Ponder This Challenge - October 2012 - Disdyakis Triacontahedrons dice

Ponder This Challenge:

There are seven polyhedral dice in a Dungeons & Dragons game. The result of throwing the 7 D&D dice is the sum of the seven numbers from each die:

1. Tetrahedron: numbers 1, 2, 3, 4
2. Cube: numbers 1, 2, 3, 4, 5, 6
3. Octahedron: numbers 1, 2, 3, 4, 5, 6, 7, 8
   4, 5. Two pentagonal trapezohedron: numbers 0,1, 2, 3, 4, 5, 6, 7, 8, 9
4. Dodecahedron: numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
5. Icosahedron: numbers 1, 2, 3, ..., 20.

Find a way to get the same distribution (same probability for each result) of tossing all seven D&D cubes by tossing four Disdyakis Triacontahedrons with integer numbers between 1 and 17 on their 120 faces.

Please supply your answer as four lines of 17 numbers. Line i describes the i-th Disdyakis Triacontahedron and Column j describes how many times the number j appears on it.

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.

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Solution

• Click here to view the solution

  The cubes can be seen as polynomials, where tossing a cube gives n with a probability proportional to the coefficient of the monomial x^n.

  Using this notation, it is easy to see that summing several cubes is the same as multiplying their polynomials.

  So all we have to do is take the seven polynomials:

  x+x^2+x^3+x^4
  x+x^2+x^3+x^4+x^5+x^6
  x+x^2+x^3+x^4+x^5+x^6+x^7+x^8
  1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+X^9
  1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+X^9
  x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+X^9+X^10+x^11+X^12
  x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+X^9+X^10+x^11+X^12+...+X^20

  Factor them and then multiply the factors together in a different way.

  There are many different ways to do this.

  A possible solution, in the format we asked for, is

  0 5 5 5 5 10 10 10 10 10 10 10 10 5 5 5 5
  3 3 6 6 6 9 9 12 12 12 9 9 6 6 6 3 3
  3 9 12 12 12 9 3 0 0 0 3 9 12 12 12 9 3
  1 2 4 6 8 10 11 12 12 12 11 10 8 6 4 2 1

