Ponder This Challenge - June 2014 - 25 intersecting Cubes

Ponder This Challenge:

Design 25 cubes so that each cube's six faces display integer numbers in the range 0,1,...,31 such that:

1. On each cube, all the numbers are different
2. Any given two cubes share exactly one common number

Please give your answer as 25 lines of six numbers.

Bonus question: Use as small a range of numbers as possible.

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Solution

• Click here to view the solution

  Using 31 different numbers, we can get 31 cubes by using the projective plane of order 5 (see http://mathworld.wolfram.com/ProjectivePlane.html) that has 31 points and 31 lines. Each line contains exactly six points, and any two distinct lines have exactly one point in common: 0 1 2 3 4 5

  0 6 7 8 9 10
  0 11 12 13 14 15
  0 16 17 18 19 20
  0 21 22 23 24 25
  1 6 11 16 21 26
  1 7 12 18 23 28
  1 9 15 17 25 29
  1 10 14 20 24 27
  2 6 13 18 24 29
  2 7 11 17 22 27
  2 8 14 16 25 28
  2 10 15 19 23 26
  3 6 12 19 25 27
  3 8 11 20 23 29
  3 9 14 18 22 26
  3 10 13 17 21 28
  4 7 13 20 25 26
  4 8 15 18 21 27
  4 9 11 19 24 28
  4 10 12 16 22 29
  5 6 15 20 22 28
  5 7 14 19 21 29
  5 8 12 17 24 26
  5 9 13 16 23 27
  0 26 27 28 29 30
  1 8 13 19 22 30
  2 9 12 20 21 30
  3 7 15 16 24 30
  4 6 14 17 23 30
  5 10 11 18 25 30
  

  If we throw away the last six lines that pass through the last point, we get 25 lines (cubes) using 30 points (numbers). To prove that this is the best solution attainable note that if a solution existed that used less than 30 different numbers, then by the pigeonhole principle at least one of the numbers must be used six times (150/29) and these six cubes would have 31 different numbers.

