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Maths Cubes Investigation

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Introduction

HIGHER TIER TASK STRUCTURES Rigid structures are constructed using unit rods. An example is shown below. The individual rods in the structures are held together using different types of joints. Some examples are shown below. 3 Joint 4 Joint 5 Joint 6 Joint Investigate structures constructed from unit rods. My Plan I have decided to investigate the number of different joints in cubes. I will draw cubes of dimensions 1x1x1, 2x2x2, 3x3x3, 4x4x4 and 5x5x5. After drawing these cubes and counting the number of different joints, I hope to find a pattern and formula for working out the number of different rods in shapes nxnxn. After finding this formula I will check it against shapes 6x6x6, 7x7x7, 8x8x8, 9x9x9 and 10x10x10, to see if it works. I will then use this theory on more complex shapes like cuboids. Collecting Results To set about collecting my results I drew the cubes on isometric paper and drew the joints in different colours so that it would be easier to count them. ...read more.

Middle

On the 1x1x1 cube there aren't any edges as there is only 8 Joints in total and these are the 3 Joints. The amount of 4 Joints increases by 12 for each extra 1 cm� you increase the dimensions by because another joint is added for each edge. The rule for 4 Joints is: 12(n-1). 5 Joints The number of 5 Joints increases in a slightly less obvious pattern than 3 Joints or 4 Joints. The pattern is actually that the number of 5 Joints increases by the following rule: 6(n-1) �. The reason for this is because 5 Joints are found on the faces of the cube. There are six faces on a cube and this explains the 6 in the rule. The reason for the square number is that each time the dimensions are increased by ncm� a square of ncm� is made on the face of the cube. ...read more.

Conclusion

= ab+a Rows + Columns = a+2ab+b Layers- c+1 Total for all layers- (a+2ab+b)(c+1) Vertical Rods For 1 Layer- (b+1)(a+1) Layers- c+1 Total for all Layers- (b+1)(a+1)c Horizontal and Vertical Rods 3abc+2ab+2ac+2bc+a+b+c JOINTS 3 Joints As in a cube there are only ever 8, 3 Joints in a cuboid so again the rule is 3J=8. 4 Joints The rule for 4 Joints is 4(a-1)+4(b-1)+4(c-1). This is relates to the edges of the cuboid. 5 Joints As in the cube, the 5 Joints are found on the faces of the cuboid. The rule for the amount of 5 Joints is this 2(a-1)(b-1)+2(b-1)(c-1)+2(a-1)(c-1). 6 Joints The 6 Joints are again found 'inside' the cube. As the dimensions of the cube are axbxc, then the amount of 6 Joints is (a-1)(b-1)(c-1). Total Joints The rule for the total number of Joints in the cuboid is this (a+1)(b+1)(c+1). This is because this is the equivalent to the cubes formula for total Joints which is (n+1)� Colin Devitt 11L Maths Coursework 1 ...read more.

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