#2             68. HIKIN’ THE HEYSEN               

John, Nick, Pete and Ralph decided to explore the unknown hazards of the Heysen Trail. They calculated that they could each carry enough provisions for just five days. Thus, if they all travelled together, they could walk for two and a half days, before returning to their starting point.

“You know,” said John thoughtfully, “maybe if we didn’t all try to travel the full distance, the rest might be able to cover more ground. For example, after one day, Ralph could give some of his food to the rest of us, and return to base, keeping of course, enough food for the return trip.”

“I don’t think much of that idea!” said Ralph. “But I wonder how far along the track we (or at least one of us) could get?”

Assuming that our intrepid explorers all want to return to base, how far can the track be explored? (Give the answer in number of days travelled.)
Hints and strategies

Hint 1 Hint 2 Solution Extensions
HINT 1

Try working in provisions for whole days.
HINT 2

Experiment. What if Ralph travelled one day (keeping a day’s provisions to get back), and then shared the remainder of his provisions?
SOLUTION

The answer is four days. After a day, R. gives one day’s ration to each of J., N. and P., and then has enough to return home. At the end of day 2, P. gives one day’s ration to J. and N., and has enough to return home. After day 3, N. gives one day's ration to J. and returns home. Hence J. can travel for a fourth day, each lad having enough to return home. The following table shows the provisions of each at the end of each day, after redistribution:

 
Day 1
Day 2
Day 3
Day 4
R
1
0
0
0
P
5
2
0
0
N
5
5
3
0
J
5
5
5
4
Total
16
12
8
4


EXTENSIONS

1. The Heysen Trail is a real walking trail in South Australia. Find out more about it. Who was Hans Heysen?

2. Try reworking the problem if for different numbers of hikers and different carrying amounts. Which choices give nice round figures?

3. This type of calculation could be very important for, say, a journey to the South Pole, or climbing Mount Everest. See if you can find any records of this.