The pebble game

In your book, Maths Mela (Chapter 1), Sanju and Mira are still on the train. Sanju teaches Mira a game called 'pile of pebbles'. There are two piles with 7 pebbles in each. On your turn you pick as many pebbles as you like, but only from ONE of the piles, and you must pick at least one. The player who picks the last pebble wins.

How can you always win? The book says: try small piles first. Start with 1 pebble in each pile. The first player must take one of them, and then the second player takes the last pebble and wins. Now try 2 in each. If the first player takes 1, the second player takes 1 from the OTHER pile, leaving 1 and 1, and wins as before. If the first player takes 2, the second player takes the other 2 and wins.

Do you see the secret? Whoever gets to leave the two piles EQUAL can always win. When the piles are equal, your friend has to make them unequal, whatever they take. Then you make them equal again by taking the same number from the other pile. The piles keep getting smaller, and in the end you are the one who picks the last pebble.

With 7 and 7 at the start, the piles are already equal. So the player who goes SECOND can win by copying: whatever the first player takes from one pile, the second player takes the same number from the other pile. If the piles are ever unequal on your turn, take pebbles from the bigger pile until the two piles are equal.