Crossing a Desert with 45 Watermelons

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.

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. 

Here are some ideas that may help you:

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/