# Ponder This

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## April 2018 - Challenge

There are nine balloons in nine different colors. Each balloon has a different lifting capability, from one through nine grams (1, 2, 3...9) respectively. Alice knows the lifting capacities of each balloon, and she wants to use the smallest possible combinations of balloons to prove to Bob the identity of the 1-gram balloon. She can tie together any subset of balloons to a 9.5-gram weight that will stay put if and only if the total lifting power of the balloons is less than 9.5 grams. Otherwise the weight will lift off and fly away.
How does Alice prove to Bob the color of the 1-gram balloon by doing as few "experiments" as possible?
Each demonstration should use a subset of the balloons that does not lift the weight. Supply your answer as a list of weights per experiment.

For example, here is a (non-optimal) solution for a similar problem with only six balloons with 1-6 gram lifting capacities and a 6.5-gram weight:
1,2
1,3
1,4
1,5

In each one of the four experiments, Alice uses a subset of the balloons that does not lift the weight and that has a total lift capacity of six grams or less. Therefore, Bob can conclude that the balloon that is common to all the experiments must be the one with the 1-gram lift capacity.
Two wrong answers for the six-balloon example include the following:

1. A single experiment: "1,2,3" - this is the only combination of three balloons that won't lift the weight, but Bob can't know from this experiment alone which of the three balloons has the 1-gram lifting capacity.
2. Three experiments: "1,2" "1,3" and "1,5" - Bob can't distinguish between that and "2,1" "2,3" and "2,4". Both sets of experiments look exactly the same to him, so he can't be sure if the repeated balloon has a 1-gram or 2-gram lifting capacity.

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Challenge: 28/03/2018 @ 12:00 PM EST
Solution: 02/05/2018 @ 12:00 PM EST
List Updated: 09/05/2018 @ 12:00 PM EST

Dan Dima (28/03/18 05:36 PM IDT)
Guillaume Escamocher (28/03/18 05:43 PM IDT)
Dieter Beckerle (28/03/18 06:02 PM IDT)
Serge Thill (28/03/18 06:54 PM IDT)
Uoti Urpala (28/03/18 08:36 PM IDT)
Bert Dobbelaere (28/03/18 09:16 PM IDT)
Pavel Glasnov (28/03/18 09:22 PM IDT)
Todd Will (28/03/18 10:39 PM IDT)
Rob Pratt (28/03/18 10:40 PM IDT)
Muralidhar Seshadri (29/03/18 10:31 AM IDT)
Hanlin Ren (29/03/18 03:13 PM IDT)
Alper Halbutogullari (29/03/18 05:10 PM IDT)
Yash Pote (29/03/18 05:25 PM IDT)
George W Barnett (29/03/18 09:44 PM IDT)
Michael Liepelt (29/03/18 11:03 PM IDT)
Paul Shaw (30/03/18 02:23 AM IDT)
Hizami Anuar (30/03/18 04:00 AM IDT)
Kang Jin Cho (30/03/18 05:30 AM IDT)
Yan-Wu He (30/03/18 05:20 AM IDT)
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Nguyen Khac Tung (30/03/18 10:29 AM IDT)
Alex Fleischer (30/03/18 12:30 PM IDT)
Daniel Chong Jyh Tar (30/03/18 01:12 PM IDT)
Tetsuo Mioka (30/03/18 04:02 PM IDT)
Dominik Reichl (30/03/18 06:15 PM IDT)
Francis Golding (30/03/18 07:34 PM IDT)
Clive Tong (30/03/18 10:14 PM IDT)
JJ Rabeyrin (30/03/18 23:22 PM IDT)
Rogerio Ponce da Silva (31/03/18 02:58 AM IDT)
Junjiajia Long (31/03/18 09:03 AM IDT)
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Rickard Björklund (31/03/18 03:25 PM IDT)
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Benjamin Lui Benjamin Lui (04/04/18 03:53 PM IDT
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Franciraldo Cavalcante (04/04/18 10:30 PM IDT)
Klaus Hoffmann (04/04/18 11:13 PM IDT)
Gerhard&Klaus Hoffmann (04/04/18 11:13 PM IDT)
Li Li (05/04/18 09:30 AM IDT)
Davin Choo (05/04/18 03:23 PM IDT)
Gumpina Rama Krishna Chaitanya (05/04/18 03:51 PM IDT)
Solomon Liu (05/04/18 06:10 PM IDT)
Jiao Zi-Nan (05/04/18 06:11 PM IDT)
Walter Sebastian Gisler (05/04/18 06:48 PM IDT)
Peter Gerritson (05/04/18 09:25 PM IDT)
Frank Mayer (06/04/18 04:34 AM IDT)
Prof.V.P. gohil (06/04/18 12:04 PM IDT)
Prashant Wankhede (06/04/18 09:55 AM IDT)
Nadia Vettier Brauner (06/04/18 07:12 PM IDT)
Tero Patana (07/04/18 12:51 AM IDT)
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Liubing Yu (12/04/18 05:39 PM IDT)
Claudio Hüni (13/04/18 10:09 AM IDT)
Robert Lang (13/04/18 12:47 PM IDT)
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Joonsoo Lee (14/04/18 08:52 PM IDT)
Jim Clare (15/04/18 03:00 AM IDT)
Govind A N (15/04/18 06:09 AM IDT)
Joehyun Kim (15/04/18 03:19 PM IDT)
Surya Kareenahalli (15/04/18 06:28 PM IDT)
Tiffany Lee (15/04/18 06:32 PM IDT)
Seo Yeong Kwag (15/04/18 08:37 PM IDT)
Adrienne Ko (15/04/18 09:55 PM IDT)
Pradeep Chandra Gorrepati (16/04/18 12:53 PM IDT)
Nir Lavee & Omer Shechter (16/04/18 03:52 PM IDT)
Yongxing Jim Wang (17/04/18 05:46 AM IDT)
Marcel Caria (17/04/18 12:58 PM IDT)
Karl D’Souza (17/04/18 08:47 PM IDT)
Jiajun Li (18/04/18 04:02 PM IDT)
Erick Lima (18/04/18 06:38 PM IDT)
Rudy Cortembert (19/04/18 11:57 AM IDT)
Esmie Jose (19/04/18 10:56 PM IDT)
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Luo Jin (23/04/18 06:58 AM IDT)
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David F.H. Dunkley (24/04/18 12:35 AM IDT)
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Buzea Alina (30/04/18 09:05 PM IDT)
Oscar Volpatti (01/05/18 06:01 AM IDT)