# Balls have finally dropped

PBE6
Posers and Puzzles 04 Oct '10 20:43
1. PBE6
Bananarama
04 Oct '10 20:43
As the new junior salesman for the Achtung!(TM) snooker ball company, you have been tasked with selling a set of snooker balls to some of the most unreasonable customers in the company database. On one such sales outing, the customer asks about the durability of the snooker balls in terms of the highest floor they can be dropped from without breaking. Unfortunately for you tech support is on lunch, and the customer notes that if you can't answer his question within the next 15 minutes, he'll become very unreasonable very quickly and storm off without purchasing the case of snooker balls you've been discussing.

The only thing you have going for you is the two demonstration model 8-balls from your salesman kit, which happen to be identical in every respect, and due to heavy-duty German engineering never change properties after impact unless they've been smashed to bits. In order to close the deal, you decide to demonstrate the durability of the snooker balls by dropping them from various floors of the closest 100-story building.

Question: What's the least number of drops required to conclusively demonstrate the highest floor the balls can be dropped from without breaking?
2. 04 Oct '10 20:561 edit
Throw one ball from the 50th floor. If it breaks, use the other ball to start from 0, if not, then throw one of the balls from the 75th floor, etc etc.

50.

Throw one ball from the 20th floor. If it breaks, use the other ball to start from 0, if not, then throw one of the balls from the 40th floor, etc etc.

24.

I don't have the answer yet.
3. PBE6
Bananarama
04 Oct '10 20:58
Originally posted by Thomaster
Throw one ball from the 50th floor. If it breaks, use the other ball to start from 0, if not, then throw one of the balls from the 75th floor, etc etc.

50.

Throw one ball from the 20th floor. If it breaks, use the other ball to start from 0, if not, then throw one of the balls from the 40th floor, etc etc.

20.

I don't have the answer yet.
You're on the right track!
4. 04 Oct '10 20:59
Originally posted by PBE6
You're on the right track!
I'm afraid it will take more than 15 minutes.
5. 04 Oct '10 21:064 edits
My final answer is 13.

2 minute to find the building, and 1 to go downstairs, take the ball, and go upstairs again ðŸ˜›
6. PBE6
Bananarama
04 Oct '10 21:21
Originally posted by Thomaster
My final answer is 13.

2 minute to find the building, and 1 to go downstairs, take the ball, and go upstairs again ðŸ˜›
Nicely done! Of course, you'll never have a successful career in sales with that kind of negative attitude. ðŸ˜‰
7. 04 Oct '10 21:321 edit
This reminds me of the film ''Angels & demons'', when the Vatican is threated with a bomb. All they can see is a live video with the bomb and a lamp.
Someone suggests to test all electricity district, if the lamp goes out they have the district.
Those idiots start by testing them one by one, while they could have known where in 80 minutes.
8. 04 Oct '10 22:26
Originally posted by PBE6
Question: What's the least number of drops required to conclusively demonstrate the highest floor the balls can be dropped from without breaking?[/b]
Will the answer not depend on which floor they will break? If they can survive up to floor 99 then it will require more drops than if they break on 51.

If they break on 51, the answer is 2.
9. 05 Oct '10 12:40
Maybe the answer is 14, actually.
10. PBE6
Bananarama
05 Oct '10 13:13
Originally posted by Tatanka Yotanka
Will the answer not depend on which floor they will break? If they can survive up to floor 99 then it will require more drops than if they break on 51.

If they break on 51, the answer is 2.
The actual number of drops you make does depend on the breaking strength, but the minimum number needed to demonstrate the breaking strength does not (i.e. not just any breaking strength can be determined conclusively in 2 drops, unless you get very lucky on your first two drops).
11. PBE6
Bananarama
05 Oct '10 13:37
Originally posted by Thomaster
Maybe the answer is 14, actually.
Oops, you're right...14 is correct (got a little too excited when I saw the other answer).

The procedures runs as follows:

1. Run up to the 14th floor and drop the first ball. If it breaks, go back down to the 1st floor and drop the second ball. Keep dropping the second ball at successively higher floors until it breaks. The maximum number of drops required here is 14 (required if the ball only breaks on the 13th floor).

2. If the ball didn't break on the 14th floor, run up to the 27th floor (14+13) and drop it again. If it breaks, go back down to the 15th floor and drop the second ball. Keep dropping the second ball at successively higher floors until it breaks. The maximum number of drops required here is 14 (1 from Step 1, plus 13 from Step 2 if the ball only breaks on the 26th floor).

3. If the ball didn't break on the 27th floor, run up to the 39th floor (14+13+12) and drop it again. If it breaks, go back down to the 28th floor and drop the second ball. Keep dropping the second ball at successively higher floors until it breaks. The maximum number of drops required here is 14 (1 from Step 1, 1 from Step 2 and 12 from Step 3 if the ball only breaks on the 38th floor).

4. If the ball didn't break on the 39th floor, run up to the 50th floor (14+13+12+11) and drop it again, etc...

...

11. If the ball didn't break on the 99th floor (14+13+12+11+10+9+8+7+6+5+4), run up to the 100th floor and drop it again. If it doesn't break, you can assure your client that the balls are indeed unbreakable under even the most unreasonable snooker conditions. If it does break, you can still assure your client of their unreasonable strength but at least you get a little bit of your frustration out after having climbed up an down every damn staircase in the building. In either case, the total number of drop required if the ball makes it to the 100th floor is only 12, but since 14 are required if the ball breaks anywhere between the 1st floor and the 99th floor, the minimum number of drops required to demonstrate the snooker balls' strength is 14.
12. 05 Oct '10 19:376 edits
(sorry this is a bit late, started typing it yesterday!)

Thinking about the best numbers of floors:

If we have 1 drop we can only cope with a 1 floor building.

With 2 drops we can do 3 floors by trying floor 2 giving 1 more floor to test whatever the result. So 3 is an "efficient" number of floors, but 2 is not.

With 3 drops, we can cope with 6 floors (by trying floor 3 first, then 5 if that doesn't break)

So if we can do F floors with D drops, we can do F + D + 1 floors with D + 1 drops.

i.e "Efficient" floors are those which are a sum of consecutive integers from 1.

This set is:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105

100 is not an efficient floor and the highest efficient floor less than 100 is 91. So one strategy is to try floor 9 first (100-91). If that smashes we need 8 more goes, and if it doesn't we have 91 possible floors, an efficient number, which takes 13 more goes max.

Once the number of floors is "efficient" the strategy is to take the possible floors down to the next highest efficient number on each go until the right floor is found.

So, I also think we need 14 tests and I think we could cope with a 105 floor building without needing more tests than that.
13. wolfgang59
Mr. Wolf
05 Oct '10 21:401 edit
Originally posted by PBE6
As the new junior salesman for the Achtung!(TM) snooker ball company, you have been tasked with selling a set of snooker balls to some of the most unreasonable customers in the company database. On one such sales outing, the customer asks about the durability of the snooker balls in terms of the highest floor they can be dropped from without breaking. Unfortu conclusively demonstrate the highest floor the balls can be dropped from without breaking?
TWO
The floor it survives
The floor above