Div2 894E — Ralph and Mushrooms

as we know we can find cycle and contract to a point to make the graph become DAG

but tutorial dont explain about finding mushrooms in the cycle by math knowledge

I have read some one's code and the math is as the follow

IF I HAVE MISTAKE WELLCOME TO TELL ME! THX

walk through the cycle me can get the max mushrooms as we could because we can still pass it though we get 0 mushroom so we can get all the mushrooms mushrooms decrease by 1,2,3...n for the initial mushrooms w i-th time you get w-i(i+1)/2 for every w suppose you can get mushrooms n time

w,w-1,w-1-2,w-1-2-3...w-(1+2..n)

which is w*(n+1)-[i(i+1)/2 i for 0 to n] -> w(n+1)-i(i+1)(i+2)/6 see here https://en.wikipedia.org/wiki/Tetrahedral_number

and the way to find n

we have w already to make it simple consider w is one of the i(i+1)/2

8*w =4*i(i+1) =4i^2+4i

8*w+1=(2i+1)^2 sqrt(8*w+1)-1=2i [sqrt(8*w+1)-1]/2=2i/2=i which is what we want,i is the max times to get mushroom for any w get floor for the ans

c++ //n is the max times to get mushroom int n= floor((sqrt(1+8*w)-1)/2); sum_mushroom=(n+1)*w — n*(n+1)*(n+2)/6;

i.e. w=9

9,8,6,3,-1

9,9,9,9,9

0,1,3,6,10

n is 4 9*(4+1)-(0+1+(1+2)+(1+2+3)+(1+2+3+4)) =9+8+6+3=26