All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Special dice (Posted on 2011-10-30) Difficulty: 4 of 5
Two six sided dice are marked with 12 different integers, so that any number from 1 to 36 can be derived when both dice are thrown and the showing face-up values are added.
What is the lowest possible value of the highest number on one of the dice faces?

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.0000 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Possible solution | Comment 2 of 18 |

Lovely problem.

Take the known solution: f(x)=(x^1+x^2+x^3+x^4+x^5+x^6)(1+x^6+x^12+x^18+x^24+x^30)

The trick is to re-partition this into two factors of six elements, something like (x.x.x.x.x.x)(x.x.x.x.x.x) with each partition having 6 factors. Trial and error suggests some guidelines:

1. Terms including a minus sign have to be multiplied by something similar to eliminate the negative terms.
2. Where possible match up the smallest available factor(s) to the largest.


These guidelines though not rigorous are sufficient for present purposes, particularly as otherwise there are hordes of possible combinations.


Start by combining (1-x+x^2) (1+x+x^2) to obtain (1+x^2+x^4), This again matches with (1-x^2+x^4) to produce 1+x^4+x^8: similarly, (1-x^3+x^6) (1+x^3+x^6)= 1+x^6+x^12, multipled by (1-x^6+x^12)= x^24+x^12+1.

We have left x (1+x) (1+x^2). The second two factors cannot be combined as they would produce too many terms, so we have either x(1+x) and (1+x^2) or  x(1+x^2) and (1+x). In due course we find: (x^25+x^24+x^13+x^12+x+1)(x^11+x^9+x^7+x^5+x^3+x).

Even though the guidelines are not rigorous, they are good enough in this case to ensure that no smaller solution is possible.

 The exponents give the numbers on the dice.
So the two dice are numbered: (1,3,5,7,9,11)( 0,1,12,13,24,25)

Note: 26 and 27 are also possible solutions.

Edited on October 30, 2011, 11:38 am
  Posted by broll on 2011-10-30 11:34:43

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information