You can win this number game by learning how to avoid math patterns

 You must avoid Math patterns in order to win this Numbers Game

It's your turn. Notice how if 4 is crossed out, you lose. That's three consecutively: 3-4-5. Crossing out 7 results in losing 7-8-9. You can only make safe plays by crossing out 1, 2, or 6. You have no other safe options than to cross out 1, 2 or 6.

This is a simple game that has interesting math. Another approach is to create spaces so your opponent doesn't have any choice but to make a three in a row pattern at the middle.

1 2 3 4 5 6 7 8 9

Another option is to play near your opponent and force them into completing three in a row, such as this:

1 2 3 4 5 6 7 8 9

It doesn't matter what style you play, mathematically speaking, it is certain that someone must win after six moves. Because it is impossible to cross out seven numbers without crossing out three. Try it, if you don’t believe me. This topic will be discussed in the final exercises. In this example, we will say that 6 is "upper bound" when it comes to the maximum number of moves that can ever be played in the game.

Quantized Academy

Patrick Honner is a nationally recognized teacher at high schools in Brooklyn, New York. He teaches basic concepts using the most current mathematical research.

View all Quantized Academy Columns

We know that even though we don't always know the best moves, our game shouldn't last longer than six moves. We can even extend this. An example of a game that includes the numbers 1-15 cannot last more than 10 moves. The rule is that if the size the game board is divided by 3, then only 2/3 can be crossed out. An upper bound such as this can help you understand your game. It could be used to guide you in figuring out winning strategies or to provide a reference point for what happens if the game board doesn't exceed 3. While it might not seem like much at first glance, the upper bound is an important tool for understanding similar games with different rules.

We can change the rules to make it so the loser is the first to complete three consecutive rows of any step size. You lose if you make 2-3-4 (as in the original game), but also if 1-3-5 (three consecutive steps of step size 2), 1-4-7 (step size 3) or 1-4-7 (7 (step size 3). These patterns are known as "arithmetic progressions" and consist of sequences that combine numbers with a common size (called the common difference).

Let's get back to the original game board and learn the new rules. Your turn is still available. And you've lost.

1 2 3 4 5 6 7 8 9

Crossing 1 makes 1-3-5; 2-5-8 makes 4-4-5; 3-4-5 makes 3-4-5; 6-3-6-9 makes 3-6-9, and 7 makes 8-9-9. It doesn't matter what you do, there's no way to make an incomplete arithmetic step. There's much more to remember here. The game is therefore more difficult than the original. It is also more difficult to find the upper bound of the safe moves.

The real joy for mathematicians is turning a simple game of numbers into a game against mathematics. On any size board, their goal is to determine how long it takes to play the game and not lose. This means that if you have a list of numbers, how many can you cross before your list contains an arithmetic progression. While the rules may seem straightforward, we aren't sure how to answer them. We also don't know anything about the variations of this game. Let's get to know the math a bit better. HOW TO WIN THE GAME