Ponder This Challenge - May 2015 - 3 people betting game

Ponder This Challenge:

Three people are playing the following betting game.

Every five minutes, a turn takes place in which a random player rests and the other two bet against one another with all of their money. The player with the smaller amount of money always wins, doubling his money by taking it from the loser.

For example, if the initial amounts of money are 1, 4, and 6, then the result of the first turn can be either 2,3,6 (1 wins against 4); 1,8,2 (4 wins against 6); or 2,4,5 (1 wins against 6). If two players with the same amount of money play against one another, the game immediately ends for all three players.

Your task is to find initial amounts of money for the three players, where none of the three has more than 255, and in such a way that the game cannot end in less than one hour.

In the example above (1,4,6), there is no way to end the game in less than 15 minutes.

Update (05/05): all three numbers should be positive integers. For example, [0,1,2], is not a valid answer since the game would end immediately.

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Solution

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  There are three solutions (up to permutations of the amounts) for the money the three players start with, so that at least 12 rounds (or one hour) of betting will take place with amounts no greater than 255: (175,199,223), (197,205,213), and (209,217,225). Incidentally, there is no solution for 13 rounds of play.

  Thanks to John Tromp for creating the relevant sequence: https://oeis.org/search?q=A256001.

  Goran Zuzic uploaded a proof (https://gist.github.com/Zuza/36b5ea4d199d14d8f22c) that for any three numbers, the game can end in finite amount of time.

