On a distant planet called Hexaris, there live two alien species: the Blipnors and the Quantoodles.
The chief alien writes on a board: “There are \(100\) aliens on this planet. Of these, \(24\) are Blipnors and \(32\) are Quantoodles.”
At first this seems confusing — the numbers do not seem to add up! Then you remember that the aliens use a different base for their numeral system.
What base are they using?
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.