The game of NIM

Your book, Maths Mela (Chapter 10), starts with a game for two players called NIM. Player 1 says 1 or 2. Player 2 adds 1 or 2 to it, and says the new total. The players keep taking turns, adding 1 or 2 each time. The player who reaches 10 first is the winner.

Is there a clever way to play? Think backwards from 10. If the total is 8 on your turn, add 2 and you reach 10. If it is 9, add 1 and you reach 10. So if the other player reaches 8 or 9, you win straight away.

Now think about 7. If YOU reach 7, the other player must add 1 or 2 and make 8 or 9. Then you reach 10 and win! So 7 is the magic number: the player who reaches 7 can always win.

So what happens if the other player has reached 6 and it is your turn? Add 1 to make 7, and you can be sure to win. But if the other player has already reached 7, you are stuck: whatever you add, you make 8 or 9, and they reach 10 first.