Ways of Counting
Here some strategies for making sure we had all of the colorings of 3x3 grids without repeats.
Dor Abrahamson contributed a template for counting 3zees (9-blocks with exactly three greens) using the method of anchors and movers.
One way to keep track of the different cells in the 9-block is to number them like this:
On the 3zees template, you will find 3x3 grids with two cells filled in with their number. The rest of the cells are blank. Below is an example.
This kind of grid is called a template because it is used to create other 9-blocks. In the template above, cells 2 and 4 are "anchors". To create a 9-block from this template, color cells 2 and 4 green. Then, you can color any one of the other cells green to create a "3zee". Here a a few 9-blocks you could generate from this template:
How many 3zees could you generate from the 2-4 template?
One important part of counting combinations is making sure that we count all possible combinations. How could this method of counting help us make sure that we get every 9-block combination?
Does the 3zee template show every possible 2zee? How do you know?
Another important part of counting combinations is making sure that we don't create duplicates which lead us to counting the same combination twice.
Could the same 3zee be created from two different templates?
Try it. See if you can find a 3zee that can be created from both of the templates below. The first template requires that the first and second cells be green. The second template requires that the first and third cells be green.
How can you make sure you don't create duplicates using templates? Suppose that your group created all the duplicates by accident. How many extra blocks would you have?
How would you use this method to create all the 4zees?