Tennis (paper game)

For a simple description, we include the numbers -3 and 3 to denote the ball being beyond the second field. This results in a playfield as (-3,-2,-1,0,1,2,3).

The draw of player i at time t will be denoted as ${\displaystyle S_{i,t}}$, the ball is at time t in ${\displaystyle B_{t}}$.

At start is ${\displaystyle B_{0}=0}$ and for both players is ${\displaystyle S_{i,0}=50}$.

Each player chooses a whole number ${\displaystyle 0\leq Z_{i,t}\leq S_{i,t-1}}$, with ${\displaystyle Z_{i,t}=0}$ only if ${\displaystyle S_{i,t}=0}$. The number reduces the points of this player (${\displaystyle S_{i,t}=S_{i,t-1}-Z_{i,t}}$).

The ball is moved as follows:

• If ${\displaystyle Z_{1,t}>Z_{2,t}}$, then the ball is moved towards player 2.
  • If the ball was on a field of player 1, it will be moved over the centre line (${\displaystyle B_{t}=1}$).
  • Otherwise, the ball will be moved one field towards player 2 (${\displaystyle B_{t}=B_{t-1}+1}$).
• Symmetrically, the ball will be moved towards player 1 if ${\displaystyle Z_{1,t}<Z_{2,t}}$.
• If ${\displaystyle Z_{1,t}=Z_{2,t}}$, the ball remains where it is.

For the result of the game, the following rules apply:

• The game ends if the ball is moved beyond the field of any player, of if both players do not have any points left.
• In both cases, the game is lost for that player on whose side the ball is.
• If the game is played repeatedly, it can be defined to give two win points if the ball is moved beyond the field, otherwise only one point.

In the first example, player one wins, but the ball is still on the playfield when both players do not have any points left.

t	Player 1 Draw ${\displaystyle Z_{1,t}}$	Player 2 Draw ${\displaystyle Z_{2,t}}$	Player 1 Status ${\displaystyle S_{1,t}}$	Player 2 Status ${\displaystyle S_{2,t}}$	Ball ${\displaystyle B_{t}}$	Comment
0			50	50	0	Start
1	5	10	45	40	-1
2	5	10	40	30	-2
3	15	10	25	20	1
4	15	10	10	10	2
5	10	10	0	0	2	Player 1 wins

In the second example, player 1 wins again, by hitting the ball beyond the other playfield using the last points he has left.

t	Player 1 Draw ${\displaystyle Z_{1,t}}$	Player 2 Draw ${\displaystyle Z_{2,t}}$	Player 1 Status ${\displaystyle S_{1,t}}$	Player 2 Status ${\displaystyle S_{2,t}}$	Ball ${\displaystyle B_{t}}$	Comment
0			50	50	0	Start
1	11	3	39	47	1
2	1	10	38	37	-1
3	15	11	23	26	1
4	1	9	22	17	-1
5	3	6	19	11	-2
6	11	3	8	8	1
7	4	3	4	5	2
8	1	5	3	0	-1
9	2	0	1	0	1
10	1	0	1	0	2
11	0	0	0	0	3	Player 1 wins