I saw this too, and it would have been much easier to use. When I noticed the pattern (1+2=3, 4+5+6=7+8...), the perfect squares stuck out and then when I came up with the recursive function I was looking at it more from a point of view of the squares than I was from the pov of a sequense...
T(n) = (T( (n^(1/2) -1 )^2 )/3 + n) * 3
T(0) = 0
T(1) = 3
T(4) = 15
T(3^2) = ( T(2^2) )/3 + 3^2) * 3 = (15/3 + 9) * 3 = 42
I think the problem was that I knew what I was thinking but then I didn't clearly state it. I was intending that T(n-1) would be the next lowest perfect square...
I think that would make it simpler too. Before I figured out the recursive function, I was using excel with the formula n + (n+1) + ... + (n+n^(1/2)), that was a pain because you can't just copy and paste...
I miss stated it, it's: T(0) = 0, T(n) = (T( (n^(1/2) -1 )^2 )/3 + n) * 3, when n^(1/2) is an integer.
so for n = 4, its T(4) = ( T(1)/3 + 4 )*3 = (3/3 + 4)*3 = 15
It can also be n + (n + 1) + ... + (n + n^(1/2)) = (n + n^(1/2) + 1) + ... + ( n + 2*n^(1/2) )
That works too, the formula I was using is T(0) = 0, T(n) = (T(n-1))/3 + n) * 3. The interesting part was that if you add consecutive numbers, they add to the next set of consecutive numbers, starting with a perfect square and going to the number before the next perfect square. Like 1+2 = 3...
0, 3, 15, 42, 90, ...
What number comes next?
There's something interesting about these numbers (at least to me). <- thats not a clue its just that this isnt just some random series, theres more to it that is mathematical, not personal or a bus route...
Feynman did some experiments with this and talks about it in one of his 'curious character' books- I cant remember which one. Anyway, he smoked pot and got in a tank that was able to remove his senses by removing external stimuli (I think the tank was designed, not by Feynman, to try to mimic...
There are 4 possible situations:
1) 3 B, 0 W
2) 3 W, 0 B
3) 2 B, 1 W
4) 2 W, 1 B
Assuming that each is equally likely, then your solution gives a 50% survival rate, same as last prisoner randomly guessing. Did I miss something that would make it 75% instead of 50%?
There is no trick to breaking the rules- two people max, both same way, one flashlight. The trick is to illiminate the bad affects of the 10 and 5 person- think of the farmer, the fox, the chicken, and the egg riddle if you know it...
3) light one rope at both ends
zero/infinity = 0 ...anything not (+,-) infinity is zero when divided by (+,-) infinity (this may actually be undefined, but its limit is 0)
infinity/zero DNE
when x approaches 0, y approaches +infinity or -infinity depending on direction... that's one of the reasons 1/0 is undefined.
there is no point at x= 0, it is untrue that "there is an imaginary point...", it does not exist.
5*0 = 4*0
If you can divide by zero, then the zeros cancel and 5 = 4 and you end up with a number system with only 1 element. Its not so much impossible to come up with a system that allows you to divide by zero, its that it would be limiting rather that useful..