# Strategy

A condition of chocolate games are determined by the following two conditions.

In the above calculation the symbol made by plus and a circle is nim-sum. Nim-sum is addition without carries for numbers expressed in base 2. Nim-sum is the same as the "exclusive or" in computer science. **If you leave chocolate whose cooridinate is nim-sum 0 to your opponent 0 each time, you will win the game.**

#### What is nim-sum？

Nim-sum is the addition without carries for numbers expressed in base 2.

#### What does it mean to express a number in base 2？

In our daily life we express a number with 0,1,2,...,9, so we use decimal numbers.

When we use binary number, we express a number with 0 and 1.0.

To convert a decimal number to a binary number is not difficult.

For example, the decinal number 22 is 2^{4}+2^{2}+2^{1} so we put 1 in 2^{4} place, 2^{2}place and 2^{1} place. We put 0 in other places. Then we have the binary number 10110.

#### Example of 10 digits number.

#### The calculation of nim-sum

We caluculate nim-sum in two steps. First express numbers in base 2, then we add them without carries.

If the number of "1" is even in each column of 2^{n} place, then the nim-sum is 0.

#### Winning way

If you leave the condition that nim-sum = 0 to your opponent each time, you finally win the game.