Below two similar problems are presented for which solutions are suggested. Certainly someone can prove/disprove the suggested solutions, and I'm fully aware that the problems most likely are well-known, or may be considered special cases of well-known problems. Apologies for any ambiguities in the statement of the problems - I'm not a mathematician.
Problem 1 --------- Consider a set of n positive integers, designed so that by combining these by means of addition - subtractions not allowed - one may obtain every integer in a range 1..m. That is, each integer in the set may contribute to in 1 out of r=2 possible ways:
1) Doesn't contribute 2) Is added
For a given n, the problem is to identify that set of integers which maximizes m (if a unique set does indeed exist). Experiments and intuition suggest the following geometric progression as the solution:
(1, 2, .. , 2^(n-1))
giving m = 2^n-1, so that each of the integers 1..m effectively may be represented by an n-digit radix r=2 number defining the combination in question.
Problem 2 --------- This time the n positive integers may be combined using either addition or subtraction, to produce the integer range 1..m. That is, each integer in the set may contribute in 1 out of r=3 possible ways:
1) Doesn't contribute 2) Is added 3) Is subtracted
For this problem another geometric progression is proposed as the solution:
(1, 3, .. , 3^(n-1))
giving m = (3^n-1)/2. So, in this case, combinations may analoguosly be given a radix r=3 representation.