Investigate the number of ways there are of arranging cubes on different sized grids.

Part 2: For the second part, I will be trying to find out how many ways there are of placing 4 cubes on top on fives cubes on a 2 x 3 grid.

I do not think I have to go further with this investigation.  I can see that there are 5 arrangements for placing 4 cubes on 5 cubes on a 2 x 3 grid.

If I multiply them together: 5 x 6 = 30, it should equal the amount of arrangements.

This has proved that my prediction will be correct.  Now I can move on to the next investigation.

Part 3: For the third part, I will be trying to find out how many ways there are of placing 6 cubes on top of 7 cubes on a 2 x 4 grid.  To make it a bit easier, I will be splitting the investigation into 2 parts.

THERE ARE 8 WAYS OF ARRANGING 7 CUBES ON A 2 x 4 GRID!

I have decided to draw out all the diagrams because I want to prove my theory fully:

In this case, if you multiply the number of arrangements for putting 6 cubes on top of 7 cubes on a 2 x 3 grid =8 BY the number of cubes on the 1st layer =7

You should get the answer of 56.  

Now I am going to plot a table of results to see if I can find any patterns in this investigation.

TABLE OF RESULTS

Grid sizes 2 x 5 and 2 x 6 are predictions I made due to my theory which I have used in other parts of the investigation.

I have also spotted a pattern, which also assisted me in making these predictions:

In the case of the 2 x 3 grid, the number of cubes is 6 so if I multiply it by the number that comes before it which is 5, it should give me the number of arrangements there are for the second layer which is equal to 30.

To prove this, I am going to draw out both 2 x 3 and 2 x 4 grids. I HOPE MY PREDICTION IS CORRECT!

• I going to find the number of arrangements for 9 cubes on a 2 x 5 grid, then I will find the number of arrangements for 8 cubes on 9 cubes on a 2 x 5 grid.

I WAS CORRECT. THERE ARE 10 WAYS OF ARRANGING 9 CUBES ON

A 2 X 3 GRID!

Now I can move on.

Instead of drawing out all of the diagrams to find how many arrangements there are, I will only draw out 3 diagrams and use my 1st theory to prove my prediction.

For each arrangement there are 9 ways of putting 8 cubes on top on 9 cubes on a 2 x 5 grid.

Number of cubes on 1st layer:                10

Number of arrangements for 2nd layer: x        9

                                                90

MY PREDICTION WAS CORRECT THERE ARE 90 ARRANGEMENTS FOR PUTTING 8 CUBES ON TOP OF 9 CUBES ON A 2 x 5 GRID!

I do not think there is any need for me to look at the 2 x 6 arrangement, as I have been successful in this.

Because I have been successful in finding the arrangements for the 1st and 2nd layers, I am going to extend my investigation by increasing the amount of layers for the 2 x 3, 2 x 4, 2 x 5 and 2 x 6 grids.

• 3 cubes on top of 4, on top of 5 cubes on a 2 x 3 grid.

        

                        

           

           

I have spotted something!  There are 16 arrangements for this grid for putting 3 cubes on top of four cubes on a 2 x 3 grid.  If there are 20 arrangements for this one there must be 20 or the next pattern and the following.  So in order to find the answer I must multiply 20 by the amount of ways there are of putting 4 cubes on top of 5 on a 2 x 3 grid.

I think there are going to be 100 arrangements.  I am going to multiply 16 x 5 and hope I am right.

To try and prove my prediction I am going to draw out the diagrams.