Layers investigation

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Layers

I am carrying out an investigation to find out the different arrangements of cubes on a specified grid size. I will first start off with a two by three grid size which means there are six squares in the grid. On these six squares I will put five cubes. Each cube must fit exactly onto one square. During the course of my investigation I will display and describe my work and findings.

I first investigated how many different arrangements of the five cubes there were on a 2 by 3 grid. The 2 by 3 grid obviously has 6 squares; as one square always has to stay blank, the other five squares will be filled in. Because there are six squares and one square is always blank, there are six different variations.

For the second layer, there are only five squares, one of which has to stay blank. Because of this four squares can be filled, and this produces 5 different arrangements.

As you can see the total number of different combinations is 30. This can also be worked out by saying that there are five different arrangements on the second layer and there are six different arrangements on the first layer. So if you calculate 5*6 you get the answer 30.

There is a theory behind this to find out the number of different arrangements without drawing the layers:

"The number of squares filled in is always -1 of the number of arrangements and the number of possible empty squares."

To test this theory out I did a made another grid of 3 by 2. Although this is the same as the 2 by 3 grid, I wanted to make the increase in grid size steady. I got the same results as the 2 by 3 grid.

The next grid size that I used was a 3 by 3 grid.

In a 3 by 3 grid, there are nine squares. This means there are 9 possible empty squares and 8 possible filled in squares. This is proof that the number of filled in squares is one less than the number of possible empty squares.

Therefore there are 9 different arrangements of cubes. To further test this theory, I used a 3 by 4 grid. In a 3 by 4 grid there are 12 squares. Of this 12 squares are possible empty squares and 11 are possible filled in squares. This also supports the theory of the number of squares that are filled in is -1 of the number of arrangement and the number of possible empty squares.

On the second layer of a 3 by 3 grid, there will be eight squares. This means that there are eight possible empty squares and there are seven possible filled in squares. Even on the second layer the rule still works.
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The reason why there are only eight square on the second layer when there were nine on the first layer is according to 'Rule 2: Each new layer is made with one cube less than the layer underneath it.'

Also on the 3 by 4 grid, there are 12 squares. On the first layer, there are 12 squares. This means that there are twelve possible empty squares and 11 possible filled in squares. On the second layer there are 11 possible empty in squares and 10 possible filled in squares. Here the theory holds true again.
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