Pop quiz!

[Draknek]

(Draknek has been working this out for far too long)

Changing the number of floors is fairly trivial (once you get how to do it at all, that is - I didn't).

With only two balls, the formula is:

y = ceil(1/2*(sqrt(1+8x)-1))

Where y is the number of attempts you need to make and x is the number of floors. (Ceil means round up.)

Showing my working backwards, that formula is produced from this one:

1/2*y*(y+1) >= x

Which is, in turn, produced by this one:

(Sum of a between a=1 and a=y) >= x

Which is the original formula for 2 balls.

For one ball, the formula is:

y >=x

But that is equivelant to:

(Sum of 1 between a=1 and a=y) >= x

Going back to 2 balls:

(Sum of a between a=1 and a=y) >= x

Is the same as:

(Sum of (Sum of 1 between a=1 and a=b) between b=1 and b=y) >= x

Extending this, for three balls, it would be:

(Sum of (Sum of (Sum of 1 between a=1 and a=b) between b=1 and b=c) between c=1 and c=y) >= x

However, at 2:20 in the morning, I'm not willing to work that out and get a y=blah equation for three balls.

(Apologies if this makes little sense, is in a stupid order, or both. Draknek is tired now.)