Consider a partizan subtraction game with the subtraction se

Consider a partizan subtraction game with the subtraction set for the first player S1 = {1, 3, 4}, and the subtraction set for the second player S2 = {1, 2}. Find all the winning positions for the first player. Justify your answer.

Solution

Subtraction set for first player S1={1,3,4}

Subtraction set for second player= S2={1,2}

Under this game

player alternate moveswith player i starts.

Last player to move wins.

So we have to find all winning position for the first player.

Hence we have to decide on which positions winning of first player is certain

If first player left 3 chips for second player. then he can remove 1 or 2 chips and after that 2 or 1 chip will be remained for player first and he will win

If first player left 4 chips to remove for the second player than second player can remove 1 or 2 chips from it, then in bothe cases 3 or 2 chips will be remained to move for 1st player thus first player will win after removing it.

Hence \"4\" is the winning position for first player.

Now we check for 5.

If first player left with 5 then second player can remove 1 or 2 chips

Case 1 : if 1 chip was removed by second player than 4 chips will be left for first player.

first player will remove 1 chip from 4 and left 3 for second player then first player will win

case 2: if 2 chips were removed by second player then first will win by removing 3 chips.

Now if first player left 6 chips then after removing 1 or 2 chips 5 or 4 chips remained for first player respectively.

Then 1st player will take away 1 chip then 4 or 3 chips will be remained for second player and from the above case we can say first player will win.

Thus winning position for first player are 4,5,6,7,