IBM Research | Ponder This | April 2016 solutions
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April 2016

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If x**3+300*x**2+a*x+b = (x-x1)(x-x2)(x-x3), then x1+x2+X3=300.
You can count the 7,500 different triplets of positive roots and use the throwing of the robocat to single out the correct one by choosing, each time, the throw that splits the space in the right proportion.
This was how most of you solved the challenge. Some of you noticed that 14 throws are sufficient, and that you could also obtain the solution using 15 throws with only 6 robocats, but the solution we were looking for was a little different:
We use the famous crystal ball method (see the <N,t,b> guessing game as described in Guessing Games and Distributed Computations in Synchronous Networks) as a tool, as follows:
Throw the robocat from the 100th floor. If it survives, then r1<r2<100. Searching for r3, allowing for four crashes, requires nine throws. Once we know r3, we know the average of r1 and r2 and can search for one of them using no more than six more throws.
If brave robocat does not survive the fall from the 100th floor, then r1 < 100 <= r2; also, r3 < 200. Since r1 is the only root below 100, we can search for it and find it in nine throws, allowing three crashes. Once we have r1, we know the average of r2 and r3, and can search for r2 using no more than six throws and three more smashed robocats.