This question is about the sum of a subset of tiles mid-game.
Consider the following boards from real games:
In each case, the top row is filled and the rest of the board (tiles on rows 2 to 4) sums to 6.
In a real game, is this sum minimal, or is it possible to reduce the sum to a number less than 6? Note that the top row must be filled.
(Please supply a proof in each answer. There's no need for spoiler tags.)
EDIT: As @Deusovi and @Trenin have shown, this can be done almost trivially.
So for something a little meatier, we will only look at the sum (of the lower 3 rows) when the top row is in strict descending order, with all top-row tiles greater than 2. We will assume that newly-spawned tiles are all '2's, and you can nominate where each tile is spawned. Once spawned, though, the tiles must move and merge according to the rules of the game. You may start with either of the above boards, or start with the standard 2-tile opening, but from there, you'll have to explain how you got to the final layout, or explain why no sequence of moves can produce a sum less than 6.