Solvers

• János Kramár (09/28/2012 02:52 PM EDT)
• Fletcher Dostie (09/28/2012 06:33 PM EDT)
• Deron Stewart (09/28/2012 07:52 PM EDT)
• Anders Kaseorg (09/29/2012 02:06 AM EDT)
• Radu-Alexandru Todor (09/29/2012 05:33 AM EDT)
• Shilei Zang (09/29/2012 06:09 AM EDT)
• Janos Csorba (09/29/2012 12:13 PM EDT)
• Peter de Rivaz (09/29/2012 02:55 PM EDT)
• Denys Kopiychenko (09/30/2012 12:45 PM EDT)
• Mark Pervovskiy (09/30/2012 02:47 PM EDT)
• Jan Fricke (09/30/2012 05:57 PM EDT)
• Jochen Voß (10/01/2012 06:31 AM EDT)
• Dan Dima (10/01/2012 10:46 AM EDT)
• Tulasi Ram Reddy (10/01/2012 01:06 PM EDT)
• Albert Stadler (10/02/2012 11:48 AM EDT)
• Rino Raj (10/02/2012 12:10 PM EDT)
• Lorenzo Gianferrari Pini (10/02/2012 03:09 PM EDT)
• Manas Joglekar & Santosh Ananthakrishnan (10/02/2012 05:44 PM EDT)
• David Friedman (10/02/2012 11:56 PM EDT)
• Rob Pratt (10/03/2012 09:48 AM EDT)
• Arthur Breitman (10/03/2012 03:55 PM EDT)
• Michael Brand (10/03/2012 07:41 PM EDT)
• Alan Murray (10/03/2012 08:55 PM EDT)
• Todd Will (10/04/2012 04:15 PM EDT)
• Peter Gerritson (10/04/2012 04:57 PM EDT)
• Matthew Charlap (10/04/2012 06:49 PM EDT)
• Joseph DeVincentis (10/05/2012 12:08 PM EDT)
• Greg McNulty (10/05/2012 01:10 PM EDT)
• Masoud Alipour (10/06/2012 01:38 AM EDT)
• Kevin Sham (10/06/2012 03:35 AM EDT)
• Daniel Kneezel (10/06/2012 04:47 AM EDT)
• Adrian Orzepowski (10/06/2012 11:49 AM EDT)
• Chuck Carroll (10/06/2012 01:32 PM EDT)
• Balakrishnan Varadarajan (10/06/2012 08:35 PM EDT)
• Guangda Huz! hang (10/06/2012 11:55 PM EDT)
• Don Reble (10/07/2012 04:02 AM EDT)
• Tomek Czajka (10/07/2012 08:17 PM EDT)
• Xianchao Xie (10/07/2012 08:45 PM EDT)
• Nis Jørgensen (10/07/2012 10:08 PM EDT)
• Dan Arnon (10/08/2012 09:01 PM EDT)
• Roberto Tauraso (10/09/2012 03:16 AM EDT)
• Øyvind Grotmol (10/09/2012 08:14 AM EDT)
• Prateek Bhakta (10/09/2012 11:37 AM EDT)
• Gautam Kamath (10/09/2012 02:43 PM EDT)
• Mark Mammel (10/09/2012 05:51 PM EDT)
• Rastko Anicic (10/09/2012 06:37 PM EDT)
• Kan Huang (10/10/2012 09:37 AM EDT)
• Armin Krauss (10/10/2012 01:00 PM EDT)
• Rogerio Ponce da Silva (10/10/2012 11:43 PM EDT)
• Ken Bateman (10/11/2012 05:00 PM EDT)
• Lewei Weng (10/11/2012 10:51 PM EDT)
• Scott Bridges & Arthur Krebsbach (10/12/2012 02:43 PM EDT)
• Eric Harley (10/14/2012 12:38 AM EDT)
• Seung Min "Brian" Park (10/14/2012 10:18 AM EDT)
• Alex Jeon (10/14/2012 03:52 PM EDT)
• Javier Rodriguez (10/14/2012 05:12 PM EDT)
• Daniel Bitin (10/14/2012 06:20 PM EDT)
• Tony Harrison (10/14/2012 06:19 PM EDT)
• Roddy Kok (10/15/2012 01:30 PM EDT)
• Yasodhar Patnaik (10/15/2012 02:22 PM EDT)
• Stéphane Higueret (10/16/2012 07:50 AM EDT)
• David Greer (10/16/2012 08:10 AM EDT)
• Andreas Stiller (10/16/2012 09:19 AM EDT)
• Dan Ismailescu (10/16/2012 10:58 AM EDT)
• Minxi Jiang (10/16/2012 12:25 PM EDT)
• Siva Dirisala (10/17/2012 02:22 AM EDT)
• Harald Bögeholz (10/17/2012 06:43 PM EDT)
• Kevin Bauer (10/17/2012 09:47 PM EDT)
• Christian Pape (10/17/2012 10:27 PM EDT)
• Marcello Herreshoff (10/17/2012 11:14 PM EDT)
• Oleksandr Pavlyk (10/18/2012 12:49 AM EDT)
• Boris Nikolaus (10/18/2012 07:50 PM EDT)
• Kartikeya Shandilya (10/19/2012 07:41 AM EDT)
• Peter Holdsworth (10/19/2012 05:32 PM EDT)
• Walter Schmidt (10/20/2012 06:36 AM EDT)
• Gyorgy Gyomai (10/23/2012 08:05 AM EDT)
• Donald Dodson (10/23/2012 06:44 PM EDT)
• John Snyder (10/24/2012 02:00 PM EDT)
• Alex Stangl (10/26/2012 08:49 PM EDT)
• Seongwoo Hong (10/27/2012 09:50 PM EDT)
• Kevin Li (10/29/2012 07:05 PM EDT)
• Anunay Kulshrestha (10/30/2012 04:45 AM EDT)
• Hugo Pfoertner (10/31/2012 01:24 AM EDT)
• Daniel Lincke (10/31/2012 08:09 AM EDT)
• Sergey Grishaev (10/31/2012 09:56 AM EDT)
• Jong Wook Kim (10/31/2012 03:43 PM EDT)