Codechef July 2013 Prob: the ultimate troll question

First, I had no idea how to solve this at all. There is no way to brute force it for such extremely large numbers.

Then, I saw this comment:

sikander_nsit @ 10 Jul 2013 09:58 PM

I was trolled by this question in two phases : First after solving for two hours I was like "oh man, awesome question. So one part of the question remains" then after 5 hours of PnC and probability, when I saw it I was like "mother of all trolls" serious mindfuck question.

I knew there had to be a shortcut somewhere. Hence, I worked out the first sample test case for t4=1. Yielded the result of 0.5. Could it be that the output is the same regardless of how many tickets Chef discards?

Investigating further, I made a program that calculates the probability of every single combination, a.k.a brute forcing the solution. Code here. Of course, I would only load the program with small inputs. The result:

Input:


Output:


I found the shortcut! Indeed, the probability is the same regardless of how many tickets Chef discarded. Happily discarded T4 from my program, submitted it, only to get a runtime error. Turns out my stack has run out of space due to too many recursive calls from T3. Hmm, I thought, I could make the calculation non-recursive, but it is still up to 1000000000 calculations for a single test. Another shortcut must exist!

Went back to the question to look for more patterns. Discovered than 0.5 = 2/(2+2) for first sample test case, and 0.4 = 2/(2+3) for second test case. T3 is also not needed!

Submitted a program consisting of just these few lines:



ANDDDDDDDDDDDDDDDDD


Ultimate troll question.