The Grid

A PuzzleCritic Original:

An empty 6-by-6 grid of squares is filled with numbers as follows. The first row contains the numbers {1,2,3,4,5,6} in some order. In each subsequent row, the kth number is equal to the position of the number k in the row above. For how many such grids are the first and last rows identical?




From the Tournament of the Towns:

A balance and a set of metal weights are given, with no two the same. If any pair of these weights is placed in the left pan of the balance, then it is always possible to counterbalance them with one or several of the remaining weights placed in the right pan. What is the smallest possible number of weights in the set?



A stunner from the Baltic Way:

A family wears clothes of three colours: red, blue and green, with a separate identical laundry bin for each colour. At the beginning of the first week, all the bins are empty. Each week, the family generates 10kg of laundry (the proportion of each colour might vary week to week). The laundry is sorted by colour and placed in the bins. Next, the heaviest bin (only one if there are several) is emptied and its contents washed.

What is the minimum requirement (in kg) of each bin?