"We Got A Great Big Convoy" (May 2024 Challenge Question)

How many trucks do you need to deliver your pallet of pillows?

Assume all the trucks are at the starting point. Siphon fuel from second truck to first. Truck 1 now has a full tank (200 mi range). Siphon fuel from truck 4 to truck 3. Truck 3 now has 200 mi range.

Trucks 1 and 3 head toward destination. At 100 miles, trucks 1 and 3 each have half a tank of fuel. Siphon remaining fuel from truck 3 into truck 1. Truck 1 can now reach the destination.

How many trucks do you need to deliver your pallet of pillows? 2

How many trucks do you need to return back to headquarters?

Assuming this means returning the original driver.

Truck 1 has 30 miles of fuel left. He drives back to 240 miles from starting point. Fill up trucks 5 and 6 from 7 and 8 at starting point. At 100 mi, fill up 5 from 7. Truck 5 (200 mi range) reaches driver of truck 1 and drives back 60 mi, 90 miles from destination and 180 miles from starting point.

Fill up truck 9 from truck 10 for 200 mi range. Drive down to truck 1 (180 mi) and back 20 miles to 160 miles from starting point. Do this again with truck 11 and 12, driving truck 11 160 miles down and 40 miles back, 120 miles from starting point. Again with trucks 13 and 14, drive truck 13 down 120 miles and back 80 miles, to 40 miles from start. One more truck, #15 can go down to pick up the original driver.

How many drivers return? (Assume the rest will walk.) 2

GA

But unfortunately the OP edited the question before you posted your answer (Getting to be a habit around here!)

Yes, OP will admit that they changed wording from half tank to full tank early on the day this question was published. Why?

It complicates the question without adding the meaning. (Just siphon petrol before you leave to increase the range to max.)

And greatly simplifies the math needed to solve the questions. (Harmonic sequence >540 with 200 instead of 100 is much easier.)

Consider me human.

Assuming that you're not allowed to carry fuel in anything other than the tanks of the trucks.

50 truck set out and drive 80 miles.

For each 5 trucks 4 empty their tanks into the fifth.

10 trucks drive the next 90 miles.

9 empty their tanks into the tenth:...

which drives the final 100 miles.

I'm not sure if this is the optimum solution???

If you want to get all the trucks back you will need more trucks than any known company has.

These things usually work best with binary. I'm thinking it's one of those things that is easier attacking backwards.

The last truck drives the last 100 miles with a full tank. Two trucks drive the previous 50 miles leaving half a tank each to pour into 1 truck 150 miles from the destination. Before this, 4 trucks drive 50 miles down to a half tank, then transfer the remaining fuel to the two trucks, and so on.

• 1 truck, 70 miles, distance = 270 miles
• 2 trucks, 50 miles, distance = 200 miles
• 4 trucks, 50 miles, distance = 150 miles
• 8 trucks, 50 miles, distance =100 miles
• 16 trucks, 50 miles, distance =50 miles

So it takes 16 trucks, you leave 8 at 50 miles, 4 at 100 miles, 2 at 150 miles, 1 at 200 miles, and the last truck still has enough fuel at the destination for 30 more miles.

How many drivers return in a truck? (Assume the rest will walk.) 2

Whoop. First one to get the progression correct.

As it turns out binary may not be the best.

Again working backwards as you suggest: the last truck drives the last 100 miles; two trucks do the previous 50 miles; three trucks do the previous 33.33..; four trucks do the previous 25 miles; etc.

Turns out that you can do it with 8 trucks.

8 set out and do 12.5 miles; one transfers 12.5 miles to each of the other seven; seven do 14 miles; one transfers 14 to each of the other six (there are 2 miles left over); six do 16.66.. .. ; five do 20; four do 25; three do 33.33..; two do 50.

The final truck has 100 miles in the tank, and only has 98.5 miles to do.

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If you want to get all the trucks back to base, which seems like a reasonable requirement, You can do it with 124 trucks.

The last truck has to do the last 50 miles there and back; the previous two have to do 25 miles then one waits for the other before they both return 25 miles; the previous three do 16.66.. etc.

Each leg done by n trucks is 50/n miles. We just need to find the number of legs so that:-

1/1 +1/2 + 1/3 + ............. + 1/n ≥ 5.4 (270/50)

I have no idea how to calculate that so I did a quick spread sheet:-

So 124 truck set out and travel 50/124 miles, before one fills the tanks of the other 123 and waits for them to return: it has just enough fuel left in the tank to get them all home when the others return empty.

123 go 50/123 miles etc. etc.

I hope that princess sleeps soundly on her feather pillows.

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This can't be the answer to the second part because it takes no account of the two seats and drivers having to walk home. But, I can't see why there should be a requirement to get the last driver and truck home (unless he's bringing the payment).

EDIT----- I see Rixter beat me to it. Reassuringly we both got the same answers.

I hope that princess sleeps soundly on her feather pillows.

Hopefully better than I did last night thinking about this puzzle...

I'm also thinking that the second part can be improved. There's a big difference between going one way 540 miles and going 270 miles in and back.

