Ponder This Challenge - July 2008 - Covering pentominoes by polyominos

Ponder This Challenge:

This month's puzzle concerns polyominoes. A polyomino is a plane figure constructed by joining unit squares along their edges. Two polyominoes are considered distinct if one cannot be mapped to the other by translations, rotations and reflections. Polyominoes consisting of 5 unit squares are known as pentominoes. It is well known that there are exactly 12 distinct pentominoes. We say a polyomino A covers a polyomino B if A can be constructed by adding unit squares to B. We ask:

1. How many distinct pentominoes can a polyomino consisting of 6 unit squares (hexomino) cover? Give an example.
2. What is the smallest size (least number of unit squares) required for a polyomino which covers all 12 pentominoes? Give an example.
   As usual we ask that you only submit answers consisting of your original work.

Clarification: Question 1 intends to ask for the maximum number of distinct pentominoes which can be covered by a single hexomino.

Note: Some of the examples submitted appear to be getting lost or garbled on the way. Unfortunately I can not give credit for these. Please ensure your answer will make sense when read in a fixed width plain text editor. Pdf, jpg, gif, bmp, xls or docbin attachments also seem to work.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  The answers are:

  1. 4; xxxx  or     x
        xx         xxx
                   xx
  
  2. 9; xxx   or   xxx
        xxxxx     xxxxx
           x         x
  

  ---

  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Dan Dima (07.01.2008 @05:57:03 PM EDT)
• Frank Yang (07.01.2008 @06:20:06 PM EDT)
• Alan O'Donnell (07.01.2008 @06:51:14 PM EDT)
• V Balakrishnan (07.01.2008 @07:33:59 PM EDT)
• Jimmy Hu (07.01.2008 @07:41:20 PM EDT)
• Sylvain Becker (07.02.2008 @06:40:06 AM EDT)
• Alex Flora (07.02.2008 @09:55:55 AM EDT)
• Jimmy He (07.02.2008 @10:41:47 AM EDT)
• Eugene Vasilchenko (07.02.2008 @12:44:40 AM EDT)
• Graham McHardy (07.02.2008 @03:48:37 PM EDT)
• Lavanya Kannan (07.02.2008 @06:05:06 PM EDT)
• Willem H. de Boer (07.02.2008 @06:39:52 PM EDT)
• Rick Shepard (07.02.2008 @11:55:01 PM EDT)
• James Dow Allen (07.03.2008 @01:34:21 AM EDT)
• Nyles Heise (07.03.2008 @04:04:31 AM EDT)
• Zhou Guang (07.03.2008 @04:39:10 AM EDT)
• Chris Hills (07.03.2008 @09:25:25 AM EDT)
• Fabio Filatrella (07.03.2008 @10:42:55 AM EDT)
• David Gower (07.03.2008 @01:09:37 PM EDT)
• Chuck Carroll (07.03.2008 @01:22:03 PM EDT)
• Ariel Flat (07.03.2008 @02:16:45 PM EDT)
• Are Hjørungnes (07.03.2008 @07:45:50 PM EDT)
• Dennis Langdeau (07.03.2008 @08:25:55 PM EDT)
• Niko Rombes (07.03.2008 @12:47:43 PM EDT)
• Fred Bone (07.04.2008 @08:43:39 AM EDT)
• Ben Tarlow (07.04.2008 @02:57:31 PM EDT)