Solvers

• *Radu-Alexandru Todor (05/30/2014 07:04 PM EDT)
• Mark Stuckel (05/30/2014 10:20 PM EDT)
• *Rogerio Ponce da Silva (05/31/2014 12:44 AM EDT)
• Guangda Huzhang (05/31/2014 03:14 AM EDT)
• *Jean-Baptiste Rouquier (05/31/2014 06:00 AM EDT)
• *Movin Jain (05/31/2014 06:12 AM EDT)
• Igor Karp (05/31/2014 08:41 AM EDT)
• *Janos Csorba (05/31/2014 08:46 AM EDT)
• *Oleg Vlasii (05/31/2014 10:21 AM EDT)
• Hakan Summakoglu (05/31/2014 10:41 AM EDT)
• William Gunther (05/31/2014 03:08 PM EDT)
• Alex Fleischer (05/31/2014 03:22 PM EDT)
• *Florian Fischer (05/31/2014 05:05 PM EDT)
• Uoti Urpala (05/31/2014 05:10 PM EDT)
• *Ioan Marius Popdan (05/31/2014 05:33 PM EDT)
• *Cynthia Beauchemin (05/31/2014 06:34 PM EDT)
• *Brian Kell (05/31/2014 09:13 PM EDT)
• *Gilles-Philippe Paillé (05/31/2014 09:20 PM EDT)
• *David Friedman (05/31/2014 09:59 PM EDT)
• Kan Shen (06/01/2014 02:29 AM EDT)
• *Anatoli Plotnikov (06/01/2014 03:16 AM EDT)
• *Kate Thatcher & Jochen Voss (06/01/2014 06:16 AM EDT)
• *Mukund R (06/01/2014 07:31 AM EDT)
• *Michael Rosola (06/01/2014 07:32 AM EDT)
• *Joseph DeVincentis (06/01/2014 09:31 AM EDT)
• *Mathias Schenker (06/01/2014 01:25 PM EDT)
• *Alan Murray (06/01/2014 01:31 PM EDT)
• *Liubing Yu (06/01/2014 07:41 PM EDT)
• *Rob Pratt (06/01/2014 09:42 PM EDT)
• *Emir Haleva (06/02/2014 02:10 AM EDT)
• Alexandre Gilotte (06/02/2014 02:57 AM EDT)
• Lev Plotnikov (06/02/2014 06:54 AM EDT)
• Zhang Shuai (06/02/2014 08:26 AM EDT)
• *Jay R. Joshi (06/02/2014 11:20 AM EDT)
• *Amy Hardesty (06/02/2014 11:22 AM EDT)
• *Fletcher Dostie (06/02/2014 11:36 AM EDT)
• *Philip Kinlen (06/02/2014 01:26 PM EDT)
• *Todd Will (06/02/2014 01:42 PM EDT)
• *Donald Dodson (06/02/2014 01:44 PM EDT)
• Benjamin Phillabaum (06/02/2014 01:46 PM EDT)
• *Benjamin Brumfield (06/02/2014 01:55 PM EDT)
• *Michel Hannoyer (06/02/2014 01:57 PM EDT)
• Ben Phillabaum (06/02/2014 02:00 PM EDT)
• Levent Ertöz (06/02/2014 02:41 PM EDT)
• *Fabio Filatrella (06/02/2014 11:36 AM EDT)
• Monica & Khalil (06/02/2014 02:47 PM EDT)
• *Daniel Bitin (06/02/2014 04:22 PM EDT)
• *Adrian Orzepowski (06/02/2014 04:47 PM EDT)
• *Motty Porat (06/02/2014 04:56 PM EDT)
• *Sean Egan (06/02/2014 05:27 PM EDT)
• *Claudiu Mihaila (06/02/2014 06:08 PM EDT)
• Kurt Hectic (06/03/2014 02:52 AM EDT)
• Serge Thill (06/03/2014 03:58 AM EDT)
• Anand Chitipothu (06/03/2014 04:05 AM EDT)
• *Matej Kollár (06/03/2014 04:48 AM EDT)
• Chuck Carroll (06/03/2014 07:20 AM EDT)
• *Lorenz Reichel (06/03/2014 08:17 AM EDT)
• Ross A. Lippert (06/03/2014 08:51 AM EDT)
• Hakan Kjellerstrand (06/03/2014 09:39 AM EDT)
• Andrew Gacek (06/03/2014 12:10 PM EDT)
• Kai Guttmann (06/03/2014 12:24 PM EDT)
• Mike MacHenry (06/03/2014 12:58 PM EDT)
• Dragos Ionescu (06/03/2014 02:21 PM EDT)
• Deepak Bal (06/03/2014 02:34 PM EDT)
• *Dan Dima (06/03/2014 02:36 PM EDT)
• *Peter Gerritson (06/03/2014 06:11 PM EDT)
• *Vineet Gupta (06/03/2014 06:52 PM EDT)
• Sameer Saurabh (06/04/2014 06:05 AM EDT)
• *Colas Kerkhove (06/04/2014 06:51 AM EDT)
• *Huafeng Jiang (06/04/2014 09:27 AM EDT)
• *Edwin Karat (06/04/2014 10:57 AM EDT)
• *Reiner Martin (06/04/2014 01:12 PM EDT)
• John Tromp (06/04/2014 01:40 PM EDT)
• *Joe Klobusicky (06/04/2014 03:49 PM EDT)
• Rudy Cortembert (06/04/2014 04:20 PM EDT)
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• Vishal Daga (06/05/2014 01:49 AM EDT)
• Puvichakravarthy Ramachandran (06/05/2014 02:43 AM EDT)
• Vikram Gupta (06/05/2014 02:48 AM EDT)
• *Michael Schuresko (06/05/2014 02:54 AM EDT)
