The game starts with a vertical list of the integers 1 through N, in increasing order (with N at top). During the game, Player-1 marks the left side, while Player-2 marks the right side.

Player-1 has the first turn, Player-2 has the second turn, and the players take turns alternately.

On his first turn, Player-1 marks a slash ("\") to the left of some integer, and on his first turn, Player-2 marks a slash ("\") to the right of some integer.

From then on, Player-1 may only mark a new slash ("\") to the left of Q if the following condition holds:
If Player-1 marked a slash ("\") to the left of P on his previous turn, then the absolute value |P-Q| has not been marked on the right by Player-2.

The rule for Player-2 is the same in reverse. After his first turn, Player-2 may only mark a new slash ("\") to the right of Q if the following condition holds:
If Player-2 marked a slash ("\") to the right of P on his previous turn, then the absolute value |P-Q| has not been marked on the left by Player-1.

If one player is no longer able to move, his opponent may continue as long as he can.

To make the game easier to follow, when a player marks a new slash ("\"), he turns the slash from his previous turn into an "X", by drawing a backward slash ("/") through it.

At the end of the game, Player-1 owns any integer with either an "\" or an "X" to the left of it, while Player-2 owns any integer with either an "\" or an "X" to the right of it.

A player's score is the sum of all the integers that he owns at the end of the game.

Here is a sample game, with 7 turns:

(1) (2) (3) (4) (5) (6) (7)

5 5 5 5 \5 \5 \5\
\4 \4\ X4\ X4X X4X X4X X4X
3 3 3 3 3 3 3
2 2 \2 \2 X2 X2\ X2X
1 1 1 1\ 1\ 1X 1X

Player-1 gets 2+4+5 = 11 points. Player-2 gets 1+2+4+5 = 12 points. Player-2 wins by 1 point.

Note that on turn-4, Player-2 could not mark a "\" to the right of 2 (because Player-1 has already marked 2 = |4-2| on the left). After turn-5, Player-1 could not move, so turns 6 and 7 were taken by Player-2.

I have not found a simple strategy for this game. Note that if Player-1 starts by marking an even integer, 2k, on his first turn, and then marks k on his second turn, then Player-2 cannot simply copy his moves.

It is easy to think of variations on this game. For instance, the goal could be to minimize your score, or to be the last player able to move, etc.

By the way, I would like to mention another math game I posted to cut-the-knot back on November 03, 2000, called "Odds vs. Evens". It is described in the archived link below:

https://www.cut-the-knot.org/cgi-bin/dcforum/forumctk.cgi?az=read_count&om=38&forum=DCForumID4

Some variations of that switching game might be: only switching neighboring pairs, or having Player-1 own 1 through N, and Player-2 own N+1 through 2N (instead of evens vs. odds), etc.

One thing I forgot to mention in the "Differences" game is: No integer may receive more than one mark on the same side. Otherwise, the game might not end. A "\" and an "X" are equivalent marks, but the "\" simply indicates a player's most recent move.

Basically, the distance you want to travel from your most recent position must not have a mark next to it on your opponent's side, and you can never visit any integer more than once.

Obviously, the spacing is wrong in that sample game. So I'll just describe the moves in each turn. The game is played on the integers <1,2,3,4,5>, listed vertically.

Turn-1: Player-1 marks a "\" to the left of 4.

Turn-2: Player-2 marks a "\" to the right of 4.

Turn-3: Player-1 marks an "X" to the left of 4, and marks a "\" to the left of 2. This is ok because |4-2|=2 has not been marked on the right.

Turn-4: Player-2 marks an "X" to the right of 4, and marks a "\" to the right of 1. This is ok because |4-1|=3 has not been marked on the left.

Turn-5: Player-1 marks an "X" to the left of 2, and marks a "\" to the left of 5. This is ok because |5-2|=3 has not been marked on the right.

Turn-6: Player-2 marks an "X" to the right of 1, and marks a "\" to the right of 2. This is ok because |2-1|=1 has not been marked on the left.

Turn-7: Player-1 cannot move, so Player-2 moves again. Player-2 marks an "X" to the right of 2, and marks a "\" to the right of 5. This is ok because |5-2|=3 has not been marked on the left.

The game is now over, because neither player can mark any "open" number on his side ("open" meaning that the number has not been marked on his side).