Ponder This Challenge - January 2016 - Efficient bicycle gears

Ponder This Challenge:

As part of IBM's efforts to make a better and greener planet, this month's challenge is to design an efficient gear system for bicycles. The front chainrings (S_1) and back cogset (S_2) are represented by two sets of numbers S_1 and S_2 where each gear g is the product of two numbers, one from each set: s_1*s_2. When designing the gears, we need sets of numbers that can produce every gear value between 1 and N up to +/- 1, i.e., for every number m between 1 and N there exists s_1 in S_1 and s_2 in S_2 such that abs(m-s_1*s_2)<=1.

So, for example, using sets of size 3, we can get to N=19 by S_1={1,3,6} and S_2={2,3,5}.

Your challenge is to find the values of S_1 and S_2 of size 6 that solve the problem for N=56 (again, using +/- 1); Earn an asterisk for a significant increase of N.

Solution

• Click here to view the solution

  There are two solutions to get N=57: S1={1,3,4,6,7,9} S2={2,5,7,8,9,13} and S1={1,5,6,7,8,13} S2={2,3,4,6,7,9}.

  A larger N can be achieved by 'cheating' and using non-integer numbers. For example,

  S1={1,1.5,2,3,4,4.5} S2={2,12,13,14,15,16.5}.

Solvers

• *Robert Gerbicz (29/12/2015 03:10 PM IDT)
• Lorenz Reichel (29/12/2015 03:36 PM IDT)
• Kang Jin Cho (29/12/2015 07:00 PM IDT)
• Daniel Johnson-Chyzhykov (29/12/2015 08:25 PM IDT)
• Dieter Beckerle (29/12/2015 09:48 PM IDT)
• *Radu-Alexandru Todor (30/12/2015 01:58 AM IDT)
• Sean Egan (30/12/2015 11:17 AM IDT)
• James Dow Allen (30/12/2015 11:59 AM IDT)
• Bruce Norskog (30/12/2015 08:25 PM IDT)
• Wayne Petrychyn (30/12/2015 10:18 PM IDT)
• Jesse Rearick (30/12/2015 10:18 PM IDT)
• Janos Csorba (31/12/2015 01:24 AM IDT)
• Jean-Baptiste Rouquier (31/12/2015 10:45 AM IDT)
• Shouky Dan & Tamir Ganor (31/12/2015 11:13 AM IDT)
• Dejan Stakić (31/12/2015 12:28 PM IDT)
• *Yan-Wu He (31/12/2015 02:17 PM IDT)
• Alex Fleischer (31/12/2015 03:05 PM IDT)
• Alexandre Ferrand (31/12/2015 03:39 PM IDT)
• Boris Kazimir Hostnik (31/12/2015 04:25 PM IDT)
• Gary M. Gerken (31/12/2015 06:45 PM IDT)
• *Rob Pratt (31/12/2015 09:19 PM IDT)
• Harald Bögeholz (31/12/2015 11:32 PM IDT)
• Matt Falk (01/01/2016 04:50 AM IDT)
• Brendan Berger (01/01/2016 07:13 AM IDT)
• Adrian Neacsu (01/01/2016 12:50 PM IDT)
• Gregory Giecold (01/01/2016 06:09 PM IDT)
• Michael Gottlieb (01/01/2016 08:38 PM IDT)
• Ji Eun Kim (01/01/2016 09:25 PM IDT)
• Todd Will (01/01/2016 10:10 PM IDT)
• Mathias Schenker (01/01/2016 10:28 PM IDT)
