# Elves and macaroons

#### absoluzation

You used DarnItJimImAnEngineer's method, but DarnItJimImAnEngineer had overlooked the final macaroon given to Anouk, so you both arrived at the solution of a slightly different problem.

Now that you know your mistake, you can start afresh. Did you notice that "at least" was italicized in the original question? Why was that done? Why are you expecting calculation? Or rather, why do you think calculations are important in obtaining the answer?
Sir, could you explain it to me step by step so I can see what I did wrong? I don't know how to get to the answer starting with 35

#### OOOVincentOOO

I am not smart enough to do it on my own. But Dr Tony Padilla (numberphile) finds the same: 3121 (12m12s).

Monkeys and Coconuts - Numberphile:

Do not want to polute this thread.

#### absoluzation

I am not smart enough to do it on my own. But Dr Tony Padilla (numberphile) finds the same: 3121 (12m12s).

Monkeys and Coconuts - Numberphile:

Do not want to polute this thread.

OOOVincentOOO

#### skipjack

Forum Staff
If there are $n - 4$ macaroons initially, the $k$th "theft" leaves $\left(\frac45\right)^kn - 4$ macaroons, where $k$ is successively $1$, $2$, $3$, $4$ and $5$, and then the remaining $\left(\frac45\right)^5n - 4$ macaroons become $\left(\frac{4^5}{5^6}\right)n - 1$ for each elf and $1$ for Anouk.
Hence the smallest possible value of $n$ is $5^6$, so that Kim gets $2499$ macaroons initially and then another $1023$, i.e., $3522$ in total.

#### absoluzation

If there are $n - 4$ macaroons initially, the $k$th "theft" leaves $\left(\frac45\right)^kn - 4$ macaroons, where $k$ is successively $1$, $2$, $3$, $4$ and $5$, and then the remaining $\left(\frac45\right)^5n - 4$ macaroons become $\left(\frac{4^5}{5^6}\right)n - 1$ for each elf and $1$ for Anouk.
Hence the smallest possible value of $n$ is $5^6$, so that Kim gets $2499$ macaroons initially and then another $1023$, i.e., $3522$ in total.
How do you go from $(4/5)^kn-4$ to $(4^5/5^6)n-1$ and how is the smallest number $5^6$? I'm sorry if I'm annoying.

#### skipjack

Forum Staff
I subtracted $1$ and then divided by $5$. For the result to be a natural number, $n$ must be divisible by $5^6$, so the smallest possible value of $n$ is $5^6$.

#### absoluzation

I subtracted $1$ and then divided by $5$. For the result to be a natural number, $n$ must be divisible by $5^6$, so the smallest possible value of $n$ is $5^6$.
Thank you so so so much. I finally get it now!

#### DarnItJimImAnEngineer

For the record, I wasn't trying to look smart. When instructors give you the answers, it's because a) we don't care, b) we don't have the time to work with you 1-on-1, or c) we don't think you're going to figure it out.
Since I hadn't seen any of your attempt so far, I optimistically assumed you could figure it out on your own. I couldn't give you any guidance without seeing what you had done so far, however.

What students (even at the university level) often see as being mean is really us being kind. We learn a lot more a lot faster by trying something, making mistakes, and then learning from our mistakes than we do by seeing something done right from the beginning.

#### DarnItJimImAnEngineer

Molikotigo, je ne suis pas sûr à qui vous parlez là, ni qu'est-ce que vous voulez dire, en fait. Je ne suis même pas convaincu que vous n'êtes pas un reaubeaut.

#### absoluzation

Molikotigo, je ne suis pas sûr à qui vous parlez là, ni qu'est-ce que vous voulez dire, en fait. Je ne suis même pas convaincu que vous n'êtes pas un reaubeaut.
Dang.

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