Years ago I wrote a book (in English) full of abstract paper games: "100 Strategic Games for Pen and Paper", (Walter Joris), Carlton, London. It was translated in several languages. Several titles also now. Most of the games were invented by me. It got a lot of response. I cannot publish them here, since the editor holds the copyright. But meanwhile you can find several on the Internet.
However, here I have a game I invented that does not occur in the book. It's simple, but that can be the charm of a game: as a pastime.
We play on a board of 10 x 10 on squared paper. In turns the players put down somewhere an own mark: a circle or a cross. They do this until each has eight marks on the board. Then the games starts. With each move you now choose a direction: up, down, left or right. If for example Cross chooses down, then he moves his whole group of crosses one square downwards. This means that under each cross he can put one extra. But this of course only if that square is free. Then the opponent is on turn. Each time the direction is freely chosen. This goes on until the whole board is filled. Whoever can't play any more has to pass, and let the other play on. Winner is the one with the most own marks.