Question

You are playing a game with a very worthy opponent named Morgan Von Neumann (VnM) -a game theorist who is familiar with all the tricks known to the people who practice that dark art. These are the rules of the game:

• When the game begins there are 22 pins planted on the ground
• You take turns playing this game.
• When it is your turn you have to take either 1, 2 or 3 pins from the ground. You cannot pass on your chance.
• VnM when it is his turn also chooses to take either 1, 2 or 3 pins form the ground.
• The person who gathers the last pin from the ground is the loser of the game.

You have won the toss and have to make the first move. What will be your first move? Will your strategy change if there are 21 pins instead of 22?

Answer 1

Solution:-

According to the rule of the game "The person who gathers the last pin from the ground is the loser of the game".

So at the second last move if I left 5 pins then VnM can take either 1,2 or 3. In that cases VnM left with either 4,3,2... pins. So at the last move I can take either 3,2,1 and left with last pin for VnM and hence I will win.

For 22 pins I will group them as follows (Digit in bold shows at which I will try to left fro VnM) .

1,2,3,4,5 ,6,7,8,9, 10,11,12,13, 14,15,16,17, 18,19,20,21, 22.

So in the first move I will take 1 pins and left with 21 for VnM. Now whatever choice VnM makes I will take in such a way that I will left with 17 pins for VnM... Next I will left at 13 then 9 and at last with 5. So At the last as explained I will surely despite of whatever choice VnM made. (So by this strategy I can dominate in each step).

So my first move will be to take 1 pin.

Now if the pins are 21 instead of 22. Then there is high chance that I will lose because VnM is a game theorist. So whatever pins I will chose he will apply the same strategy as explained above and he will left with 17 , 13, , 9 and at last 5. So I have to chose last pin and will lose.

My first move could be any (1,2 or 3), VnM (game theorist) will win.