One way to approach the game is to build a list of boards that force your opponent to lose. The simplest of these boards is, of course, 1. Your opponent has no choice but to take the last stick and lose. Now let's look at the board 2 2. Since the order of the rows doesn't matter, your opponent has only two choices: to take one or two sticks. If he takes one, then the board becomes 1 2, and you can take two leaving him with the losing board of 1. On the other hand, if he takes two, then the board becomes 2 and you can take one leaving him again with the losing board of 1. Now we can add 2 2 to the list of losing boards. We now have 1 and 2 2. From there we can keep adding losing boards by checking that every possible move by your opponent gives you the opportunity to create another losing board on your turn.

If you continue with this method it is possible to show that the following boards are all losing boards.

1 1 1, 1 2 3, 3 3, 4 4, 1 4 5, 1 1 2 2, 2 2 2 2, 1 1 3 3, 2 2 3 3, 1 1 4 4, 2 2 4 4, 1 1 1 1 1, 1 1 1 2 3, 1 2 2 2 3, 1 2 3 3 3, 1 2 3 4 4, 1 2 2 4 5, 2 3 4 5

It turns out that there is a more general method for determining whether a board is a losing board. For all boards except those with only one stick in each row, a board is losing if after converting the number of sticks in each row to a binary number and summing the numbers in each column, each column sums to an even number. You can see that this holds true for all the boards in the above list. For example, let's take the board 2 3 4 5. Converting the numbers to binary numbers yields: 010 011 100 101. If we then sum up each place separately we get 222. All the places sum to even numbers; therefore, it is a losing board.