Math optimization riddle

Question

You have been given the task of transporting 3,000 apples 1,000 miles from Appleland to Bananaville. Your truck can carry 1,000 apples at a time. Every time you travel a mile towards Bananaville you must pay a tax of 1 apple but you pay nothing when going in the other direction (towards Appleland). What is highest number of apples you can get to Bananaville

The answer is $\boxed{833}$. However, I don't understand how to obtain this answer. Also, is there any way to show that the answer is optimal?

I have seen a riddle similar to this before, and I think the trick is to move $1000$ apples towards a stopping point, come back, get more to that stopping point, and so on. I can't figure this one out, though.

Doing a quick Google search will get you a method for $833$. See for example http://goodriddlesnow.com/riddles/hard/page:5/sort:Riddle.difficulty/direction:DESC . This doesn't prove optimality though — pwerth, Jan 17 '19 at 20:22

score 2 · Accepted Answer · answered Jan 17 '19 at 20:26

2

The truck should be as full as possible, so as soon as you can combine multiple truckloads into one, you should do so. The first time that happens is after 333 1/3 miles. You can get the apples to the 333 1/3 mile marker in 3 truckloads, pay 1000 in tax, and be left with 2000 apples. Then you can go another 500 miles before you can combine truckloads again. Hauling 2000 apples in two truckloads over 500 miles costs you 1000 in tax, leaving you with 1000 apples at the 833 1/3 mile marker. The remaining 166 2/3 miles can be driven with one truckload, leaving you with 1000-166 2/3=833 1/3 apples.

answered Jan 17 '19 at 20:26

LinAlg

19,822

how do i show this is optimal? – Jan 17 '19 at 21:31
@stackofhay42 the rationale is in the first sentence. – LinAlg Jan 17 '19 at 21:44
does this method extend to this related question (same as this)? Using 3000→2000→1000 way gives 416.67. – muser Dec 28 '22 at 11:07

Math optimization riddle

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