Math Write Up

Consider the case of Gary Gorilla—an enterprising, albeit eccentric, owner of a small banana plantation in a remote desert oasis.


Gary’s harvest consists of 3000 bananas.  The market place where the stash can be cashed in is 1,000 miles away.  However, Gary must walk to the market, and can only carry up to 1,000 bananas at a time.

Furthermore Gary eats one banana during each and every mile that he walks (so Gary can never walk anywhere without bananas).

What is the maximum amount of bananas Gary can get to market?

I think Gary is pretty fucked

1000

333

Assuming he only walks in 1 mile intervals...

To transport 2001 to 3000 bananas, he must carry 1000 bananas for 1 mile, go back, carry the second thousand, go back, and carry whats left of the third thousand.  Advancing one mile in this method consumes 5 bananas, thus he can travel 200 miles until he is left with 2000 bananas.

Similarly, it costs 3 bananas to travel 1 mile when carrying 1001 to 2000 bananas.  He can travel an additional 333 miles during this period.

Finally, it costs 1 banana to travel 1 mile when carrying 1000 bananas or fewer.  Since he has 1000 - (200 + 333) = 467 miles to go, he will consume 467 out of his remaining 1000 bananas, leaving him 533 to sell.

Why doesn't he just get a truck?

What if he could walk in half-mile intervals?

if permitted part banana part mile solution

3000 bananas, 1000 miles to go
taking first 1000 bananas to 200 miles allows 600 bananas placed (going there and back)
taking second 1000 bananas to 200 miles allows 600 plus earlier 600 = 1200 bananas placed (going there and back)
taking third 1000 bananas to 200 miles allows 800 plus earlier 1200 = 2000 bananas placed (going there only)

2000 bananas left, 800 miles to go
taking first 1000 bananas to additional 333 1/3 miles allows 333 1/3 bananas placed (going there and back)
taking second 1000 bananas to additional 333 1/3 miles allows 666 2/3 plus earlier 333 1/3 = 1000 placed (going there only)

1000 bananas left, 466 2/3 miles to go
taking 1000 bananas to additional 466 2/3 miles allows 533 1/3 bananas placed (going there only)