## Solving Puzzles

June 5, 2011

I was reading ergodicity.net the other day and saw this post, where Anand put up a puzzle and received a solution within minutes.  This whole collective intelligence thing is really exploding, isn’t it?  Anyway, I thought I might also put up a puzzle and see if you or anyone else out there can provide a solution.  Alright, here we go.  I have a sequence of numbers and I want to find the next number in the sequence.

$0$
$-2 + \pi\left(\frac{1}{2}\right)$
$-2 + \pi\left(\frac{2\sqrt{3}}{9}\right)$
$-4 + \pi\left(\sqrt{2}-\frac{1}{2}\right)$
$-4 + \pi\left(\frac{2}{5}\sqrt{5-\frac{2}{5}}\right)$

Do let me know what you think the next number is. Of course anything would fit, but I have something particular in mind for the next number.  I don’t however know whether there is a simple general form for the whole sequence.

1. I have to admit I am a bit stumped on this one. I think I’m getting tripped up on your not knowing if there is a simple general form…

2. To be less mystic about it, I do know one general form for this sequence, but was hoping for one that doesn’t involve integration and somehow is more “number theoretic”.

$\int_0^1 \log\left[ x^n + (1-x)^n \right]dx$

3. Thanks for the puzzle. The good thing upon reading the solution is that I know that I wouldn’t have guessed it in any amount of time.