Chess or Mathematics? Staircase

While generally Chess and Mathematics have a vast difference, sometimes they can be very similar. Some chess puzzles give us the striking impression of a pleasingly geometrical sequence.

These puzzles are taken from an article written by the famous Bruce Pandolfini (coach of Fabiano Caruana earlier). See if you can solve them, and you will find an interesting pattern used in all of these puzzles! It is called the Staircase, since the idea is reaching the final destination by taking steps! (Remember! Sometimes you need to descend the Staircase as well! )

At the end, there is a study featured in Oliver Reeh's Chessbase column, which is an amazingly brilliant one! Can you find it? (SPOILER ALERT: Use the "rook roller" theme!")

White to play and win in all of the puzzles! THE FIFTH ONE IS A MATE IN 8 PUZZLE. I will post the answers in about 20 hours from now. (EDIT: The blog is updated with solutions!!) section. Enjoy!

(EDIT: You might still like to try out these problems, and that's why I have added the solutions in puzzle form. You can move pieces. If you want to see the solution, just click on that "?" button down the board.

I would still urge you to try the final puzzle out. If you really want to check the solution, SPOILER ALERT, @dugdugdug has mentioned it in the comments. Well done!!)

ATTEMPT THIS POSITION ONLY ONCE YOU HAVE SOLVED THE ABOVE ONES!!! White to play and win. Black's pawns are almost there..

Good luck! Please post your solutions in the comments section below.