Shove Off

Shove Off is a 1984 game by Hank Atkins, published at Ideal.

The game is played on a 4x4 board. Each player has eight pieces, and there's also 16 neutral pieces.

Rules:

• Initially players place all 16 neutral pieces on the board
• On her turn, the player adds a friendly stone at the board's edge, by pushing all pieces on that row or column (so, the last pieces is pushed off the board)
• however, it is illegal to push off an opponent's piece
• if the player does not have a friendly piece to push (all eight pieces are on the board), then she uses one neutral piece instead
  
• Wins the player that makes a 4 in-a-row orthogonally or diagonally
  

The moku ludeme is quite common, and the pushing element is not original (cf. Overboard). What is different here is that players are unable to push off enemy pieces.

Here are low quality pics of the original rules:

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A similar idea was proposed independently by Steve Bateman, when he posted at rec.games-abstract, in Aug 25, 1995. Here's the message (slighty edited):

I was looking through the games on Richard's PBeM server when I had an idea for a new game, called "Push". I have attached a copy of the rules, and would appreciate any feedback on the game, especially on:
1. Has anyone ever heard of, encountered, played a game like this before?
2. Is the game concept OK - would you want to play a game?
3. How well does it work in two-player, three-player or four-player mode?

The Game of Push
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Description
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Push is a two-, three- or four-player game played on an 8x8 square board, each player playing with counters of a single color. The aim of the game is to have a line of your color appear on the board.

Rules
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1) The board is an 8x8 square, and is empty at the start of the game.

2) The game may be played by two, three or four players.

3) Each player has an unlimited supply of counters of a single color. With two players, player one plays white, player two plays black. With three players, the third player plays red. With four players, the fourth player plays green.

4) Play begins with player one, and proceeds in turn through player(s) two, three and four, then back to one again (and so on).

5) Each player must make one move on their turn (there is NO passing).

6) A move consists of "pushing" a counter onto the board from an edge position. The push may be horizontal, vertical or diagonal, but must be away from the pushing edge toward another edge of the board (so from the top, the push can be toward the right, the bottom or the left - SE, S, or SW on a compass - but cannot be parallel to the edge - E, W - nor away from any other edge - NE, N, NW).

7) "Pushing" a counter onto the board may move other counters already on the board as follows:
  a) If the edge position is empty, then no other counters are moved.
  b) If the edge position is not empty, then all connected counters in the
     direction of the push are moved one square in that direction.
  c) Only connected counters are affected by the push.

8) Any counter pushed off the edge of the board during a move is returned to the player playing that color (there are no captures).

9) After each move, the board is examined for lines (horizontal, vertical or diagonal) of a single color. With two players, the lines must be five counters long, with three or four players, the lines must be four counters long. A player wins when a line of their color appears on the board, as follows:

a) If a player makes a line of their color as a result of their move, that player wins.
b) If a line of another other color is made as a result of the move, then the player playing that color wins.
c) If lines of two or more colors are made at the same time, then the player whose turn comes next wins, starting with the player who made the most recent move (so if player one moves, and makes two lines, one for player three and one for player four, player three wins - players one and two do not have a
line so do not win).