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You want to buy some soda, and there is a special offer saying that if you bring $7$ crown corks of soda, they will give you an extra soda! You notice that you have enough money to buy at most $30$ of them:

In most efficient way, how many soda you should buy to maximize your profit out of this offer?

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  • $\begingroup$ How do you define profit here? $\endgroup$ – Lawrence Oct 15 '17 at 12:36
  • $\begingroup$ @Lawrence as you know, you are supposed to pay money for each soda, so you need to make each soda as cheap as possible. The question asks how many soda you need to buy to make the most profit out of it, by using lids. $\endgroup$ – Oray Oct 15 '17 at 19:00
  • $\begingroup$ Ah, thanks for explaining. Normally, profit = sale price - buy price for the same person. $\endgroup$ – Lawrence Oct 15 '17 at 22:47
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Assuming

The objective is to drink as much soda as possible, as cheaply as possible per bottle (which means not having more lids left over than necessary)

Then buy

25 soda

And drink them. You now have

25 lids.

Trade in

21 lids, keep 4, get +3 soda, drink them too.

You now have

4+3=7 lids.

Trade them in:

you now have 1 soda. Drink it, or don't.

You now have:

1 lid, and enough money to buy 5 more soda.

If you could

borrow one lid, you could use that to get one more bottle of soda, then return the borrowed lid.

My profit is

If I can borrow a lid: I get to drink 35 bottles for the price of 30, a 16.666% 'profit' margin.

If buy 25, then cannot borrow a lid: I get to drink 29 bottles for the price of 25, a 16% 'profit' margin.

If however, I were buy 30 bottles, only to find that cannot borrow a lid: I would now drink 34 bottles for the price of 30, a 'profit' of only 13.333%. So I don't do that.

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    $\begingroup$ Why not just buy all 30 at once? You'll get 30 lids; trade 28 for +4, with your final trick still applicable for an extra +1 to give the same total of 35. $\endgroup$ – Lawrence Oct 15 '17 at 14:51
  • $\begingroup$ Thanks for the comment. An explanation of why buying 30 up front could lead to lower profit has been added. $\endgroup$ – Ben Aveling Oct 15 '17 at 22:27
  • $\begingroup$ Thanks for explaining. I suppose you could check with your friend before doing business with the shop. :) $\endgroup$ – Lawrence Oct 15 '17 at 22:53
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    $\begingroup$ I hate to think how long ago I first met a version of this puzzle - perhaps I should have left it for someone who hasn't met it before. But I figured, maybe there's a twist - and the obvious twist is that it is asking "how many to buy to maximise profit", implying that maximising profit (however defined) is more important than maximising consumption. So I was looking for a reason not to buy 30 - and the lack of clarity about being able to borrow the extra cap provided that. $\endgroup$ – Ben Aveling Oct 16 '17 at 12:17

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