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Benford process tends to minimize product

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Consider the simple question, what is the largest product of two numbers that can be formed using all the ten digits of the decimal system, each digit used exactly once. We have two answers \{96420, 87531\} and \{9642, 875310\}. Continuing on the same line, we can seek three numbers that give the maximum product and so on. In general we can ask: Given a set A of n single digits numbers, not necessarily distinct, form a set of k numbers using all the given numbers i.e. of the digit 2 occurs thrice in the set A then we can use the digit 2 exactly three times while forming our set with the maximum product. Trivially we can see that there can be one of more answer depending on weather the number 0 is a element of A because  we can form more and one set with same product by placing the 0 in the unit’s place of any number that does not end in a 0 .

Suppose that a certain random process can generate one set of k number per run using all the elements of a given a set of numbers A. Hence an exhaustive run of this process will generate all possible set of numbers with k elements. Let the set of all numbers generated by this random process satisfy the conditions of Benford Law. We shall call such a process as a Benford process and formally define it as:

Benford Process: A Benford process is a random process which generates numbers that satisfy the conditions of Benford Law.

Now if we run a Benford process once, how likely is that the process generates a set of numbers that give the maximum product. In this article we shall see that under the conditions of Benford Law, the probability that the k numbers so generated will have the largest product should be the least. In other words, numbers generated a random process which satisfies the conditions of Benford Law are least likely give the maximum product.

Theorem 1. For any give k and a set of single digits numbers A, not necessarily distinct, the numbers generated by a Benford process, using all the elements of A, is least likely to have the maximum product.

Proof: The proof of this theorem follows trivially form the following lemma.

Lemma 1Let A=\{d_1, d_2, ... , d_n\} be a set of n digits in base 10 notation, not necessarily distinct, such that 0\le d_i \le d_{i+1}\le 9. If a set M_k (A) of k elements is formed using only and all the elements of the set A then the elements of M_k (A) are given by

a_j = d_{n-k+j}d_{n-2k+j}...d_{n-rk+j}

where n-rk+j \ge 1, j=1, 2, ... , k.

Thus larger a digit, the more significant position it occupies. According to the General Benford Law, the probability of a number starting with a larger digit in the most significant place is lower than that of number starting with a smaller number in the most significant place. Hence the theorem follows.


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