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UT Math Club Problem of the Month February 25, 2009 1 Puzzle 1: Is There a Sequence . . . Does there exist a sequence {a1 , a2 , ...} = {an |n, an ∈ Z>0 } of positive integers such that for every positive integer m ∈ Z>0 there exist unique indices i, j ∈ Z>0 such that m = ai − aj , i.e. every positive integer can be uniquely expressed as a difference of terms in the sequence? Either produce an example of such a sequence or prove that no such sequence can exist. 2 Puzzle 2: Which Disc Doesn’t Belong? You have twelve discs. They are all identical in shape, size and feel. All of the discs have exactly the same weight except for one whose difference in weight is so slight that it is detectable only by a balance scale. The puzzle is, using only a balance scale, to figure out the least number of weighings it takes to determine (without a doubt)not only which disc has a different weight and but also whether it is heavier or lighter than the rest. Due Date: March 18 at 11:59 PM 1