First, doing some simple image editing can help you see your goals more clearly.

1. Take a screenshot.
2. Drop it into MSpaint or an image editor of your choice.
3. Use rectangle select on the areas of the board that need to be made a particular color (choose either blue or orange)
4. Invert the colors (I used GIMP and Hue/Saturation to just rotate the hues 180 degrees, same effect.)

Now the puzzle looks like a single color puzzle! You'll find that the moves to solve this modified puzzle are the same as the original.

The game runs on process of elimination, and if you've gotten midway into the game, you'll have had some practice here. To keep track of my moves, I diagrammed screenshots as follows:

Definite moves: If there is a square that needs to be flipped at least once, and only one move of one number can reach it, that move is a definite move. Find all of these moves and make them before continuing.

Assigned moves: If there is a square that needs to be flipped at least once, and only one number (with two or more of its moves) can reach it, that move is assigned to that number. (Marked in red lines, with dots over the needy square)

Avoid: If any move would flip a square to a wrong color, which could not be unflipped by any other move, that move should NOT be taken. (Marked with red X's) This can help determine limits for a number.

Busy numbers: Suppose there are three assigned moves from three different directions of a 3 square. Then it's safe to put a red X on the fourth direction. Likewise, if a 2 has two assigned moves in two directions, it will not go in the other two directions.

Parity: Certain isolated-ish numbers will have an even or odd number of wrong tiles immediately to all four (or three or two) sides of it. Since every move from that number must flip one of these adjacent tiles, it's reasonable for the even/oddness (parity) of the number to match the number of wrong tiles. If this is not the case, then you know that some other number has to make a move that flips one of those wrong tiles.

Wasting a move: Sometimes this is necessary to solve a puzzle. If there's a big number near an edge, making two moves toward that edge will reduce the number by two and leave the puzzle unchanged. Always count this as a possibility.

Leftover tiles: You'll find that you do not have to spend all the numbers to complete the puzzle! If the colors match the borders and you still have some 2's out there, the puzzle will still be counted finished. With that in mind, you can save 2's, 4's or 6's, leaving them on the board in case they are needed to fix a color.

Flexible tiles: (cyan circles) This is a big one, and the leftover 2's is the simplest example. Any tile two tiles away from a leftover 2 is automatically a flexible tile. Often, big numbers can be exhausted in two or more different ways, but the difference between the outcomes is a single tile. You can note this tile and continue the puzzle until you know whether or not that tile should be flipped.

Once you've mapped out all the moves you can, you should be able to logically work out some moves. Then repeat this process - screenshot, invert colors, diagram it up. After enough of these, the rest of the puzzle should be solvable by instinct.

Here was my puzzle 49, which I spent quite some time on, and it serves as an example of all of the above strategies. (All puzzles are randomly generated so this isn't a spoiler.)


I started by inverting the colors (and kept them inverted for the whole puzzle)..
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Then I did some definite moves. The 3 in the second row would flip a square on its first move, and that square can only be flipped back by the same number, ONLY when it's a 1. Since this 3 is also assigned to move down, it must spend its 2 move down. Then, the 2 in the next-to-last row has only one assigned move, but it must make another move in a different direction (otherwise it will undo its assigned move.) Elimination leads to moving this 2 up, then left.



The logic here goes row by row. In the first row, the 5 is forced to make four moves left. Once this is done, it becomes clear that the 2 in the third row now has two assigned squares, and therefore cannot move down. It's a busy number. In the fourth row, the 6 must move left, and in the fifth row, the 5 must move left (since the 2 above it cannot move down.)
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From here, there are no definite moves. The other strategies come into play. That means lots of diagramming. :P
This is a great time to mention: If you've followed all the logic and can't solve it, one of your assumptions is wrong! I was certain I'd gotten an unsolvable puzzle all because I made a faulty assumption about the rightmost 4. It was very easy to suppose it could only be cleared by moving down, right, down, left. But in fact, it could affect tiles further to the left. This was the key to the solution, and a good example of flexible tiles.
Other things to notice in this picture:
W's indicate directions in which a tile can waste a move.
X's indicate tiles that are off-limits to all moves. Most of them are because the only number that can reach it cannot also flip it back. They also help to fence in the puzzle.
The 3 in the bottom row is assigned to move left and right. But its remaining move MUST be up! This is because if it moved left or right again, it would unflip a tile that nothing else could reach.


I wanted to focus on the leftmost 4. Its 2 and 1 moves were assigned, so it had only to spend its 4 and 3 moves in the other directions. Branching through possibilities...
If it went 4 up, it would need to move 3 up immediately after. (A good example of how, in general, a number N can flip a tile N squares away from it using its first two moves.) The problem here is that the 5 in the fourth row would need to unflip it by moving left twice, and now it no longer has a 4 move to flip its assigned tile 4 squares below it. So this possibility is out.
If it went 4 down, it would also have to move 3 down to unflip the tile it flipped, in effect, wasting a move. This is a possibility.
If it went 4 right, again, it would need to move 3 right. This flips a tile otherwise reachable only by the 3 in the bottom row. So right-right isn't strictly necessary, but it could be useful. The tile it points to is now a flexible tile. Either it will flip this tile or waste a move before clearing to the left. So we set it aside.


I tried many different paths from here. The first one that led to results was the 3 in the bottom row. It could not move 3 left, because as I discovered, it throws off the parity of the 5 and 6. So should it move 3 right? To answer that, I investigated the 6 in the sixth row.
I'd stared at it long enough to determine that its first two moves would decide its next few moves. It was also in a good position to waste moves. So I tried out all sixteen combinations of this six. I'll spare you the calculations, but what I found was that there were only a few options that didn't lead to a mess:

-6 left, 5 left
-6 left, 5 right
-6 right, 5 left
-6 right, 5 right (Waste a move)

Not only did this prove the 3 in the bottom row has to move right (and therefore the leftmost 4 can be cleared), this also showed that two of the tiles to the left of the 6 were independently flexible!

Also, the 4 in the fifth row also was great for dissecting, since it could not move 4 up or left, and it was assigned a tile 3 to its right. Moving 3 up was also off limits since it throws off the parity of the 5 above it. So this left the following options:
-4 right, 3 down
-4 down, 3 right.
-4 right, 3 left,
The first two possibilities create a flexbile tile, while the third leads to its own thread which I chased for a while but couldn't make it work nor prove it incorrect.

This is enough to demonstrate all the strategies. As in many puzzles, the remainder of the solution was mainly "throw attempts at the wall until you stumble on the right answer and have no idea how you did it." I've got the screenshot and achievement to show that I eventually did, but I can't recall the moves. :P

I hope this is helpful and mostly straightforward! If something doesn't make sense, I'll be glad to answer questions.