Mathematical model of an unbiased and random arrangement of elements in an inherently biased die

Abstract

Systems and methods are disclosed herein to a die comprising a plurality of faces (N); and a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing an element from the unbiased set of elements after every (N−n)/n occurrences of an element from the set of biased elements, when a count (c i ) of each unique element in the unbiased set is equal to 1.

What is claimed is: 
     
         1 . A die comprising:
 a plurality of faces (N); and   a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing an element from the unbiased set of elements after every (N−n)/n occurrences of an element from the set of biased elements, when a count (c i ) of each unique element in the unbiased set is equal to 1.   
     
     
         2 . The die of  claim 1 , wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased. 
     
     
         3 . A die comprising:
 a plurality of faces (N); and   a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing ((N/n)−c i ) elements from the biased set of elements after ever (N−n)/n occurrences of an element from the unbiased set of elements, if a count of the elements in the unbiased set less the elements having an identical value as the elements in the biased set of elements is greater than a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set.   
     
     
         4 . The die of  claim 3 , wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased. 
     
     
         5 . A die comprising:
 a plurality of faces (N); and   a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing an element from the unbiased set of elements after every (N−((c i +1)*n))/n occurrences an element from the biased set of elements, if a count of the elements in the unbiased set less the elements having an identical value as the elements in the biased set of elements is less than or equal to a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set.   
     
     
         6 . The die of  claim 3 , wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased. 
     
     
         7 . A die comprising:
 a plurality of faces (N); and   a plurality of elements (n), including an unbiased set of elements and a biased set of elements, that are labeled on the plurality of faces, wherein each face is labeled with one of the plurality of elements, at least one element is labeled on more faces than the other elements, and all the faces are labeled in an unbiased manner such that the faces are labeled by placing ((N/n)−n) elements from the biased set of elements after every (N−n)/n occurrences of an element from the unbiased set of elements, when a count (c i ) of each unique element in the unbiased set is equal to n.   
     
     
         8 . The die of  claim 7 , wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased. 
     
     
         9 . A computer-implemented method of labeling elements on an inherently biased die having a plurality of faces (N), comprising:
 determining, by a computer, whether the count of each unique element (c i ) in an unbiased set of elements is greater than 1;   applying, computer, a first placement means if the c i  of the unbiased set of elements is equal to one;   determining, by a computer, whether the count of each unique element (c i ) in an unbiased set of elements is equal to a number of elements (n);   applying, by a computer, a fourth placement means if the c i  of the unbiased set of elements is equal to n;   determining, by a computer, whether a count of the elements in the unbiased set less the elements having an identical value as the elements in an biased set of elements (c1) is greater than a count of the elements in the biased set plus all elements in the unbiased set having the identical value as the elements in the biased set (c2);   applying, by a computer, a second placement means if c1is greater than c2; and   applying, by a computer, a third placement means if c1 is less than or equal to c2.   
     
     
         10 . The method of  claim 9 , wherein the first placement means comprises a first placement module that places an element from the unbiased set of elements after every (N−n)/n occurrences of an element from the set of biased elements, where N is the number of faces on the die and n is the number of unique elements included on the die. 
     
     
         11 . The method of  claim 9 , wherein the second placement means comprises a second placement module that places ((N/n)−c i ) elements from the biased set of elements after ever n)/n occurrences of an element from the unbiased set of elements, where N is the number of faces on the die and n is the number of unique elements included on the die. 
     
     
         12 . The method of  claim 9 , wherein the third placement means comprises a third placement module that places an element from the unbiased set of elements after every (N−((c i +1)*n))/n occurrences of an element from the biased set of elements, where N is the number of faces on the die and n is the number of unique elements included on the die. 
     
     
         13 . The method of  claim 9 , wherein the fourth placement means comprises a fourth placement module that places ((N/n)−n) elements from the biased set of elements after ever (N−n)/n occurrences of an element from the unbiased set of elements. 
     
     
         14 . The method of  claim 9 , wherein the unbiased set of elements is the set of elements that all have the same probability of being rolled, and the biased set of elements includes the elements that make the die biased. 
     
     
         15 . The method of  claim 9 , wherein the N is exactly divisible by n, and [N<=(n2+n)].