Monday 10 November 2014

riddle - The story of green apples and red herrings


Let's say we are in a magic land with lots of fantastical animals and plants and whatever...
Imagine you have way too much time, really like apples and love collecting and growing them.
In fact, there are two rare kinds of apples and their colours are blue and green.


Just by looking at your table, you can easily tell that the number of blue apples is not a lot more than the number of green apples; let's say less than 10 times.
From what you've red in your spellbook, 5 out of 6 blue apples are poisonous. I would take care of that. You don't have to, just saying. Anyway, you go ahead and add one blue apple, which is your favourite kind.

As a magician (did I forget to mention that?), you now use a very old spell that doubles the number of green apples immediately.

Next, you feel like adding another blue apple right after you cautiously use that old spell again but this time for both kinds at once (you are a very audacious person).


magic sound


You totally underestimated the power of this spell and you can feel that your body can't take such strong magic easily. You accidentally drop your wand, which is taking action on its own and changes the colour of one apple and also scorches your hat. But no problem, the hat was kind of maroon anyway.
Left with two kinds of apples and a burnt hat, you divide the bigger group in half. Now you are ready and decide to eat one of each kind.


eating sound


If done correctly you should have the same number of green and blue apples now. I'm wondering how much money you could get by selling all of them...


By the way, look at all those beautiful colours! Wait...take a few seconds...


How can this be real?


I think the question is "Given the described operations and the final assertion that there are the same number of green and blue apples, how many green apples and blue apples did you start with?" – apsillers




♫ Math, huh, yeah! What is it good for, absolutely nothing ♫
I will also accept the mathematical solution (for the sake of Emrakul - may God rest his soul).
Note: apples are integers and cutting them is not allowed.




Answer



I'm going on from @Togashi's answer, which seems to be mostly correct except that he's overlooked one possibility. Anyone who's seen fallacious proofs such as this one, or who knows about pathological (often trivial) examples in pure mathematics, will know



not to exclude the case $n=0$ unless you're specifically told $n$ is nonzero.




This is why Togashi's statement



at this point we have $\frac{n}{4}$ green apples and $6n$ blue apples. This means we have 24 times as many blues and greens, so that can't be true [because we have the same number of blues and greens]



immediately rang warning bells for me. Here's my solution.


Let $(G,B)$ denote the ordered pair (number of green apples, number of blue apples).


Just by looking at your table, you can easily tell that the number of blue apples is not a lot more than the number of green apples; let's say less than 10 times.



$(G,B)=(0,0)$. [Lynch mob!, division by 6 is optional ;)]




you go ahead and add one blue apple [...] you now use a very old spell that doubles the number of green apples immediately.



$(G,B)=(0,1)$.



you feel like adding another blue apple right after you cautiously use that old spell again but this time for both kinds at once



$(G,B)=(0,3)$.



your wand, which is taking action on its own and changes the colour of one apple




$(G,B)=(1,2)$.



you divide the bigger group in half



$(G,B)=(1,1)$.



you [...] eat one of each kind



$(G,B)=(0,0)$ again!




If done correctly you should have the same number of green and blue apples now. I'm wondering how much money you could get by selling all of them...


The answer is



0 pounds, dollars, euros, deutschmarks, francs, rubles, tenges, zlotys, or any other currency you care to mention.



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