You should be able to leave some gas in the trucks that are jettisoned on the way in to use on the way back, just enough to get our delivery guy to the last one jettisoned.

In jettisoning empty trucks, we redistribute fuel when the gauge is at (N-1)/N full. If you leave enough fuel in the jettisoned truck to get to the previous jettisoned truck, the amount of fuel necessary for the return trip (270 miles) should be just 3 or maybe 4 trucks worth. I haven't run the numbers.

If for some reason you want to get the pallet back, after delivering the pillows: it has to take more trucks than just getting there.

I make it 22

I tried to include a screen capture of my spread sheet, but, CR4 won't let me AGAIN.

The final leg is done by the one truck there and back 50 miles each way; the previous leg by 2 trucks has to be done in both directions by only one of the two (with both drivers), so that's 100 for the final leg and 100/3 for two there and one back; etc.

You leave trucks strewn all over the place, and have to pay drivers to walk 1400 miles. Better to pay two guys to carry the pillows, walking there and back. (At least they'll be comfortable when they need to rest on the way there.)

One way trip, 8 trucks

Drive 100/8 miles all tanks at 7/8, siphon gas into 7 trucks, leave 1 truck empty

Drive 100/7 miles all tanks at 6/7, siphon gas into 6 trucks, leave 1 empty

Drive 100/6 miles all tanks at 5/6, siphon gas into 5 trucks, leave 1 empty

Drive 100/1 miles total distance 271.79 miles

Round trip requires 22 trucks, start at HQ.

Instead of leaving empty trucks, leave enough gasoline in parked truck to get back to HQ or the previous abandoned truck.

Drive 22 trucks 100/23 miles, all tanks at 22/23, siphon gas into 21 trucks leaving 1/23 gas in last truck

Drive 21 trucks 100/22 miles, all tanks at 21/22, siphon gas into 20 trucks leaving 1/22 gas in last truck

Drive 7 trucks 100/8 miles, all tanks at 7/8, siphon gas into 6 trucks leaving 1/8 gas in last truck

Drive 6 trucks 100/7 miles, all tanks at 6/7, siphon gas into 5 trucks leaving 1/7 gas in last truck

Drive 5 trucks 100/6 miles, all tanks at 5/6, siphon gas into 4 trucks leaving 1/6 gas in last truck

Drive 4 trucks 100/5 miles, all tanks at 4/5, siphon gas into 3 trucks leaving 1/5 gas in last truck

Drive 3 trucks 100/4 miles, all tanks at 3/4, siphon gas into 2 trucks leaving 1/4 gas in last truck

Drive 2 trucks 100/3 miles, all tanks at 2/3, siphon gas into 1 truck leaving 1/3 gas in last truck

Drive 1 truck 100/2 miles, tank at 1/2, deliver feather bed, return 100/2 miles to location of last truck.

On the way back, siphon fuel from trucks #2 - #22 to get back to HQ

I agree with Randall, 22 is the number of trucks for a round trip, 273.429 miles each way.

One more thing...

On the way back, you can pick up the drivers of the trucks that were parked along the side of the road for refueling. Riding in the back of the truck has got to be better than walking.

How many drivers return in a truck? (Assume the rest will walk.)

22

Zero trucks: for one pallet of pillows its much cheaper to hire an estate car.

Getting there would be easy. I believe that there were a few diesel estates with a range of 540 miles. Not sure about the ranges of hybrids, and plug in hybrids.

You don't need no stinkin' convoy! You need a conveyor belt system like this: https://jpt.spe.org/atlas-energys-dune-express-proppant-conveyor-rolling-into-the-permian-basin This one will be 42 miles long.

Seems like an awful lot of bother for <...one pallet of feather pillows...> for a <...trucking company...>.

Couldn't Amazon deliver instead (usual disclaimer)?

So Truck 1 is sitting in the HQ with a range of 100miles. Truck two is sitting 100 miles up the road at Stage 2, truck 2 then transports the load 100 miles to Stage 3. Truck 3 then drives 70 miles, delivers the load and then drives 30 miles towards HQ. Truck 4 then drives 100 miles towards HQ. Truck 5 then drives another 100 miles towards HQ. Truck 6 then carries the driver from Truck 5 with him for the remaing 40 miles back to HQ !
At no point in the question is it stated that all the trucks are based at HQ.
No. of trucks to deliver. 2.7
No. Trucks to get back to HQ 2.7
No. drivers returning in last truck 2.

Just need this guy...

This may sound simple minded:

Option 1: if a full tank will take us 100 miles with a round trip of 540 miles, we could use 6 trucks. The pallet would be in the 3rd truck. The first truck tows the remaining 5. At 100 miles we disconnect truck 1 and continue for the second 100 miles with the 2nd truck towing the remaining 4 trucks. At the 200-mile mark we disconnect truck 2 and deliver our pallet then begin the return trip following the same plan. This way we can get away with using only 1 driver and no one needs to walk back.

Option 2: 6 fifth wheel pickup trucks and a car carrier. Follow the same basic plan as option 1.