• *Joaquim Neves Carrapa (06/05/2014 03:50 AM EDT)
• Abhishek Gupta (06/05/2014 04:33 AM EDT)
• Kursad Albayraktaroglu (06/05/2014 08:45 AM EDT)
• *Jamie Jorgensen (06/05/2014 12:18 PM EDT)
• Lv Jianhao (06/05/2014 01:36 PM EDT)
• Boaz Menuhin & Shahar Lev (06/05/2014 02:31 PM EDT)
• *Sri Mallikarjun J (06/05/2014 04:07 PM EDT)
• *Gil Citro (06/05/2014 07:37 PM EDT)
• Nagendra Gd (06/06/2014 03:50 AM EDT)
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• *Daniele Bovi (06/06/2014 12:52 PM EDT)
• Barbara Jones (06/06/2014 04:18 PM EDT)
• *Tristan Launay (06/06/2014 06:14 PM EDT)
• *Petr Capek (06/06/2014 11:19 PM EDT)
• Astrid Rossat-Mignod (06/07/2014 04:14 AM EDT)
• Kwonil Kobe Ko (06/07/2014 11:49 AM EDT)
• *Jin Hyup Hong (06/07/2014 01:31 PM EDT)
• *Traian Delca (06/07/2014 04:50 PM EDT)
• *Seung Hwan (Sonny) An (06/08/2014 07:11 AM EDT)
• Ronen Rojas (06/08/2014 01:10 PM EDT)
• *Tim Cieplowski (06/08/2014 02:46 PM EDT)
• *Peter Shim (06/08/2014 02:46 PM EDT)
• *Gyorgy Gyomai (06/08/2014 04:13 PM EDT)
• Michael Liepelt (06/09/2014 03:16 AM EDT)
• *Hugo Pfoertner (06/09/2014 03:39 AM EDT)
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• Leandro Araújo (06/09/2014 08:25 AM EDT)
• Emilio Del Tessandoro (06/09/2014 09:00 AM EDT)
• *Evert van Dijken (06/09/2014 01:56 PM EDT)
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• *Nozomu Yachie (06/09/2014 09:41 PM EDT)
• *Jesper Kjær (06/10/2014 09:35 AM EDT)
• *You SoonHo (06/10/2014 06:59 AM EDT)
• Adam Wysocki (06/10/2014 01:19 PM EDT)
• Adam Rubin (06/10/2014 04:06 PM EDT)
• *Zoey Zhou (06/10/2014 06:22 PM EDT)
• Jiri Navratil (06/10/2014 10:04 PM EDT)
• *Tulasi Ram Reddy (06/10/2014 11:52 PM EDT)
• *Jiri Hrdina (06/11/2014 03:43 AM EDT)
• *William Heller (06/11/2014 08:48 PM EDT)
• *Thomas Egense (06/12/2014 12:35 AM EDT)
• *Philippe Fondanaiche (06/12/2014 01:56 AM EDT)
• *Olivier Mercier (06/12/2014 08:49 AM EDT)
• Matthew Salac (06/13/2014 09:53 AM EDT)
• Tim Joseph Clark (06/13/2014 10:02 PM EDT)
• *Simon Yang (06/14/2014 07:30 AM EDT)
• *William Kang (06/14/2014 03:16 PM EDT)
• Terence Tarvis (06/14/2014 06:16 PM EDT)
• *Cristian Yones (06/14/2014 10:04 PM EDT)
• *Balakrishnan Varadarajan (06/14/2014 10:28 PM EDT)
• *Aliaksei Sanko (06/15/2014 03:40 AM EDT)
• *Antonio García Astudillo (06/15/2014 11:46 AM EDT)
• *David Yang (06/15/2014 06:54 PM EDT)
• Srivathsa Acharya (06/16/2014 01:09 AM EDT)
• *Daniel Blanchard (06/16/2014 03:39 AM EDT)
• *Matthias Bussas (06/16/2014 06:16 AM EDT)
• *Jan Fricke (06/17/2014 05:50 AM EDT)
• *Aharon Malachi (06/17/2014 08:21 AM EDT)
• *Daniel Matras (06/17/2014 11:46 AM EDT)
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• *Alok Menghrajani (06/17/2014 03:25 PM EDT)
• *David Jakopac (06/17/2014 04:53 PM EDT)
• *Armin Krauss (06/18/2014 02:44 AM EDT)
• Said Mohamed (06/18/2014 07:25 AM EDT)
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• Lindsey Puryear (06/18/2014 05:15 PM EDT)
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• *Norbert Koszó (06/21/2014 03:59 AM EDT)
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• *Woosung (William) Jung (06/21/2014 12:35 PM EDT)
• *Michael Liepelt (06/21/2014 02:43 PM EDT)
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• *Aviv Nisgav (06/22/2014 03:47 AM EDT)
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• *Da Young (Clara) Hong (06/22/2014 06:56 PM EDT)
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• Vivek Ranjan Nema (06/30/2014 07:17 AM EDT)
• Andrew Scherbakov (06/30/2014 09:13 AM EDT)