  There are $n$ bags with coins that look the same. Each bag has an infinite
number of coins and all coins in the same bag weigh the same amount. Coins in
different bags weigh 1, 2, 3, and so on to $n$ grams exactly. There is a unique
label from the set 1 through $n$ attached to each bag that is supposed to
correspond to the weight of the coins in that bag. The task is to confirm all
the labels by using a balance scale once.
  We study weighings that we call downhill: they use the numbers of coins from
the bags that are in a decreasing order. We show the importance of such
weighings. We find the smallest possible total weight of coins in a downhill
weighing that confirms the labels on the bags. We also find bounds on the
smallest number of coins needed for such a weighing.
