So I got one submission for an answer to my Modern Word Problem from my brother Steve. I’ll excerpt part of it here:

I guess the problem is how to effectively use that $100. Knowing that you are going to perpetually get 1/6 BOGO and thus have to spend $1 more dollar you can find your initial buy-in.

x + (1/6)x + (1/6)^2x + …. + (1/6)^nx = 100 [but of course you want that last term to be a whole number]

He talked through a bit more of the math and different ways that you may solve that equation. His final answer: 38 free songs.

My solution is less thorough but yields similar results. 1/6 of your original 100 gives you 16 more bottles. 1/6 of that gives you 2. You can keep going but you aren’t going to get any more whole bottles. So your total number of bottles is 118 bottles. 1/3 of those would be winners which leaves you with 39.

There is a whole bunch of things that I am ignoring here and I think Steve’s answer is much better. If I were asking this as an interview question (which I have considered doing), I would have accepted either answer.

Steve got the extra credit as well. You have no chance whatsoever of winning an iPod mini. Sorry.

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Pepsi is currently running a promotion with the caps of its 20oz bottled sodas. They are giving away free songs on iTunes. So I came up with a word problem. I’ll post the solution tomorrow:

Under each cap you can get one of three things: “Free Song” [FS], “Try Again” [TA], “Buy One Get One Free” [BOGO]. Pepsi (as required by law) gives you the odds of winning. You get a FS 1/3 of the time and BOGO 1/6 of the time.

So the question is, given $100 (and assuming that each bottle costs only $1), how many Free Songs can you win? And what algorithm would you follow to maximize your return?

Extra Credit: How many iPod Mini’s will you have won when you run out of money?

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