The questions were assigned to the students with the instructions that they
were to find which of the problems were IMPOSSIBLE, and to prove it. Wups, I
didn't tell you that part, did I?

1) In this simple problem, it is easy to draw a right triangle with sides 5,
10, and root-125, with a smaller triangle of 5, 3, and root-34 as the sides.

The villagers travel the root-125 side at 2 kmph, which works out to be 5.59
hours.

The explorers travel the root-34 side at 2kmph, and the remaining 7 km on the
road at 4kmph, which works out to be 4.67 hours total.

The explorers get there with 0.92 hours to spare, or 55 mins, 28 seconds.

2) The dimensions of the box being 30x20x15 make it 9,000 cubic centimeters.
The interior dimensions of the hollow box are 26x16x11, making it 4576 cubic
centimeters. The difference of 4424 cubic centimeters is how much gold is in
the box itself.

Gold weighs 19.23 grams per cubic centimeter, which yes, we did do in class.
That makes the box weigh a total of 85.074 kg, which in turn, is more than the
4 jugs completely filled with water (4 kg for the jugs and 80kg for the
water).

Done as intended (no trick questions, no smartass answers, no fifth jug) the
problem is unsolvable.

3) Because the water mixture is being drained at 1 liter, out of 100 liters,
per minute, it is losing its contents constantly at the rate of one percent
per minute. Since no more basin water is being added, this means that the
basin water is dropping at one percent per minute as well. This is an interest
rate problem.

100 liters of original basin water, losing 1% of its value per minute for 40
minutes and constantly being mixed with fresh water, is
100e^(-0.01*40)=67.03 liters remaining
While not technically as accurate, you could use instead
100*(1 - 0.01)^40 = 66.90 liters remaining
Or, you could have used instead "how much fresh water was in the basin?" using
the continuously compounded with interest formula
$ = Ce^(rt)-A/r where C = Dr+A
which totals 32.97 liters of fresh water in the basin.
Either way, the rest of the basin water (either 32.97 liters or 33.10 liters)
must be in the jugs.

4) Because I did not phrase the problem on the website as thuroughly as I did
in class, people were able to incorrectly assume that, as soon as someone
stopped running, they would be captured, and no longer require water. Because
of the bad phrasing, I would have accepted that as a correct answer.

However, here's what my students had to come up with:
-- the object is to get one person as far into the desert as possible. That
means, get them a maximum distance with full water before letting them go.
-- All four travellers travel for day 1. Now, all four have 8 liters of water.
The first explorer tops up everyone else's jug, which leaves 2 liters for
himself. This allows him to return to the temple safely.
-- The remaining three explorers travel for one more day. Now they all have 8
liters of water again. The second explorer tops up the remaining 2 jugs,
leaving him with 4 liters of water left in his own. This allows him to make
the trek back.
-- THe last 2 explorers go for a third day and now both have 8 liters left.
The third explorer tops up the fourth's jug, then has 6 liters, exactly enough
to make the return trip.
-- The fourth explorer has five days of water, and can travel for days four
through eight. Now his jug is empty.

Anyone who got this far in the problem and tried to find a clever way out got
some credit, but missed the point. No matter what you try, the problem has
such heavy restrictions (one jug per person limit, for example) that takes him
all the way to the ninth day of travel.

As it was intended (no trick questions, no smartass answers), the problem is
unsolvable.

Here, however, is the correct answer:
--The gold box holds 4,576 cubic centimeters of empty space.
--No matter how it opens/closes, it should be able to hold at least half of
that in water.
--This happens to be just barely over two liters of water, enough to make the
ninth day of travel.

Shinarae Lluminus