Radioactive River

Question

You, Alice, Bob, Charlie, Daniel, Elisa, Fred, and Greg are playing a cooperative game called Radioactive River. They have to get across a 20 meter carpet/radioactive river, and they only have: 3 small spots that only one person can fit on, 2 mats that 2 people can fit on, and an electronic scooter. Here are a few things you need to know before reading on:

Think of these things as a single point with no length or width (Or height)

Once you place the mats down, you can not lift them up

You can not slide on something while you are on it

The electronic scooter moves at a pace of 1 meter per second

Everyone has a 50% chance of missing a throw that is more than 2 meters, and when they do, it goes into the river, never to be found again (Unless they start over)

Everyone has a 75% chance of making a throw less than 2 meters

Everyone can jump at most 1 meter with 100% certainty, from 1 meter to 2 meters 50%, and they can not make something past 2 meters

If someone touches the river everyone has to take everything back to the beginning and start over (Their time doesn't reset)

Each jump takes 2 seconds

They can pass things at most 1 meter away from them

Only 1 person can fit on the scooter at a time

If you throw something but it gets lost, you will have to start over to get it back

What is the fastest strategy to get across to the other side?

Answer 1

I did a recalculation as I had inadvertently required the last person to throw the scooter back across the river:

It depends on what you mean by "the fastest strategy". Do you mean the best possible time, the best average time, the best worst-case time, or something altogether different. Without knowing that I would choose the following:

Step 1) Place the mats at 1 meter intervals from the near edge of the river, leaving 18 meters to the other side. Step 2) Move 2 people to each mat and the scooter to the outer mat. Step 3) Repeat [One person rides across and throws the scooter back.] until all over (or all back to step 2 if the scooter is dropped)

This gives a best possible time of

18*8+4=148 seconds.

And an average time of

(approximately) 4580 seconds with standard deviation 4438 seconds). On average it would take about 256 rides before all eight get across

and the distribution of completion time measured over 100,000 trials is as follows

90% of trials better than 10,370 seconds, 99% better than 20,524 seconds, 99.9% better than 30,8400 seconds (the probability of a trial exceeding a time T falls approximately as exp[-T/4570]). Longest time to cross in 100,000 trials was 48,640 seconds.

I believe this is about as good as can be done as it minimizes the probability of any specific river crossing failing and requiring a restart.

Answer 2

My strategy would be to

Grab the scooter and go straight to the other side of the radioactive river. Once I get to the other side, I'll try to send it back across empty, if I can. Those other people can use their carpet squares to build a few steps out into the river, so they can get the scooter faster, and then ride the scooter across themselves.

This is a good strategy because

If there needs to be someone on the scooter to move it, then to get everyone across the river we'd need to deploy carpet squares at 4m, 8m, 12m, 16m, and 18m. When crossing between those squares, the scooter could be parked in the middle of each of those gaps, which means there are eight 2m gaps and four 1m gaps. One person can get past each gap riding the scooter, but that still leaves seven people that need to jump each gap. With a 50% chance of failure on each 2m jump, the chances of everyone making it are 1 in 2^56. (7 people * 8 2m gaps) If we took 2^56 attempts and each attempt took 1 second, we'd be stuck at the river for about 2.2 billion years. By stealing the scooter and just getting myself across, I could avoid that.