Ponder This Challenge - April 2008 - Polygons formed from squares / triangles

Ponder This Challenge:

Suppose you are given a collection of squares and equilateral triangles (of unit side). You are asked to form convex polygons by sticking the squares and triangles together with their sides aligned. These polygons must also have unit sides (so that for example sticking two squares together to form a domino does not count). How many distinct polygons can be formed in this way (including polygons formed by a single square or triangle)? List them, for each giving the
length of the perimeter, the number of squares and the number of triangles used to form it. We don't expect (nor will we provide) a proof that the list is exhaustive but please try to find all the examples before submitting a solution.

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

  April 2008 Solution:

  There are 11 distinct polygons that can be formed in this way.
  They are listed with the requested parameters below.
        Case         Perimeter       Squares       Triangles
          1              3              0              1
          2              4              0              2
          3              4              1              0
          4              5              1              1
          5              6              0              6
          6              6              1              2
          7              7              2              3
          8              8              2              6
          9              9              3              7
         10             10              4              8
         11             12              6             12

Solvers

• Dan Dima (04.01.2008 @04:53:07 PM EDT)
• David W. Cantrell (04.01.2008 @09:24:57 PM EDT)
• Vladimir Sedach (04.02.2008 @03:36:32 PM EDT)
• Bogdan Banu (04.02.2008 @08:16:38 AM EDT)
• Chris Hills (04.02.2008 @08:30:39 AM EDT)
• Eugene Vasilchenko (04.02.2008 @09:19:27 AM EDT)
• Bojan Basic (04.02.2008 @10:23:58 AM EDT)
• Joseph DeVincentis (04.02.2008 @10:48:53 AM EDT)
• Carlo Piovesan (04.02.2008 @12:10:05 PM EDT)
• John T. Robinson (04.02.2008 @01:27:48 PM EDT)
• James Boyce (04.02.2008 @01:59:36 PM EDT)
• Samantha Casanova (04.02.2008 @02:07:31 PM EDT)
• Gary Gerken (04.03.2008 @01:40:00 AM EDT)
• Zhou Guang (04.03.2008 @12:02:25 PM EDT)
• V Balakrishnan (04.03.2008 @03:40:09 AM EDT)
• Joe Fendel (04.03.2008 @12:41:52 PM EDT)
• Gale Greenlee (04.03.2008 @01:22:53 PM EDT)
• Ole Martin (04.03.2008 @01:22:54 PM EDT)
• Michael Quist (04.03.2008 @06:16:46 PM EDT)
• Fred Batty (04.04.2008 @11:28:00 PM EDT)
• Dave Parashar (04.04.2008 @01:59:28 PM EDT)
• Jens Voz (04.04.2008 @08:54:33 PM EDT)
• John Fletcher (04.04.2008 @10:12:54 PM EDT)
• Daniel Bitin (04.04.2008 @01:28:22 PM EDT)
• Sriram Durbha (04.04.2008 @03:27:48 PM EDT)
• Reinder Verlinde (04.04.2008 @04:01:17 PM EDT)
• Robert Szafranski (04.04.2008 @11:18:23 PM EDT)
• Hengfeng Tian (04.06.2008 @02:46:35 AM EDT)
• Javier Rodrigues Soler (04.06.2008 @06:21:58 AM EDT)
• Henry Bottomley (04.06.2008 @05:44:44 PM EDT)
• Frank Yang (04.06.2008 @08:39:52 PM EDT)
• Edna Mantsour (04.07.2008 @03:26:01 AM EDT)
• Tim Mercure (04.07.2008 @12:43:43 AM EDT)
• Maurizio (04.07.2008 @03:32:29 AM EDT)
• Stefan Znam (04.07.2008 @04:22:28 PM EDT)
• Shankar Ananthanarayanan (04.07.2008 @04:12:39 PM EDT)
• Jochen Hoenicke (04.07.2008 @03:08:37 PM EDT)
• Jeff Steele (04.08.2008 @10:03:40 AM EDT)
• Almudena Moratinos (04.08.2008 @10:34:21 AM EDT)
• Chris Lomont (04.08.2008 @05:53:17 PM EDT)
• Wolfgang Kais (04.09.2008 @08:23:36 AM EDT)
• Arno Candel (04.09.2008 @05:21:27 PM EDT)
• Piotr Komisarski (04.10.2008 @03:27:39 PM EDT)
• Mathias Schenker (04.10.2008 @08:11:20 AM EDT)
• Alvaro Martinez Echevarria (04.11.2008 @04:55:37 AM EDT)
• Anneke Van Steirteghem (04.11.2008 @05:08:14 AM EDT)
• Alan Murray (04.11.2008 @12:52:47 AM EDT)
• Farid Lian (04.12.2008 @10:15:19 AM EDT)
• Christian Blatter (04.12.2008 @07:14:23 AM EDT)
• Guler Amos (04.13.2008 @05:45:00 AM EDT)
• John Naud (04.13.2008 @09:20:45 AM EDT)
• Andrea Andenna (04.13.2008 @06:55:59 PM EDT)
• Kevin Williamson (04.13.2008 @12:25:05 PM EDT)
• Kalyana Chakravarthy (04.13.2008 @09:15:21 AM EDT)
• Lukasz Bolikowski (04.14.2008 @11:20:57 AM EDT)
• Matt Hickford (04.14.2008 @02:00:37 PM EDT)
• Jan Fricke (04.16.2008 @06:52:39 AM EDT)
• Stephen Merriman (04.17.2008 @12:39:41 AM EDT)
• Vishal Babu Somani (04.17.2008 @11:38:52 AM EDT)
• William C. Hasenplaugh (04.17.2008 @11:47:49 AM EDT)
• Michael Schaaf (04.17.2008 @03:29:31 PM EDT)
• Mindaugas Zaleckis (04.17.2008 @03:45:59 PM EDT)
• Leroy Junker (04.18.2008 @09:13:48 PM EDT)
• Nathan Carlson (04.19.2008 @06:24:54 PM EDT)
• Greg McNulty (04.20.2008 @01:32:24 PM EDT)
• Tim Cooper (04.21.2008 @07:47:34 AM EDT)
• Johannes Pelzer (04.21.2008 @03:29:07 PM EDT)
• Fabio Filatrella (04.22.2008 @11:04:28 AM EDT)
• William Darnieder (04.29.2008 @10:43:42 AM EDT)
• Steven V. Thacker (04.29.2008 @07:47:51 PM EDT)
• Phil Muhm (04.29.2008 @04:07:37 PM EDT)
• Albert Stadler (04.30.2008 @03:34:39 AM EDT)
• Boris Nikolaus (04.30.2008 @01:06:36 PM EDT)
• Dion Harmon (04.30.2008 @09:49:23 AM EDT)