Every Tapdown board can be solved by reasoning, not guessing. These are the ideas that get you to par, from quick wins to the method the game's own solver uses.
A tile's number is exactly how many taps it needs from itself and its neighbours put together. A tile showing 3 needs three taps among: itself, the tile above, below, left and right. Nothing else can touch it.
So if a 3 sits next to two empty tiles, those two can't be tapped, and the three taps must come from the 3 itself or its remaining neighbours.
A corner tile has only two neighbours, so only three taps can ever lower it. Edges have four, and middle tiles five. Fewer options means less guessing, so read the corners first, then the edges.
Wall rule. If a corner shows 1 and one of its neighbours is empty, the corner can't be tapped (it would touch the empty tile). Only its other neighbour can clear it.
A tap in the middle removes 5 from the board, a tap on an edge removes 4 and a corner tap removes 3. Add up every number on the board and divide by 5: par can never be lower than that. If a board adds up to 40, you need at least 8 taps. If par is much higher than the total ÷ 5, expect a lot of edge and corner taps.
This is how the game's solver finds par, and you can do it in your head on small boards.
There are only a few sensible top rows, because a tile can't be tapped more times than the smallest number around it. That keeps the search short. Among all the top rows that work, the one with the fewest total taps gives par.
Top row of taps here is 2, 0, 0, 0. Once that is fixed, every other tap in the picture follows from rows above it, which is why there is nothing to guess after the first row.
A tile with a number but every neighbour empty can only be cleared by tapping itself, which is blocked by the empty neighbours. That board is now stuck and needs undos. Before you empty a tile, check that none of the tiles around it still depend on it.
Because order doesn't matter, a tap you know belongs in the solution is safe to make now. Lock in the forced taps first, and the rest of the board gets smaller and easier to read.
Read the corners and walls first for forced taps, then use the row-by-row method: fix the top row, and every row below is forced.
A tile still has a number, but it, or a tile it depends on, is next to an empty tile. Undo the taps that emptied that neighbour.
Not always. Some boards have more than one set of taps that clears them. Par is the smallest of them.