Solvers

• Robert Gerbicz (04/30/2015 07:08 PM EDT)
• Radu-Alexandru Todor (04/30/2015 09:09 PM EDT)
• Tobias Zurell (04/30/2015 10:38 PM EDT)
• James Dow Allen (05/01/2015 12:53 AM EDT)
• Chuck Carroll (05/01/2015 03:06 AM EDT)
• Kang Jin Cho (05/01/2015 04:21 AM EDT)
• Eli Biham (05/01/2015 07:35 AM EDT)
• Jan Fricke (05/01/2015 08:54 AM EDT)
• Don Dodson (05/01/2015 11:36 AM EDT)
• Ofir Sasson (05/01/2015 12:51 PM EDT)
• Andrei Malashevich (05/01/2015 01:18 PM EDT)
• Joseph DeVincentis (05/01/2015 01:35 PM EDT)
• Hakan Summakoğlu (05/01/2015 01:51 PM EDT)
• Graham Hemsley (05/01/2015 02:01 PM EDT)
• Andreas Stiller (05/01/2015 03:24 PM EDT)
• Walter Schmidt (05/01/2015 04:34 PM EDT)
• Alex Fleischer (05/01/2015 04:35 PM EDT)
• Rob Sessions (05/01/2015 05:22 PM EDT)
• Al Zimmermann (05/01/2015 06:01 PM EDT)
• Cadence Marseille (05/01/2015 06:05 PM EDT)
• Ariel Landau (05/01/2015 06:58 PM EDT)
• Alex Wagner (05/02/2015 05:50 AM EDT)
• Dave Gower (05/02/2015 06:04 AM EDT)
• Daniel Bitin (05/02/2015 11:17 AM EDT)
• Todd Will (05/02/2015 11:35 AM EDT)
• Alexandre Gilotte (05/02/2015 12:36 PM EDT)
• Guang Gao (05/02/2015 12:39 PM EDT)
• Philip Anderson (05/02/2015 04:25 PM EDT)
• Jitesh Gandhi (05/02/2015 06:37 PM EDT)
• Caili Shen (05/02/2015 06:48 PM EDT)
• David Greer (05/03/2015 12:35 AM EDT)
• Ajay Mallepally (05/03/2015 03:01 AM EDT)
• Rajendran Thirupugalsamy (05/03/2015 05:23 AM EDT)
• Abraham Elmahrek (05/03/2015 05:56 AM EDT)
• Yoav Kaempfer (05/03/2015 07:40 AM EDT)
• Harald Bögeholz (05/03/2015 02:58 PM EDT)
• Liubing Yu (05/03/2015 07:21 PM EDT)
• Benjamin Phillabaum (05/03/2015 10:59 PM EDT)
• Serge Batalov (05/04/2015 04:10 AM EDT)
• Lorenz Reichel (05/04/2015 04:17 AM EDT)
• Martin Husar (05/04/2015 06:21 AM EDT)
• Di Luo (05/04/2015 06:56 AM EDT)
• Piotr Nadczuk (05/04/2015 10:13 AM EDT)
• Dan Ismailescu (05/04/2015 10:44 AM EDT)
• Alphan Karacaer (05/04/2015 12:45 PM EDT)
• Peter Gerritson (05/04/2015 01:44 PM EDT)
• Mark Pervovskiy (05/04/2015 05:46 PM EDT)
• Victor Chang (05/04/2015 06:03 PM EDT)
• Zhizhong Zhou (05/04/2015 06:53 AM EDT)
• Hengfeng Tian (05/04/2015 10:13 PM EDT)
• José Eduardo Gaboardi de Carvalho (05/04/2015 10:19 PM EDT)
• Michael Rosola (05/04/2015 10:59 PM EDT)
• Tim Cieplowski (05/05/2015 01:12 AM EDT)
• Emir Haleva (05/05/2015 05:59 AM EDT)
• Børge Nordli (05/05/2015 07:03 AM EDT)
• Karim Alaoui (05/05/2015 09:43 AM EDT)
• John Tromp (05/05/2015 01:56 PM EDT)
• Susan Brommer (05/05/2015 04:02 PM EDT)
• Hendrik Nigul (05/05/2015 10:02 AM EDT)
• Roy St john (05/05/2015 10:11 AM EDT)
• Jochen Voß (05/05/2015 04:18 PM EDT)
• Aviv Nisgav (05/05/2015 04:28 PM EDT)
• Louis Slothouber (05/05/2015 04:58 PM EDT)
• David Friedman (05/05/2015 05:46 PM EDT)
• João (05/05/2015 08:28 PM EDT)
• Gary M. Gerken (05/06/2015 01:13 AM EDT)
• Levent Erkok (05/06/2015 02:57 AM EDT)
• LaurV (05/06/2015 05:33 AM EDT)
• Stéphane Higueret (05/06/2015 01:12 PM EDT)
• Nicolas Miret (05/06/2015 01:35 PM EDT)
• Timo Denk (05/06/2015 04:15 PM EDT)
• Ratmir Panov (05/06/2015 04:31 PM EDT)
• Jason Scott (05/06/2015 06:27 PM EDT)
• Álvaro Begué (05/06/2015 08:16 PM EDT)
• Shirish Chinchalkar (05/06/2015 11:03 PM EDT)
• Michael Perrone (05/07/2015 11:42 AM EDT)
• Mathias Schenker (05/07/2015 13:12 PM EDT)
• John Jones (05/07/2015 14:45 PM EDT)
• Arthur Vause (05/07/2015 15:21 EDT)
• Saverio Maresca (05/07/2015 22:36 PM EDT)
• Arsene Lui (05/08/2015 12:56 AM EDT)
• Christian Blatter (05/08/2015 06:14 AM EDT)
• Tim Nordenfur (05/08/2015 09:40 AM EDT)
• Rob Pratt (05/08/2015 11:07 AM EDT)
• Huafeng Jiang (05/08/2015 12:33 PM EDT)
• Leandro Araújo (05/08/2015 01:55 PM EDT)
• Motty Porat (05/08/2015 07:29 PM EDT)
• He Yanwu (05/08/2015 10:00 PM EDT)
• Franciraldo Cavalcante (05/09/2015 02:18 AM EDT)
• Yochay Lustmann (05/10/2015 03:25 AM EDT)
• Gil Citro (05/10/2015 04:33 AM EDT)
• Michael Liepelt (05/10/2015 05:45 AM EDT)
• ByeongYeon (Jackie) Ryu (05/10/2015 10:22 AM EDT)
• Joaquim Neves Carrapa (05/10/2015 01:15 PM EDT)
• Paolo Farinelli (05/10/2015 08:45 PM EDT)
• Thomas Egense (05/11/2015 10:44 AM EDT)
• Torbjörn Karfunkel (05/11/2015 11:40 AM EDT)
• Daniel Johnson (05/11/2015 05:07 PM EDT)
• Armin Krauss (05/11/2015 05:41 PM EDT)
• Grant Boudreaux (05/12/2015 04:20 PM EDT)
• Muralidhar Seshadri (05/12/2015 04:17 PM EDT)
• Ehud Schreiber (05/13/2015 04:19 AM EDT)
• Jesus Sanz (05/13/2015 03:03 PM EDT)
• Ariel Ish-Shalom (05/13/2015 05:59 PM EDT)
• Sri Mallikarjun J (05/13/2015 05:59 PM EDT)
• Joe Klobusicky (05/14/2015 12:40 PM EDT)
• Lawrence Hon (05/15/2015 02:15 AM EDT)
• Nihat Kocyigit (05/15/2015 04:46 AM EDT)
• Li Li (05/15/2015 10:52 PM EDT)
• Dieter Beckerle (05/15/2015 12:19 PM EDT)
• Rajesh Sathiyanarayanan (05/16/2015 07:41 AM EDT)
• Sandipan Dey (05/17/2015 10:18 AM EDT)
• Daniel Lincke (05/19/2015 12:23 PM EDT)
• Emilio Del Tessandoro (05/21/2015 09:41 AM EDT)
• Matthew Salac (05/21/2015 03:01 PM EDT)
• Erik Wünstel (05/22/2015 03:16 PM EDT)
• Christopher Black (05/22/2015 08:46 PM EDT)
• Sean Egan (05/23/2015 03:30 PM EDT)
• Kyoung A Lee (05/24/2015 11:24 AM EDT)
• Jhiseung Daniel Hahn (05/24/2015 12:37 PM EDT)
• Gundars Kokts (05/24/2015 01:15 PM EDT)
• Alessandro Marzo (05/24/2015 07:32 PM EDT)
• Jakub ŁopuszaÅ„ski (05/25/2015 04:06 AM EDT)
• Shouky Dan & Tamir Ganor (05/25/2015 04:36 AM EDT)
• Reiner Martin (05/25/2015 04:55 AM EDT)
• Rudy Cortembert (05/26/2015 09:03 AM EDT)
• Tommaso Diotalevi (05/25/2015 12:10 PM EDT)
• Dario Serino (05/27/2015 03:40 AM EDT)
• Danny Lynch (05/27/2015 05:39 AM EDT)
• Carl Meijer (05/27/2015 04:19 PM EDT)
• Richard George (05/28/2015 07:19 AM EDT)
• Colm Bhandal & Michael Clear (05/28/2015 10:24 AM EDT)
• Nis Jørgensen (05/28/2015 08:30 PM EDT)
• Francis Golding (05/29/2015 09:04 AM EDT)
• Kwonil Kobe Ko (05/30/2015 11:14 AM EDT)
• Christian Pape (05/30/2015 04:28 PM EDT)
• Denys Kopiychenko (05/31/2015 04:39 PM EDT)
• Chintan Maggu (05/31/2015 10:10 PM EDT)
• Phil Benamy (05/31/2015 11:40 PM EDT)
• Balakrishnan Varadarajan (06/01/2015 11:28 PM EDT)