You are given a grid filled with numbers. If a number $n$ is orthogonally adjacent (horizontally or vertically) to another number $n$ then you can pick it up and place it on top of the second number. When you do that, the two numbers will merge into $n+1$ and the original cell will become empty. Starting with the following grid, can you make a series of moves to obtain a single number?
There are 2x2 solutions.
In the picture below, each column represents a solution that leads to a pair of fives, and each pair of fives can be resolved in 2 ways (depending on which 5 you move on the other 5).
In the first row, the red arrows show the first moves that merge the 1s into 2s, the blue arrows show the second moves to merge the 2s into 3s. In the second row, the red arrows merge the 3s into 4s and the blue arrows merge the 4s into 5s.
I'll update this answer later with my reasoning which proves these are the only solutions.
I believe I can do it:
First, we realise that all the ones will merge into 2's where the 2's will merge into 3's and etc. So, we first merge the 1's in a way that the 2 created touches another 2, so we can make it into 3.
So first, the original picture:
I believe 5 moves.