For now I'll share with first one, as I didn't make image for second - I'll do it later.
Here it is: SOCCER!
(EDIT 2: After I made an explanation, I found this: http://en.wikipedia.org/wiki/Paper_Soccer it may help provide additional explanation...)
Sequential/alternating, turn-based, two-player board game, map can be in different sizes but there are some limits (goal-post is exactly 2 units, from goal post to corner minimum 2 units, so it can't be slimmer by 6 units in total)
Goals:
1. Enter enemy's goal-post
2. Create situation in which enemy cannot move anymore
Mechanics:
1. Field is filled with matrix of dots, player moves from dot to dot
2. Moving player leaves a trace or "path"
3. Player cannot move along the existing "path"...
4. But he can "bounce" from it
5. He can also bounce from field/map boundaries
6. If he gets into corner (from where he cannot move anymore), he loses
7. If he cannot move at all, he loses
8. When player bounce, he can move farther. This way he can block return path for enemy (for example, he can create such lattice around his own goal post, so enemy can't enter it. There is still possibility that enemy drives him into corner - so it's not an immediate winning condition - but he has higher probability of winning) or run away from his goalpost.
EDIT: I can't explain it in a better way. When player meets with already made "path" or hits field boundary, he can (or rather he must) continue to move. So in one turn he can issue more than one move. When he goes into empty space, he cannot move anymore.
Here's an image:
Row 1, beginning of the game:
1. Initial, empty map, 8 units wide and 12 units long
2. TURN 1: Blue player starts, he might want to get closer to red's goal post
3. TURN 2: Red goes down, away from his goal post
4. TURN 3: Blue goes left, "bounces" from his old path and goes up diagonally
Row 2, more advanced game closer to red's goal post (it's not an real example since it doesn't show path from beginning of the game, but still a good example):
1. Blue enters a hole in the lattice
2. Possible move: If red goes into the corner, he LOSEs...
3. ...he cannot do it so his only way to continue game is to break free just by his goal post (at this point he didn't yet lose)
4. blue can enter goal directly or he can bounce from post into it. Either way, blue WINs (but he can bounce away from goal, which wouldn't make sense at this point...)
So, what do you think?
(I'll make image for second game later today or tomorrow, it is a bit more complicated... maybe I'll find some archival notepad with real example of the game)