Take a common die (1 to 6 dots).
Assign to each vertex a number corresponding to the product of numbers denoted by the intersecting faces.
Clearly, the sum of those 8 products is 343.

I have modified the die by changing the number of dots on some (or all) of the faces and the new sum of the products is now 1001.

a. (d2) Find the sum of the new numbers (given they are distinct) - unique answer.
b. (d3) What possible sets of 6 distinct positive integers could enable the new sum?
c. (d3) How should the new numbers be distributed on the modified die?
d. (d3) How many distinct "new dice" exist?

Charlie's done all the hard work, so I am just swooping in and answering part b.

1 2 3 4 10 11 
1 2 3 4 9 12  
1 2 3 5 8 12  
1 2 3 6 8 11 
1 2 3 6 9 10  
1 2 4 5 7 12  
1 2 4 5 9 10  
1 2 4 6 7 11 
1 2 5 6 7 10 
1 2 5 6 8 9 
1 3 4 5 6 12  
1 3 4 5 8 10  
1 3 4 6 7 10
1 3 4 6 8 9
1 4 5 6 7 8
2 3 4 5 6 11 
2 3 4 5 7 10
2 3 4 5 8 9
2 3 4 6 7 9
2 3 5 6 7 8 

Posted by Steve Herman on 2015-02-18 14:57:20