• Seung Hwan (Sonny) An (02/01/2016 12:24 AM IDT)
• Shirish Chinchalkar (02/01/2016 02:00 AM IDT)
• Philip Adams (02/01/2016 03:40 AM IDT)
• Rahul Upakare (02/01/2016 10:16 AM IDT)
• Steven Langerwerf (03/01/2016 12:46 AM IDT)
• Sergey Koposov (03/01/2016 12:14 PM IDT)
• Amos Guler (03/01/2016 12:23 PM IDT)
• *Andreas Stiller (03/01/2016 09:23 PM IDT)
• Boris Rakus (03/01/2016 10:06 PM IDT)
• Jesus Sanz (03/01/2016 10:28 PM IDT)
• Igor Karp (03/01/2016 11:58 PM IDT)
• Liubing Yu (04/01/2016 01:56 AM IDT)
• *Phil Benamy (04/01/2016 03:18 AM IDT)
• Joseph DeVincentis (04/01/2016 04:29 AM IDT)
• Ajay Mallepally (04/01/2016 05:52 AM IDT)
• Shaleen Deep (04/01/2016 09:47 AM IDT)
• Alex Teich (04/01/2016 02:38 PM IDT)
• Timothy S. Lewis (04/01/2016 05:20 PM IDT)
• Ali Can Keserel (04/01/2016 06:13 PM IDT)
• Curtis Watson (04/01/2016 07:11 PM IDT)
• Rogerio Ponce da Silva (04/01/2016 10:35 PM IDT)
• Michael Liepelt (04/01/2016 11:19 PM IDT)
• Tom Sirgedas (05/01/2016 01:43 AM IDT)
• Victor Chang (05/01/2016 09:02 AM IDT)
• Umut Uludag (05/01/2016 09:42 AM IDT)
• Lorenzo Gianferrari Pini (05/01/2016 09:59 AM IDT)
• Reiner Martin (05/01/2016 10:11 AM IDT)
• Raymond Lo (05/01/2016 01:23 PM IDT)
• Pål Hermunn Johansen (05/01/2016 01:54 PM IDT)
• *Donald T Dodson (05/01/2016 05:05 PM IDT)
• Clive Tong (05/01/2016 11:04 PM IDT)
• Rajesh Sathiyanarayanan (06/01/2016 10:22 AM IDT)
• Hendrik Nigul (06/01/2016 01:10 PM IDT)
• Mark J Murphy (06/01/2016 10:06 PM IDT)
• Erdinc Ozturk (06/01/2016 11:56 PM IDT)
• Minxi Jiang (07/01/2016 02:34 AM IDT)
• Ajit Thyagarajan (07/01/2016 07:01 AM IDT)
• Francis Golding (07/01/2016 12:04 PM IDT)
• Roberto Tauraso (07/01/2016 12:46 PM IDT)
• Tero Patana (07/01/2016 11:17 PM IDT)
• Jan Rothkegel (08/01/2016 09:47 AM IDT)
• Cody Pfau (08/01/2016 11:42 PM IDT)
• Yuval Salant (09/01/2016 12:01 AM IDT)
• Levent Ertoz (09/01/2016 01:49 AM IDT)
• Jimmy Waters (09/01/2016 04:58 AM IDT)
• Claudio Hüni (09/01/2016 08:41 PM IDT)
• Di Luo (10/01/2016 05:12 AM IDT)
• Aviv Nisgav (10/01/2016 11:06 AM IDT)
• *Paolo Farinelli (10/01/2016 11:07 AM IDT)
• Daniel Bitin (10/01/2016 05:07 PM IDT)
• Kwonil Kobe Ko (10/01/2016 08:26 PM IDT)
• Fabio Michele Negroni (10/01/2016 09:43 PM IDT)
• Peter Gerritson (10/01/2016 11:32 PM IDT)
• Kipp Johnson (11/01/2016 12:41 AM IDT)
• Christian Pfaffenzeller (11/01/2016 12:40 PM IDT)
• Dan Salajan (11/01/2016 05:31 PM IDT)
• Phillip Ahn (12/01/2016 04:17 AM IDT)
• Viola Lee (12/01/2016 05:00 AM IDT)
• Heidi Kim (12/01/2016 08:09 AM IDT)
