## The Four Numbers Game

Today in class I gave the students a problem based lesson centered around the four number game.  This game starts with a square and four non-negative integers placed, in any order, at the vertices of the square.  One then constructs a new numbered square by connecting the midpoints of each edge.  Next, place the absolute value of the difference of the adjacent vertices at the midpoint of each edge.  Repeat and continue with each new numbered square.  Here is an exampe:

I used this game to reinforce the idea of how mathematics is developed.  After giving the students an example of the game, I asked them to spend about 15 minutes playing the game.  At the end of the 15 minutes, the students had to present three things to the class:

1. A new question about the game;
2. A conjecture about the game;
3. Some result showing why their question or conjecture is true.

While no one really had results to share, the  class did come up with some very good questions.  Here are some examples:

• Will the game always end?
• What is the largest number of squares you will draw? (i.e. the above game has 6 squares)
• Is there a pattern to the numbers that appear in the game?

These types of questions may seem like the obvious ones to ask, however, I would like to point out that many older students fail to see the obvious.  I find encouragement in seeing these students spending time to work the problem and think about how to expand it with meaningful questions.  If given more time, I am sure the students could take this game  much further.  This may be my them for the next couple of weeks.

As to the answers to the questions, the first and third have affirmative answers and the second has a linear bound.