UT Math Club Problem of the Month 1 February 25, 2009

UT Math Club Problem of the Month
February 25, 2009
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.
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