• Evert van Dijken (12/01/2016 04:56 PM IDT)
• ByeongYeon (Jackie) Ryu (12/01/2016 05:33 PM IDT)
• Yeonjune Kang (12/01/2016 11:54 PM IDT)
• Hyun Jin Kim (13/01/2016 02:55 AM IDT)
• Jason Lee (13/01/2016 10:55 AM IDT)
• Emir Haleva (13/01/2016 05:14 PM IDT)
• Myong Claire Yun (13/01/2016 05:36 PM IDT)
• Emilio Del Tessandoro (13/01/2016 06:33 PM IDT)
• Anton Čotar (14/01/2016 01:17 AM IDT)
• Nadir S (14/01/2016 11:52 AM IDT)
• Tomáš Jurík (14/01/2016 03:38 PM IDT)
• Michael Blaszczyk (15/01/2016 07:19 PM IDT)
• Albert Stadler (16/01/2016 10:45 PM IDT)
• Ayaskant Pani (18/01/2016 03:37 AM IDT)
• Benjamin Lui (18/01/2016 04:19 AM IDT)
• Hamidreza Bidar (18/01/2016 09:00 PM IDT)
• Fabio Filatrella (19/01/2016 03:42 PM IDT)
• Franciraldo Cavalcante (20/01/2016 06:44 AM IDT)
• Thomas Egense (20/01/2016 10:04 AM IDT)
• Tilo Anschütz (20/01/2016 03:11 PM IDT)
• Simon Foster (21/01/2016 10:52 AM IDT)
• Guillem Collell Talleda (21/01/2016 11:44 AM IDT)
• Zachary Powers (21/01/2016 07:53 PM IDT)
• Anders Bisbjerg Madsen (21/01/2016 10:17 PM IDT)
• Ioan Marius Popdan (22/01/2016 02:45 PM IDT)
• Abhishek Gupta (23/01/2016 12:24 AM IDT)
• Lawrence Hon (23/01/2016 05:10 PM IDT)
• Junjiajia Long (24/01/2016 06:17 PM IDT)
• Hyun Min Victoria Woo (24/01/2016 06:37 PM IDT)
• Alex Zdan (24/01/2016 08:48 PM IDT)
• Guenter Nowak (25/01/2016 03:44 AM IDT)
• David Dunkley (25/01/2016 09:16 AM IDT)
• Amit Bhatia (25/01/2016 10:05 PM IDT)
• Tim Cieplowski (26/01/2016 01:21 AM IDT)
• *Chris Shannon (26/01/2016 09:01 AM IDT)
• Andreas Wohlfrom (26/01/2016 11:14 AM IDT)
• Mihail Plesa (26/01/2016 08:31 PM IDT)
• Gyorgy Gyomai (26/01/2016 08:45 PM IDT)
• Loukas Sidiropoulos (26/01/2016 09:34 PM IDT)
• Rudy Cortembert (26/01/2016 10:05 PM IDT)
• Stéphane Higueret (27/01/2016 12:33 PM IDT)
• Jitesh Jain (28/01/2016 01:47 AM IDT)
• *Nyles Heise (28/01/2016 04:51 AM IDT)
• Armin Krauss (28/01/2016 03:31 PM IDT)
• Motty Porat (29/01/2016 02:24 AM IDT)
• Nicolas Bitouze (29/01/2016 04:42 AM IDT)
• Mike Chirozzi (29/01/2016 05:44 AM IDT)
• Toke Eskildsen (30/01/2016 11:04 AM IDT)
• Martin Sergio H. Faester (31/01/2016 04:44 PM IDT)
• Cindy Sewon Park (31/01/2016 07:10 PM IDT)
• Ryunhyung Jessica Kim (31/01/2016 07:35 PM IDT)
• Colas Kerkhove (31/01/2016 07:53 PM IDT)
• Erik Wünstel (31/01/2016 11:33 PM IDT)
• David Greer (31/01/2016 11:30 PM IDT)
• Yun Jin Jeong (01/02/2016 03:31 AM IDT)
• Muralidhar Seshadri (01/02/2016 05:34 AM IDT)
• Oscar Volpatti (01/02/2016 08:07 AM IDT)