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

To win the Numbers Game, learn how to avoid mathematical patterns

It is your turn again. Notice that if you cross out 4, you lose. This makes it three in a row: 3–4–5. Crossing out 7 also results in loss, which makes 7–8–9. The only safe play is to cross out 1, 2, and 6. No matter which number you choose I can cross out 1, 2, or 6.

This is a simple, but interesting game. One way to do this is to leave gaps for your opponent so they have no other choice but complete a 3-in-1 row pattern in middle.

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Another approach is to play with your opponent, and force them to do three consecutive sets on either side.

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No matter how well you play, one mathematical fact is certain: After six moves, somebody has to have won. It is impossible to get seven out of nine numbers without completing three consecutive moves. If you aren’t convinced, just give it another try. This is what we’ll be discussing in the exercises. In this case, 6 is the “upper bound”, which refers to the number of moves you can play in the game.

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Patrick Honner from Brooklyn, New York is a highly-respected high school teacher. He introduces concepts from the latest mathematics research.

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Although we may not always be able to decide the best move, we do know that our game should last no more than six moves. You can go further. A game using numbers 1 through 15 cannot last more then 10 moves. In general, the upper bound can only be used if the game board’s size is divisible with 3. Understanding our game requires an upper bound. This is useful for a number of reasons. Even though it might seem small, we know a lot more about similar games using slightly different rules than what the upper bound can tell us.

Let’s take an example: Change the rules to make the winner the first person to complete 3 in a row for any step size. This means that you lose if 2–3–4 is made, just like in the original game. However, it also applies if you make 1–3–5 or 1–4–7 (step sizes 2 and 3). These patterns can be called “arithmetic progesss” or sequences of numbers with the same step size, also known as common difference.

Let’s take a look at the rules and return to our original game board. It’s your turn. And you’ve lost.

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Crossing 1 out creates 1–3–5, 2–5–8, 4–4–5, 3–4–5 and 6–3–6–9. 7 make 7–8–9. You cannot do anything without completing a cross-out arithmetic sequence. It’s important to note that this game has a lot more. This makes the game more difficult than its predecessor. It’s even more difficult than the original to find the upper limit on the number of safe moves.

For mathematicians, it’s all about turning a simple game involving numbers into a game against itself. They want to see how long you can keep playing this game on any size game board before someone loses. So, given a number list, how many numbers can be crossed out before the arithmetic progression is applied to your list? We don’t really know the answer to these simple questions. We know very little about the game’s variations. Let’s see what the mathematics can do for us. **HOW TO WIN THE GAME**