Three ideas - a picture on the splash screen that shows swipes and above and below taps and what they do with a continue button that shows up the first few times it runs, maybe with a button also for the tutorial. Two is to put a picture like this at the start of the tutorial. Third idea is to shade the row and column on the first tap with some + and - indicators overlaid (maybe) or a text box at the bottom that indicates to tap again for +_. This may need to have a way to turn it off if the user doesn't like it.

One other thought is to make the "Solve New Puzzle" more visually distinct. It took me a few taps to figure out the screen.

One other thought is to use some sort of picture (instead of a fill color) for the tile backgrounds like many of the square puzzles do on the app store.