# Problem 99 Solution

The way to deliver as much grain as possible is not not waste the camel's movement with light loads. However moving heavy loads too small of a distance can often leave a small remainder in the last trip, which is a waste of grain.

From the beginning you have enough grain to make 19 one-way trips to some cache point along the way. The first question is, far out should you transport that grain? The answer is far enough so that you leave a cache somewhere of 9000 pounds of grain. Why 9000? Every cache point should have some amount of grain evenly divisible by 1000. This way the camel can start out every trip with 1000 pounds, and not have a remainder on the last trip. So you have 1000 pounds of grain to let the camel eat, divided by 19 one-way trips. 1000 pounds divided by 19 trips is 1000/19=~52.63 pounds, or 52.63 miles per trip. Each trip you will eat 52.63 miles along the way there, deposit 894.73 pounds of grain, and take back 52.63 pounds of grain to eat on the way home, except the last trip in which you don't need to return. The total deposit works out to (10*894.73)+52.63 = 9000 pounds, as arranged. So you will be able to cache 9000 pounds of grain 52.63 miles.

Next repeat the above but only allowing 17 one-way trips. 1000 divided by 17 is 1000/17 =~ 58.82 . So you will be able to cache 8000 pounds of grain 111.46 miles from city A.

Below is a summary of the entire journey:

1. Cache 9000 pounds over 19 one-way trips, moving 52.63 miles, or 52.63 miles from city A
2. Cache 8000 pounds over 17 one-way trips, moving 58.82 miles, or 111.46 miles from city A
3. Cache 7000 pounds over 15 one-way trips, moving 66.67 miles, or 178.12 miles from city A
4. Cache 6000 pounds over 13 one-way trips, moving 76.92 miles, or 255.04 miles from city A
5. Cache 5000 pounds over 11 one-way trips, moving 90.91 miles, or 345.95 miles from city A
6. Cache 4000 pounds over 9 one-way trips, moving 111.11 miles, or 457.07 miles from city A
7. Cache 3000 pounds over 7 one-way trips, moving 142.86 miles, or 599.92 miles from city A
8. Cache 2000 pounds over 5 one-way trips, moving 200.00 miles, or 799.92 miles from city A
9. Take 1000 pounds directly to city B, comsuming 200.08 pounds on the way, leaving 599.84 pounds and take back 200.08 pounds to eat on the way back.
10. Take the last 1000 pounds directly to city B, comsuming 200.08 pounds on the way, and depositing an additional 799.92 pounds of grain, for a total deposit of 1399.77 pounds.

I would like to thank Scott Morris for this problem.

Michael Shackleford, A.S.A., January 14, 1999