The Riddler - Series on FiveThirtyEight

[TheRealJoeFish]

If you only get 2 phones, 9 and 9 I think is a guaranteed solution. (18 moves max)

Tthere are other riskier moves with less potential total moves, but with no guarantee of an absolute answer.

I tend to agree.  This should then mean that, for 1000 stories, the max should be 2 * (square root of 1000, rounded down) = 62.

The strategy's the same, except instead of going S10, S20... S90 with the first phone and using the second phone to fine tune the nine stories between each decade, you go S32, S64... S960, S992 (thirty-one) and then use the second phone to fine tune the 31 stories between each thirty-second.  You end up not quite even (at floor 992, with only 7 possible floors above) because 1000 is not a perfect square.

Let's see what the solution on the site says:

NO WE ARE WRONG!!! We can do better!!!

Discussion of the correct solution, and why we are wrong:

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Having seen the solution, I understand what our problem is.  From a broad standpoint, we were looking at it geometrically when we should've been looking at it triangularly!  Our solutions were a good effort, but were inefficient inasmuch as there were too many times there would only be a couple of drops needed, and we could have better leveraged this into a solution that lowered the ceiling of how many drops we'd need (which is what we want to minimize) at the expense of raising the floor (which we don't care about).

The lowest number is 14.  This is because 1 + 2 + ... + 14 > 100, and you can make it so that, if phone 1 breaks after x drops, you only need 14 - x drops with the second phone to fine-tune it.

The strategy is this:

Drop from S14, S27, S39, S50, S60, S69, S77, S84, S90, S95, S99.  If phone 1 breaks after the first drop - S14 - then you need 13 drops with phone 2 to fine-tune.  If it's the second drop that breaks phone 1, well, there are now only twelve stories from S15 to S26.  And so on: every extra drop with phone 1, you set it up so that there's one fewer floor to test.  

That means that the strategy for 1000 gets us 45 drops, as 1 + 2 + ... + 45 = 1035 > 1000.

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So, open challenge to anyone who saw my and Ignoramus's reasonable but ultimately incorrect efforts: POST YOUR BETTER SOLUTION BELOW (without cheating, sucka) AND YOU WILL GET MAJOR PROPS FROM ME AND MAYBE A REP  

I'll post The Riddler No. 2 later today!