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        riddles >> medium >> Liquid Lunch
        
(Message started by: ThudanBlunder on Jan 30th, 2009, 5:52pm)
Title: Liquid Lunch
Post by ThudanBlunder on Jan 30th, 2009, 5:52pm
Tom, Dick, and Harry were having a drink at lunchtime. But they must get back to the office 9 miles away by 3.00 pm. Tom has a bike, Dick has a moped, and Harry has no transport at all. So they decided to share. Each of them can walk at 5mph, cycle at 9mph, and ride the moped at 15mph. Although neither vehicle can take more than one person, they can be left safely by the road for somebody else. The three friends left the bar together at the last minute and arrived back just in time for work.

At what time did they leave the bar?
Title: Re: Liquid Lunch
Post by 1337b4k4 on Jan 30th, 2009, 11:29pm
I can have them leaving at [hide]1:52[/hide] and thats probably the optimum.
Title: Re: Liquid Lunch
Post by Grimbal on Jan 31st, 2009, 2:33am
I find [hide]the same[/hide].
How?
[hide]
#1: ride 5 miles @ 15 mph =  20 min.
  walk 4 miles @ 5 mph = 48 min.
  total: 68 min.
#2: cycle 1.5 miles @ 9 mph = 10 min.
  walk 3.5 miles @ 5 mph = 42 min.
  ride 4 miles @ 15 mph = 16 min.
  total: 68 min.
#3: walk 1.5 miles @ 5 mph = 18 min.
  cycle 7.5 miles @ 9 mph = 50 min.
  total: 68 min.
[/hide]
Title: Re: Liquid Lunch
Post by ThudanBlunder on Jan 31st, 2009, 6:32am
At a later date Tom, Dick, and Harry decide to go for lunch at a local bistro 10 miles from their office. Dick's moped can still do 15mph, they still walk at 5mph, and Harry still has no transport of his own. But Tom's bike has been damaged and, as it can now do no more than 6mph, he doesn't care what happens to it. As before, they leave the bistro together at the last minute and arrive back just in time for work.

At what time did they leave the bistro?


Title: Re: Liquid Lunch
Post by 1337b4k4 on Jan 31st, 2009, 10:47pm
[hide]As before, the theoretical limit is found by considering that every vehicle must make the journey at least once, which will need, in this case, 9/5 + 9/6 + 9/15 = 3.9 man-hours, thus at best we can have each person take 1.3 hours. However, I'm having much more difficulty gerrymandering them all in, probably for some important reason.[/hide]
Title: Re: Liquid Lunch
Post by ThudanBlunder on Feb 1st, 2009, 1:06am

on 01/31/09 at 22:47:23, 1337b4k4 wrote:
However, I'm having much more difficulty gerrymandering them all in, probably for some important reason.

Indeed, as they must now travel 10 miles. ;)
So try shoehorning them into 260/3 mins.
Title: Re: Liquid Lunch
Post by Immanuel_Bonfils on Mar 1st, 2009, 8:41pm
Assume n persons traveling the total distance n*d with n-1 vehicles and by foot, on n different speeds, without stopping.
For which values of n can they do it with each vehicle traveling distance d?
Title: Re: Liquid Lunch
Post by ThudanBlunder on Mar 23rd, 2009, 7:51am

on 03/01/09 at 20:41:57, Immanuel_Bonfils wrote:
Assume n persons traveling the total distance n*d with n-1 vehicles and by foot, on n different speeds, without stopping.
For which values of n can they do it with each vehicle traveling distance d?

Let L = the length of the journey
Let M = the number of travellers
Let N = the number of vehicles
Let T = the time time taken by the last  traveller to arrive at the destination
Let v0 = the speed with which they travel on foot
Let vr = the speed of the rth vehicle (r = 1 to N - 1)

Then
t0 = L/v0
t1 = L/v1
t2 = L/v2
t1 = L/v3

If there were no vehicles available, the total of the times taken by the M travellers would be Mt0.
Making one vehicle available with speed vr can reduce this total by at most t0 - tr.
Making N vehicles available reduces this total by at most (t0 - t1) + (t0 - t2) +  (t0 - t3) + ........ +  (t0 - tN)
That is, the total time for M travellers using N vehicles is at least (M - N)t0 + t1 + t2 +  t3 + ........ + tN

Let T* = [(M - N)t0 + t1 + t2 +  t3 + ........ + tN]/M
Then we have T http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/geqslant.gif T* where T is the time taken by the last traveller to arrive at the destination.
Also, if T = T* then every traveller takes T* to arrive at the destination, as T* is the average time they take and T is the time taken by the last arrival. That is, T = T* when t1,t2,t3,........,tN http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/leqslant.gif T* because, when no vehicles are abandoned, if any vehicle takes more than T* to complete the journey then its last user must also take more than T*.

In the first problem
L = 9
v0 = 5  
v1 = 15
v2 = 9
t0 = 108
t1 = 36
t2 = 60

Hence T* = [108*(3 - 2) + 36 + 60]/3 = 68mins.
As t1 and t2 are both less than T*we have T = T*

In the second problem
L = 10
v0 = 5  
v1 = 15
v2 = 6
t0 = 120
t1 = 40
t2 = 100

Hence T* = [120*(3 - 2) + 40 + 100]/3 = 86.666... mins.
But as t2 > T* we have T > T* and there is no 'easy' solution.

But they are allowed to abandon a vehicle, the bike. Suppose they abandon the bike after D miles from the start, taking 10D minutes, and ride the rest of the way using the moped left by the others, taking 4(10 - D) minutes. So total time = 10D + 4(10 - D) = 6D + 40 minutes. The other two will have cooperated to get the moped to the dropping-off point, each riding D/2 miles and walking D/2 miles, and then walking together to the destination, taking a further 12(10 - D) = 120 - 4D miles.

So the last of these three to arrive will have taken max{6D + 40, 120 - 4D}
This is minimised when the two amounts are equal. That is, when D = 8
Putting D = 8 gives T = 88 minutes as the common time for all travellers to arrive.


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