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

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Introduction

Atinuke Odunsi 10L Maths Coursework - (LAYERS) Mr. Roberts In this piece of coursework I'm going to investigate the number of ways there are of arranging cubes on different sized grids. I will then use these results to see if there is any correlation between the size of the grids and the amount of cubes. Rule 1:the number of cubes on the 1st layer is 1 less than the number of squares on the grid. Rule 2:in each layer there is 1 cube less than on the layer below KEY~ ~ 1st layer ~ 2nd layer Part 1: For the first part I will be trying to find out how many ways there are of arranging 5 cubes on a 2 x 3 grid. I HAVE FOUND OUT THAT THERE ARE 6 WAYS OF ARRANGING 5 CUBES ON A 2 x 3 GRID! 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. ...read more.

Middle

Now I am going to plot a table of results to see if I can find any patterns in this investigation. TABLE OF RESULTS Number of arrangements Grid size No. of squares 1st layer 2nd layer 1. 2 x 3 6 6 30 2. 2 x 4 8 8 56 3. 2 x 5 10 10 90 4. 2 x 6 12 12 132 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. ...read more.

Conclusion

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. ...read more.

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