• Mate Bartalos (07.04.2008 @07:18:48 AM EDT)
• Andreas Eisele (07.04.2008 @12:21:17 PM EDT)
• Dan Colestock (07.05.2008 @03:29:42 AM EDT)
• Brett E. Grothouse (07.05.2008 @03:32:17 AM EDT)
• Laszlo Kozma (07.05.2008 @07:02:56 AM EDT)
• John Tromp (07.05.2008 @12:17:49 AM EDT)
• Joseph DeVincentis (07.05.2008 @12:58:04 AM EDT)
• Gale Greenlee (07.05.2008 @07:33:35 PM EDT)
• Maris van Sprang (07.06.2008 @08:54:16 AM EDT)
• John G. Fletcher (07.06.2008 @11:29:24 PM EDT)
• Ernst Jaeger (07.06.2008 @12:25:28 PM EDT)
• Andrew Jones (07.06.2008 @12:31:51 PM EDT)
• John Douma (07.06.2008 @12:33:46 PM EDT)
• Ole Martin (07.06.2008 @12:58:16 PM EDT)
• Dave Parashar (07.07.2008 @01:17:34 AM EDT)
• Gary M Gerkin (07.07.2008 @08:16:16 AM EDT)
• Amos Guler (07.07.2008 @09:09:59 AM EDT)
• Ashutosh Mahajan (07.07.2008 @12:39:28 AM EDT)
• Victor Balacescu (07.07.2008 @01:51:12 AM EDT)
• Joe Barnett and Bryan Jacobs (07.07.2008 @02:07:11 PM EDT)
• Hadrien Cambazard (07.07.2008 @05:26:06 PM EDT)
• Matt Hickford (07.07.2008 @06:43:42 PM EDT)
• Bhargav Kanagal (07.07.2008 @07:10:25 PM EDT)
• Jeff Steele (07.08.2008 @06:19:54 AM EDT)
• Niraj Jha (07.08.2008 @01:12:48 PM EDT)
• Michael Vilim (07.08.2008 @02:55:57 PM EDT)
• Alvaro Martinez Echevarria (07.08.2008 @04:34:29 PM EDT)
• Mithil Ramteke (07.08.2008 @11:59:16 PM EDT)
• Albert Stadler (07.09.2008 @02:55:41 AM EDT)
• Vivek Nema (07.09.2008 @05:05:32 AM EDT)
• Mitchell Joe (07.09.2008 @04:14:04 PM EDT)
• Will Hasenplaugh (07.10.2008 @01:02:52 AM EDT)
• Radu Borza (07.10.2008 @07:45:04 AM EDT)
• Jochen Hoenicke (07.10.2008 @03:05:00 PM EDT)
• Jesse Kolman (07.11.2008 @04:02:18 PM EDT)
• Dmitry Litvinenko (07.12.2008 @02:47:21 AM EDT)
• Clive Tong (07.12.2008 @02:00:36 PM EDT)
• Stephen Steffes (7.14.2008 @05:56 PM EDT)
• Michael Liepelt (07.14.2008 @07:38:21 PM EDT)
• George Saon (07.15.2008 @04:20:57 PM EDT)
• Graham Hesketh (07.16.2008 @06:26:28 PM EDT)
• Jonathan Wildstrom (07.16.2008 @06:33:59 PM EDT)
• Neil Smeby (07.17.2008 @10:25:18 PM EDT)
• Andrea Andenna (07.17.2008 @12:29:55 PM EDT)
• J K Viswanath Raju (07.18.2008 @06:45:28 AM EDT)
• Rickard Björklund (07.18.2008 @03:14:14 PM EDT)
• Steve Trevorrow (07.20.2008 @05:51:44 AM EDT)
• Abhisek Ukil (07.20.2008 @11:41:23 AM EDT)
• Ian Smith (07.21.2008 @06:58:18 PM EDT)
• Kalyana Chakravarthy (7.22.2008 @12:18 AM EDT)
• Boris Nikolaus (07.22.2008 @07:34:00 PM EDT)
• Li Zou (07.23.2008 @05:44:00 AM EDT)
• Tom Rice (07.23.2008 @08:58:00 PM EDT)
• Ben Yackley (07.24.2008 @02:36:00 AM EDT)
• Jan Rubak (7.25.2008 @08:28 PM EDT)
• Alok Jain (7.25.2008 @12:00 PM EDT)
• Michael Schaaf (7.28.2008 @01:58 PM EDT)
• Don Dodson (7.29.2008 @09:31 AM EDT)
• Travis Craig (7.31.2008 @06:45 PM EDT)
• Phil Muhm (7.31.2008 @06:59 PM EDT)
• Tony Maric and Amanda Nethers (7.31.2008 @08:21 PM EDT)