# [seqfan] Re: puzzle suitable to make a sequence ?

franktaw at netscape.net franktaw at netscape.net
Tue May 11 01:28:03 CEST 2010

```I can't quite tell whether you already know the answer or not. If not,
I don't see how you can suggest it as a puzzle.

-----Original Message-----
From: Alexander P-sky <apovolot at gmail.com>

Hello,

Here is the puzzle suitable IMHO to make a sequence ....
Given are the "n" story building and two balls, which are made of some
breakable material (say glass). The balls are such that they break
being thrown down from the certain height (and above), while being
thrown at lesser height, the ball remains to be in tact.
What is the minimum number of throwing attempts "k" is needed to
experimentally but systematically and successfully determine (for the
the worst case scenario ) starting from which floor of the "n" story
building  (if from any) the ball will start to break while being
thrown down. Again it is important to stress that only two balls are
available for experimental testing and that obtaining the definite
result is required.

I am leaving to the audience (recipients of this list) to come up
with the strategy of determining the minimum number of throwing
attempts "k" for given "n".

Is k(n) sequence in OEIS ?

Regards,
ARP
_______________________________________________

The answer is A002024, n repeated n times. This is both the minimum
number of throws, and the first story we throw from.

First, consider the case where only 1 ball is available. The only
viable strategy is to throw the ball from floors in increasing order, 1
to n; and so it will take n throws in the worst case.

Now with 2 balls, throwing the ball from story i, if the ball breaks,
we are reduced to the previous case for the first i-1 stories, hence
the worst case is i-1+1 throws (i-1 remaining throws, +1 we just did).
If the ball does not break, we can ignore the first i stories, and
proceed as above with the remaining n-i stories. To get the optimum
result, we must take a(n-i) < i, hence the "repeated n times".

```