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### Crossing a Desert with 45 Watermelons

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Date: 07/08/98 at 17:05:45
From: Ryan Williams
Subject: A boy in a desert with 45 watermelons

A boy has 45 watermelons in the desert. He needs to get them across to
the Oasis fair, 15 miles away. He can only carry 15 watermelons at a
time, and he eats one watermelon every mile he walks, including walking
back to where he started from. He can also leave watermelons at any
mile he has walked, but no fractions of a mile. How many watermelons
can he possibly take to the fair? How did you arrive at your
conclusion?

All I know is that the answer isn't zero.
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Date: 07/10/98 at 12:09:30
From: Doctor Peterson
Subject: Re: A boy in a desert with 45 watermelons

Hi, Ryan.

I like this problem! It took me a little while to figure it out, and a
little longer to think of some hints that wouldn't take away the fun of
discovering the answer for yourself. I first played with the problem a
bit to see how various strategies would work, like carrying everything
nearly halfway and then carrying what's left the rest of the way, and I
was able to have 2 left when I arrived. Then I tried a three-stage
strategy (with two stopping places) and had 6 watermelons left. Then I
thought more about what made one way work better than another, and came
up with a nice solution that finishes with more than 6.

1. No matter how you solve it, in the end the number of watermelons
left will be 45 minus the number of miles you have walked. So the
point of the question is to find a way to carry the watermelons to
the fair with the least possible amount of walking.

2. No matter what you do, you have to take multiple trips. You want to
make as few trips back as possible.

3. In order to use your walking miles most efficiently (not wasting
any watermelons by walking too far) you should never carry less than
you can when you are going forward (that is, you should always start
out with 15), and you should never carry more than you have to when
you are going backwards (that is, you should always end up with 0
when you get back to a starting place).

4. It seems best to carry all the melons to some temporary place, then
carry them all to the next, and so on, rather than to scatter them.

5. If you can get to some place with just the right number of
watermelons, you can avoid having to make any trips in the next
stage with less than your full load of 15. (This is the big hint:
how can you choose the best place to leave them in each stage of the
trip?)

See if that helps you. Let me know if you need more help.

It's a fun problem, and I'm going to do some more thinking to see
how it would work if you have a different distance or number of
watermelons, or can carry a different amount. Once you've solved it,
you might want to do the same. What is it about the numbers 45 and 15
that makes everything come out neatly, for instance? And can you
rephrase the question to ask what is the farthest you can go before
using up the watermelons?

By the way, this is a good example of the unreality of some math
puzzles. I would never make a trip in the desert planning to exactly
run out of water as I get to my destination! In real life, we always
plan on something going wrong, and leave a margin for error.

Have fun!

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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