Take the numbers \(0,1,2,\dots,3^k-1\), where \(k\) is a whole number.
Show that you can pick \(2^k\) of these numbers so that, among the numbers you picked, no number is the average of two other chosen numbers.
What is the smallest number of weights that allows us to weigh any whole number of grams of gold from \(1\) to \(100\) on a two-pan balance? The weights may be placed only on the left pan.