesn't matter if part of the route is on the flat, because he would be walking at the same speed in both directions, thus having the same thing on both sides of the equation, which would cancel each other out.  So the 3mph on the flat is a red herring.



I stand corrected again!! 

But, looking at it again. Whilst everything stacks up with the numbers given. 3mph on the flat works if the entire route is flat too (i.e. 3hrs there and 3hrs back). But, if he walks at 4.5mph on the flat? That no longer stacks up if the flat portion of the route cancels out. I think I'm missing some maths here!!

Andy


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## robert@fm (Sep 23, 2013)

*My apologies!*

Sorry, I got the initial statement of the puzzle wrong!  So I don't know if the given solution is correct (it seems that it might be, but...)

Here is the actual classic puzzle as it should have been stated:



> At 3PM one fine summer's day, a hiker sets off from the inn where he is staying to climb a nearby hill. He averages 4mph on level ground, 3mph uphill and 6mph downhill. On reaching the hilltop, he pauses for the briefest of moments to admire the view, then starts back the way he came, reaching his inn at 9PM.
> 
> 1) How far is it from the inn to the hilltop?
> 
> 2) To the nearest half-hour, when did he reach the top of the hill?



Note also that the puzzle is completely solvable as-is; you have all the required information, it is not necessary to make any assumptions (justified or not).


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## Redkite (Sep 23, 2013)

OK, in that case the total distance from the inn to the hilltop is 12 miles, and he gets there at 7.30pm.  I think


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## robert@fm (Sep 23, 2013)

Redkite said:


> OK, in that case the total distance from the inn to the hilltop is 12 miles



Correct, but you haven't explained your working. 



> and he gets there at 7.30pm.  I think



Sorry, wrong.


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## Andy HB (Sep 23, 2013)

I agree with the 12 miles --> Woo Hoo!!

T = Total time = 6hrs
Tf = Time on flat
Tu = Time going up
Td = Time going down

T = 2Tf + Tu + Td

Dh = Distance going up or down = 3Tu = 6Td
Df = Distance on flat = 4Tf

So, 
Tu = Dh/3
Td = Dh/6
Tf = Df/4

So,
T = Df/2 + Dh/3 + Dh/6 = Df/2 + Dh/2 = (Df+Dh)/2

Or
6 = (Df+Dh)/2

And
(Df+Dh) = 12 .... The distance up the hill.

Now on to the next bit!!

Andy


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## robert@fm (Sep 23, 2013)

Well, the original solution to the first question was to note that to pass over a flat mile takes 15 minutes in either direction, so to pass over that mile in both directions (going out and coming back) takes 30 minutes. To go a mile uphill takes 20 minutes, but returning the same way takes 10 minutes. So crossing each mile in both directions takes 30 minutes, regardless of slope.

Equivalently, one could note that the average sloping-ground speed over both directions is 2/(1/3+1/6)=4 mph, the same as the flat-ground speed.


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## Andy HB (Sep 23, 2013)

Well, I am stuck on the next bit!

If the land has no flat bit, then it's 12 miles up and 12 miles back. Time to go up being 12/3 (4hrs) and the time to get back being 12/6 (2hrs).

If the land is entirely flat, then it's 12/4 (3hrs) both ways.

I can't see any information to tell me how to split the flat to non-flat distances.

Ah! Hang on, I think I've got it. I think it's 6:30pm (on the basis that it has to be some time between 6:00pm and 7:00pm).


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## Redkite (Sep 23, 2013)

Andy HB said:


> Well, I am stuck on the next bit!
> 
> If the land has no flat bit, then it's 12 miles up and 12 miles back. Time to go up being 12/3 (4hrs) and the time to get back being 12/6 (2hrs).
> 
> ...


Yaaay Andy I think you're right!

Robert, my maths for the 12 miles answer was:

x = flat distance
y = slope distance

So x/4 + y/3 + x/4 + y/6 = 6

Which reduces to 6(x+y)/12 = 6

i.e. x+y = 12 miles

I couldn't make a mathematical formula for the time taken to the top, so guessed!!!


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