First screen should be a nice full screen image to show while the game is loading. I will leave this to your imagination.
Second screen is first puzzle type. In this puzzle, you have two containers full of ingredients (I am not sure how the containers should look, feel free to come up with some idea, maybe a cooking pot, or a book with alchemist runes).
Both containers with ingredients are very similar, there is only a slight difference (one of the ingredients could be different, or flipped around, or with different colour). The point of this puzzle is to spot the difference between the two in a time limit. Think of it as an alchemist learning a new potion recipe and he has to get all the ingredients right. This is a classic "spot the difference puzzle". In order for this to be enjoyable, you will need to design lots of ingredients and their variations so the puzzle is different and fun over and over again.
Feel free to design as many alchemist ingredients and their variations as you can. I will leave to you how you will approach this. Some inspirations:
Third screen will be a second type of puzzle. (sliding puzzle) It will use the same ingredients / symbols designed for the first puzzle. In this puzzle the ingredients in one of the containers will be shuffled randomly and you will be required to reshuffle the ingredients to get the correct structure by moving one at a time.
This puzzle will work on a similar concept as sliding puzzle: http://mypuzzle.org/sliding
Both puzzles should look fairly similar. I have created some wireframes to explain what I mean. The wireframes are very simple and ugly, please don't follow them:
Puzzle 1: http://i39.tinypic.com/24bud8p.png
Puzzle 2: http://i40.tinypic.com/24v486h.png
Both screens with the puzzles should have a count down (there will be a limited time to solve each puzzle) and score indicator. There might be difficulty and level as well (the puzzles will get more complicated over time). Feel free to add anything else if you think it makes sense or it would look good. Also add some nice background if possible.
I might need more screens and designs in the future as I develop the game so I will work with the creator of the winning design in the future as well.